Split (1)

train · 213k rows

state stringlengths 0 159k	srcUpToTactic stringlengths 387 167k	nextTactic stringlengths 3 9k	declUpToTactic stringlengths 22 11.5k	declId stringlengths 38 95	decl stringlengths 16 1.89k	file_tag stringlengths 17 73
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V H H' : Subgraph G x✝¹ x✝ : V h : Adj H x✝¹ x✝ ⊢ x✝ ∈ H.verts ∧ x✝¹ ∈ H.verts ∧ Adj H' x✝¹ x✝ ↔ Adj H' x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h	simp [H.edge_vert h, H.edge_vert h.symm]	lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h	Mathlib.Combinatorics.SimpleGraph.Subgraph.1034_0.BlhiAiIDADcXv8t	lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) a b : V ⊢ (G'.Adj \ Sym2.ToRel s) a b → (G'.Adj \ Sym2.ToRel s) b a	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by	simp [G'.adj_comm, Sym2.eq_swap]	/-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1046_0.BlhiAiIDADcXv8t	/-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) ⊢ deleteEdges (deleteEdges G' s) s' = deleteEdges G' (s ∪ s')	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by	ext	@[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1072_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) x✝ : V ⊢ x✝ ∈ (deleteEdges (deleteEdges G' s) s').verts ↔ x✝ ∈ (deleteEdges G' (s ∪ s')).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;>	simp [and_assoc, not_or]	@[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1072_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) x✝¹ x✝ : V ⊢ Adj (deleteEdges (deleteEdges G' s) s') x✝¹ x✝ ↔ Adj (deleteEdges G' (s ∪ s')) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;>	simp [and_assoc, not_or]	@[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1072_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ deleteEdges G' ∅ = G'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by	ext	@[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1078_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝ : V ⊢ x✝ ∈ (deleteEdges G' ∅).verts ↔ x✝ ∈ G'.verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;>	simp	@[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1078_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝¹ x✝ : V ⊢ Adj (deleteEdges G' ∅) x✝¹ x✝ ↔ Adj G' x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;>	simp	@[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1078_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ SimpleGraph.deleteEdges (Subgraph.spanningCoe G') s = Subgraph.spanningCoe (deleteEdges G' s)	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by	ext	@[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1083_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝¹ x✝ : V ⊢ SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.spanningCoe G') s) x✝¹ x✝ ↔ SimpleGraph.Adj (Subgraph.spanningCoe (deleteEdges G' s)) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext	simp	@[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext	Mathlib.Combinatorics.SimpleGraph.Subgraph.1083_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) ⊢ SimpleGraph.deleteEdges (Subgraph.coe G') s = Subgraph.coe (deleteEdges G' (Sym2.map Subtype.val '' s))	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by	ext ⟨v, hv⟩ ⟨w, hw⟩	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts ⊢ SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.coe G') s) { val := v, property := hv } { val := w, property := hw } ↔ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' (Sym2.map Subtype.val '' s))) { val := v, property := hv } { val := w, property := hw }	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩	simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff]	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts ⊢ Adj G' v w → (⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s ↔ ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧)	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff]	intro	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff]	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w ⊢ ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s ↔ ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro	constructor	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w ⊢ ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s → ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor ·	intro hs	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s ⊢ ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs	refine' Sym2.ind _	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s ⊢ ∀ (x y : ↑G'.verts), ⟦(x, y)⟧ ∈ s → ¬Sym2.map Subtype.val ⟦(x, y)⟧ = ⟦(v, w)⟧	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _	rintro ⟨v', hv'⟩ ⟨w', hw'⟩	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp.mk.mk ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s v' : V hv' : v' ∈ G'.verts w' : V hw' : w' ∈ G'.verts ⊢ ⟦({ val := v', property := hv' }, { val := w', property := hw' })⟧ ∈ s → ¬Sym2.map Subtype.val ⟦({ val := v', property := hv' }, { val := w', property := hw' })⟧ = ⟦(v, w)⟧	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩	simp only [Sym2.map_pair_eq, Quotient.eq]	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp.mk.mk ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s v' : V hv' : v' ∈ G'.verts w' : V hw' : w' ∈ G'.verts ⊢ ⟦({ val := v', property := hv' }, { val := w', property := hw' })⟧ ∈ s → ¬(v', w') ≈ (v, w)	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq]	contrapose!	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq]	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp.mk.mk ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s v' : V hv' : v' ∈ G'.verts w' : V hw' : w' ∈ G'.verts ⊢ (v', w') ≈ (v, w) → ⟦({ val := v', property := hv' }, { val := w', property := hw' })⟧ ∉ s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose!	rintro (_ \| _)	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose!	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp.mk.mk.refl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s hv' : v ∈ G'.verts hw' : w ∈ G'.verts ⊢ ⟦({ val := v, property := hv' }, { val := w, property := hw' })⟧ ∉ s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;>	simpa only [Sym2.eq_swap]	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mp.mk.mk.swap ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s hw' : v ∈ G'.verts hv' : w ∈ G'.verts ⊢ ⟦({ val := w, property := hv' }, { val := v, property := hw' })⟧ ∉ s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;>	simpa only [Sym2.eq_swap]	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w ⊢ (∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧) → ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∉ s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] ·	intro h' hs	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ : Set (Sym2 V) s : Set (Sym2 ↑G'.verts) v : V hv : v ∈ G'.verts w : V hw : w ∈ G'.verts a✝ : Adj G' v w h' : ∀ x ∈ s, ¬Sym2.map Subtype.val x = ⟦(v, w)⟧ hs : ⟦({ val := v, property := hv }, { val := w, property := hw })⟧ ∈ s ⊢ False	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs	exact h' _ hs rfl	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs	Mathlib.Combinatorics.SimpleGraph.Subgraph.1090_0.BlhiAiIDADcXv8t	theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s : Set (Sym2 V) ⊢ Subgraph.coe (deleteEdges G' s) = SimpleGraph.deleteEdges (Subgraph.coe G') (Sym2.map Subtype.val ⁻¹' s)	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by	ext ⟨v, hv⟩ ⟨w, hw⟩	theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1107_0.BlhiAiIDADcXv8t	theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s)	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.mk.h.mk.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s : Set (Sym2 V) v : V hv : v ∈ (deleteEdges G' s).verts w : V hw : w ∈ (deleteEdges G' s).verts ⊢ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' s)) { val := v, property := hv } { val := w, property := hw } ↔ SimpleGraph.Adj (SimpleGraph.deleteEdges (Subgraph.coe G') (Sym2.map Subtype.val ⁻¹' s)) { val := v, property := hv } { val := w, property := hw }	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩	simp	theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩	Mathlib.Combinatorics.SimpleGraph.Subgraph.1107_0.BlhiAiIDADcXv8t	theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s)	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ deleteEdges G' s ≤ G'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by	constructor	theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1113_0.BlhiAiIDADcXv8t	theorem deleteEdges_le : G'.deleteEdges s ≤ G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case left ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ (deleteEdges G' s).verts ⊆ G'.verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;>	simp (config := { contextual := true }) [subset_rfl]	theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1113_0.BlhiAiIDADcXv8t	theorem deleteEdges_le : G'.deleteEdges s ≤ G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ ∀ ⦃v w : V⦄, Adj (deleteEdges G' s) v w → Adj G' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;>	simp (config := { contextual := true }) [subset_rfl]	theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1113_0.BlhiAiIDADcXv8t	theorem deleteEdges_le : G'.deleteEdges s ≤ G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) h : s ⊆ s' ⊢ deleteEdges G' s' ≤ deleteEdges G' s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by	constructor	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case left ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) h : s ⊆ s' ⊢ (deleteEdges G' s').verts ⊆ (deleteEdges G' s).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;>	simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl]	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) h : s ⊆ s' ⊢ ∀ ⦃v w : V⦄, Adj (deleteEdges G' s') v w → Adj (deleteEdges G' s) v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;>	simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl]	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set (Sym2 V) h : s ⊆ s' ⊢ ∀ ⦃v w : V⦄, Adj G' v w → ⟦(v, w)⟧ ∉ s' → ⟦(v, w)⟧ ∉ s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl]	exact fun _ _ _ hs' hs ↦ hs' (h hs)	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl]	Mathlib.Combinatorics.SimpleGraph.Subgraph.1117_0.BlhiAiIDADcXv8t	theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ deleteEdges G' (edgeSet G' ∩ s) = deleteEdges G' s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by	ext	@[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1124_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝ : V ⊢ x✝ ∈ (deleteEdges G' (edgeSet G' ∩ s)).verts ↔ x✝ ∈ (deleteEdges G' s).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;>	simp (config := { contextual := true }) [imp_false]	@[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1124_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝¹ x✝ : V ⊢ Adj (deleteEdges G' (edgeSet G' ∩ s)) x✝¹ x✝ ↔ Adj (deleteEdges G' s) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;>	simp (config := { contextual := true }) [imp_false]	@[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1124_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ deleteEdges G' (s ∩ edgeSet G') = deleteEdges G' s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by	ext	@[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1130_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝ : V ⊢ x✝ ∈ (deleteEdges G' (s ∩ edgeSet G')).verts ↔ x✝ ∈ (deleteEdges G' s).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;>	simp (config := { contextual := true }) [imp_false]	@[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1130_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) x✝¹ x✝ : V ⊢ Adj (deleteEdges G' (s ∩ edgeSet G')) x✝¹ x✝ ↔ Adj (deleteEdges G' s) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;>	simp (config := { contextual := true }) [imp_false]	@[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1130_0.BlhiAiIDADcXv8t	@[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) ⊢ Subgraph.coe (deleteEdges G' s) ≤ Subgraph.coe G'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by	intro v w	theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1136_0.BlhiAiIDADcXv8t	theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts)	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set (Sym2 V) v w : ↑(deleteEdges G' s).verts ⊢ SimpleGraph.Adj (Subgraph.coe (deleteEdges G' s)) v w → SimpleGraph.Adj (Subgraph.coe G') v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w	simp (config := { contextual := true })	theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w	Mathlib.Combinatorics.SimpleGraph.Subgraph.1136_0.BlhiAiIDADcXv8t	theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts)	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V s : Set V ⊢ SimpleGraph.induce s G = Subgraph.coe (induce ⊤ s)	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by	ext	theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1166_0.BlhiAiIDADcXv8t	theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V s : Set V x✝¹ x✝ : ↑s ⊢ SimpleGraph.Adj (SimpleGraph.induce s G) x✝¹ x✝ ↔ SimpleGraph.Adj (Subgraph.coe (induce ⊤ s)) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext	simp	theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext	Mathlib.Combinatorics.SimpleGraph.Subgraph.1166_0.BlhiAiIDADcXv8t	theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V hg : G' ≤ G'' hs : s ⊆ s' ⊢ induce G' s ≤ induce G'' s'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by	constructor	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case left ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V hg : G' ≤ G'' hs : s ⊆ s' ⊢ (induce G' s).verts ⊆ (induce G'' s').verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor ·	simp [hs]	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V hg : G' ≤ G'' hs : s ⊆ s' ⊢ ∀ ⦃v w : V⦄, Adj (induce G' s) v w → Adj (induce G'' s') v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] ·	simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp]	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V hg : G' ≤ G'' hs : s ⊆ s' ⊢ ∀ ⦃v w : V⦄, v ∈ s → w ∈ s → Adj G' v w → v ∈ s' ∧ w ∈ s' ∧ Adj G'' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp]	intro v w hv hw ha	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp]	Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V hg : G' ≤ G'' hs : s ⊆ s' v w : V hv : v ∈ s hw : w ∈ s ha : Adj G' v w ⊢ v ∈ s' ∧ w ∈ s' ∧ Adj G'' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha	exact ⟨hs hv, hs hw, hg.2 ha⟩	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha	Mathlib.Combinatorics.SimpleGraph.Subgraph.1176_0.BlhiAiIDADcXv8t	theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ induce G' ∅ = ⊥	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by	ext	@[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1194_0.BlhiAiIDADcXv8t	@[simp] theorem induce_empty : G'.induce ∅ = ⊥	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝ : V ⊢ x✝ ∈ (induce G' ∅).verts ↔ x✝ ∈ ⊥.verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;>	simp	@[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1194_0.BlhiAiIDADcXv8t	@[simp] theorem induce_empty : G'.induce ∅ = ⊥	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ⊢ Adj (induce G' ∅) x✝¹ x✝ ↔ Adj ⊥ x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;>	simp	@[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1194_0.BlhiAiIDADcXv8t	@[simp] theorem induce_empty : G'.induce ∅ = ⊥	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ induce G' G'.verts = G'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by	ext	@[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t	@[simp] theorem induce_self_verts : G'.induce G'.verts = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝ : V ⊢ x✝ ∈ (induce G' G'.verts).verts ↔ x✝ ∈ G'.verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext ·	simp	@[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t	@[simp] theorem induce_self_verts : G'.induce G'.verts = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ⊢ Adj (induce G' G'.verts) x✝¹ x✝ ↔ Adj G' x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp ·	constructor	@[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t	@[simp] theorem induce_self_verts : G'.induce G'.verts = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ⊢ Adj (induce G' G'.verts) x✝¹ x✝ → Adj G' x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;>	simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff]	@[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t	@[simp] theorem induce_self_verts : G'.induce G'.verts = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ⊢ Adj G' x✝¹ x✝ → Adj (induce G' G'.verts) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;>	simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff]	@[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t	@[simp] theorem induce_self_verts : G'.induce G'.verts = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ⊢ Adj G' x✝¹ x✝ → x✝¹ ∈ G'.verts ∧ x✝ ∈ G'.verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff]	exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩	@[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff]	Mathlib.Combinatorics.SimpleGraph.Subgraph.1199_0.BlhiAiIDADcXv8t	@[simp] theorem induce_self_verts : G'.induce G'.verts = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ induce G' s ⊔ induce G' s' ≤ induce G' (s ∪ s')	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by	constructor	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case left ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ (induce G' s ⊔ induce G' s').verts ⊆ (induce G' (s ∪ s')).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor ·	simp only [verts_sup, induce_verts, Set.Subset.rfl]	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ ∀ ⦃v w : V⦄, Adj (induce G' s ⊔ induce G' s') v w → Adj (induce G' (s ∪ s')) v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] ·	simp only [sup_adj, induce_adj, Set.mem_union]	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ ∀ ⦃v w : V⦄, v ∈ s ∧ w ∈ s ∧ Adj G' v w ∨ v ∈ s' ∧ w ∈ s' ∧ Adj G' v w → (v ∈ s ∨ v ∈ s') ∧ (w ∈ s ∨ w ∈ s') ∧ Adj G' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union]	rintro v w (h \| h)	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union]	Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right.inl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V h : v ∈ s ∧ w ∈ s ∧ Adj G' v w ⊢ (v ∈ s ∨ v ∈ s') ∧ (w ∈ s ∨ w ∈ s') ∧ Adj G' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;>	simp [h]	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right.inr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V h : v ∈ s' ∧ w ∈ s' ∧ Adj G' v w ⊢ (v ∈ s ∨ v ∈ s') ∧ (w ∈ s ∨ w ∈ s') ∧ Adj G' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;>	simp [h]	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1212_0.BlhiAiIDADcXv8t	lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ induce G' s ≤ induce G' (s ∪ s')	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by	exact (sup_le_iff.mp le_induce_union).1	lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1218_0.BlhiAiIDADcXv8t	lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V ⊢ induce G' s' ≤ induce G' (s ∪ s')	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by	exact (sup_le_iff.mp le_induce_union).2	lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1221_0.BlhiAiIDADcXv8t	lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v : V ⊢ SimpleGraph.singletonSubgraph G v = induce ⊤ {v}	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by	ext	theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1224_0.BlhiAiIDADcXv8t	theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v x✝ : V ⊢ x✝ ∈ (SimpleGraph.singletonSubgraph G v).verts ↔ x✝ ∈ (induce ⊤ {v}).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;>	simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false]	theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1224_0.BlhiAiIDADcXv8t	theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v x✝¹ x✝ : V ⊢ Adj (SimpleGraph.singletonSubgraph G v) x✝¹ x✝ ↔ Adj (induce ⊤ {v}) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;>	simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false]	theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1224_0.BlhiAiIDADcXv8t	theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v}	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w ⊢ subgraphOfAdj G hvw = induce ⊤ {v, w}	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by	ext	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝ : V ⊢ x✝ ∈ (subgraphOfAdj G hvw).verts ↔ x✝ ∈ (induce ⊤ {v, w}).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext ·	simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ ↔ Adj (induce ⊤ {v, w}) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp ·	constructor	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝ → Adj (induce ⊤ {v, w}) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor ·	intro h	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V h : Adj (subgraphOfAdj G hvw) x✝¹ x✝ ⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h	simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mp ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V h : v = x✝¹ ∧ w = x✝ ∨ v = x✝ ∧ w = x✝¹ ⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h	obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mp.inl.intro ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w ⊢ Adj (induce ⊤ {v, w}) v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;>	simp [hvw, hvw.symm]	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mp.inr.intro ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w ⊢ Adj (induce ⊤ {v, w}) w v	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;>	simp [hvw, hvw.symm]	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V ⊢ Adj (induce ⊤ {v, w}) x✝¹ x✝ → Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] ·	intro h	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] ·	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V h : Adj (induce ⊤ {v, w}) x✝¹ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h	simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v w : V hvw : SimpleGraph.Adj G v w x✝¹ x✝ : V h : (x✝¹ = v ∨ x✝¹ = w) ∧ (x✝ = v ∨ x✝ = w) ∧ SimpleGraph.Adj G x✝¹ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h	obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inl.intro.inl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V w x✝ : V hvw : SimpleGraph.Adj G x✝ w ha : SimpleGraph.Adj G x✝ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	first \|exact (ha.ne rfl).elim\|simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inl.intro.inl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V w x✝ : V hvw : SimpleGraph.Adj G x✝ w ha : SimpleGraph.Adj G x✝ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	exact (ha.ne rfl).elim	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inl.intro.inr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ha hvw : SimpleGraph.Adj G x✝¹ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	first \|exact (ha.ne rfl).elim\|simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inl.intro.inr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ha hvw : SimpleGraph.Adj G x✝¹ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	exact (ha.ne rfl).elim	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inl.intro.inr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ha hvw : SimpleGraph.Adj G x✝¹ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|	simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inr.intro.inl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ha : SimpleGraph.Adj G x✝¹ x✝ hvw : SimpleGraph.Adj G x✝ x✝¹ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	first \|exact (ha.ne rfl).elim\|simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inr.intro.inl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ha : SimpleGraph.Adj G x✝¹ x✝ hvw : SimpleGraph.Adj G x✝ x✝¹ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	exact (ha.ne rfl).elim	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inr.intro.inl ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V x✝¹ x✝ : V ha : SimpleGraph.Adj G x✝¹ x✝ hvw : SimpleGraph.Adj G x✝ x✝¹ ⊢ Adj (subgraphOfAdj G hvw) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|	simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inr.intro.inr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v x✝ : V hvw : SimpleGraph.Adj G v x✝ ha : SimpleGraph.Adj G x✝ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	first \|exact (ha.ne rfl).elim\|simp	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a.mpr.intro.inr.intro.inr ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' G'' : Subgraph G s s' : Set V v x✝ : V hvw : SimpleGraph.Adj G v x✝ ha : SimpleGraph.Adj G x✝ x✝ ⊢ Adj (subgraphOfAdj G hvw) x✝ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	exact (ha.ne rfl).elim	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|	Mathlib.Combinatorics.SimpleGraph.Subgraph.1228_0.BlhiAiIDADcXv8t	theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w}	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V u v : V ⊢ Adj (deleteVerts G' s) u v ↔ u ∈ G'.verts ∧ u ∉ s ∧ v ∈ G'.verts ∧ v ∉ s ∧ Adj G' u v	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by	simp [and_assoc]	theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1258_0.BlhiAiIDADcXv8t	theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set V ⊢ deleteVerts (deleteVerts G' s) s' = deleteVerts G' (s ∪ s')	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by	ext	@[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1263_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set V x✝ : V ⊢ x✝ ∈ (deleteVerts (deleteVerts G' s) s').verts ↔ x✝ ∈ (deleteVerts G' (s ∪ s')).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;>	simp (config := { contextual := true }) [not_or, and_assoc]	@[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1263_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s✝ s s' : Set V x✝¹ x✝ : V ⊢ Adj (deleteVerts (deleteVerts G' s) s') x✝¹ x✝ ↔ Adj (deleteVerts G' (s ∪ s')) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;>	simp (config := { contextual := true }) [not_or, and_assoc]	@[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1263_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s')	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V ⊢ deleteVerts G' ∅ = G'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by	simp [deleteVerts]	@[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1269_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V ⊢ deleteVerts G' s ≤ G'	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by simp [deleteVerts] #align simple_graph.subgraph.delete_verts_empty SimpleGraph.Subgraph.deleteVerts_empty theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by	constructor	theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1274_0.BlhiAiIDADcXv8t	theorem deleteVerts_le : G'.deleteVerts s ≤ G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case left ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V ⊢ (deleteVerts G' s).verts ⊆ G'.verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by simp [deleteVerts] #align simple_graph.subgraph.delete_verts_empty SimpleGraph.Subgraph.deleteVerts_empty theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;>	simp [Set.diff_subset]	theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1274_0.BlhiAiIDADcXv8t	theorem deleteVerts_le : G'.deleteVerts s ≤ G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
case right ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V ⊢ ∀ ⦃v w : V⦄, Adj (deleteVerts G' s) v w → Adj G' v w	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by simp [deleteVerts] #align simple_graph.subgraph.delete_verts_empty SimpleGraph.Subgraph.deleteVerts_empty theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;>	simp [Set.diff_subset]	theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1274_0.BlhiAiIDADcXv8t	theorem deleteVerts_le : G'.deleteVerts s ≤ G'	Mathlib_Combinatorics_SimpleGraph_Subgraph
ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V ⊢ deleteVerts G' (G'.verts ∩ s) = deleteVerts G' s	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by simp [deleteVerts] #align simple_graph.subgraph.delete_verts_empty SimpleGraph.Subgraph.deleteVerts_empty theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;> simp [Set.diff_subset] #align simple_graph.subgraph.delete_verts_le SimpleGraph.Subgraph.deleteVerts_le @[mono] theorem deleteVerts_mono {G' G'' : G.Subgraph} (h : G' ≤ G'') : G'.deleteVerts s ≤ G''.deleteVerts s := induce_mono h (Set.diff_subset_diff_left h.1) #align simple_graph.subgraph.delete_verts_mono SimpleGraph.Subgraph.deleteVerts_mono @[mono] theorem deleteVerts_anti {s s' : Set V} (h : s ⊆ s') : G'.deleteVerts s' ≤ G'.deleteVerts s := induce_mono (le_refl _) (Set.diff_subset_diff_right h) #align simple_graph.subgraph.delete_verts_anti SimpleGraph.Subgraph.deleteVerts_anti @[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by	ext	@[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by	Mathlib.Combinatorics.SimpleGraph.Subgraph.1289_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case verts.h ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V x✝ : V ⊢ x✝ ∈ (deleteVerts G' (G'.verts ∩ s)).verts ↔ x✝ ∈ (deleteVerts G' s).verts	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by simp [deleteVerts] #align simple_graph.subgraph.delete_verts_empty SimpleGraph.Subgraph.deleteVerts_empty theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;> simp [Set.diff_subset] #align simple_graph.subgraph.delete_verts_le SimpleGraph.Subgraph.deleteVerts_le @[mono] theorem deleteVerts_mono {G' G'' : G.Subgraph} (h : G' ≤ G'') : G'.deleteVerts s ≤ G''.deleteVerts s := induce_mono h (Set.diff_subset_diff_left h.1) #align simple_graph.subgraph.delete_verts_mono SimpleGraph.Subgraph.deleteVerts_mono @[mono] theorem deleteVerts_anti {s s' : Set V} (h : s ⊆ s') : G'.deleteVerts s' ≤ G'.deleteVerts s := induce_mono (le_refl _) (Set.diff_subset_diff_right h) #align simple_graph.subgraph.delete_verts_anti SimpleGraph.Subgraph.deleteVerts_anti @[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by ext <;>	simp (config := { contextual := true }) [imp_false]	@[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1289_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s	Mathlib_Combinatorics_SimpleGraph_Subgraph
case Adj.h.h.a ι : Sort u_1 V : Type u W : Type v G : SimpleGraph V G' : Subgraph G s : Set V x✝¹ x✝ : V ⊢ Adj (deleteVerts G' (G'.verts ∩ s)) x✝¹ x✝ ↔ Adj (deleteVerts G' s) x✝¹ x✝	/- Copyright (c) 2021 Hunter Monroe. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Hunter Monroe, Kyle Miller, Alena Gusakov -/ import Mathlib.Combinatorics.SimpleGraph.Basic #align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b" /-! # Subgraphs of a simple graph A subgraph of a simple graph consists of subsets of the graph's vertices and edges such that the endpoints of each edge are present in the vertex subset. The edge subset is formalized as a sub-relation of the adjacency relation of the simple graph. ## Main definitions * `Subgraph G` is the type of subgraphs of a `G : SimpleGraph V`. * `Subgraph.neighborSet`, `Subgraph.incidenceSet`, and `Subgraph.degree` are like their `SimpleGraph` counterparts, but they refer to vertices from `G` to avoid subtype coercions. * `Subgraph.coe` is the coercion from a `G' : Subgraph G` to a `SimpleGraph G'.verts`. (In Lean 3 this could not be a `Coe` instance since the destination type depends on `G'`.) * `Subgraph.IsSpanning` for whether a subgraph is a spanning subgraph and `Subgraph.IsInduced` for whether a subgraph is an induced subgraph. * Instances for `Lattice (Subgraph G)` and `BoundedOrder (Subgraph G)`. * `SimpleGraph.toSubgraph`: If a `SimpleGraph` is a subgraph of another, then you can turn it into a member of the larger graph's `SimpleGraph.Subgraph` type. * Graph homomorphisms from a subgraph to a graph (`Subgraph.map_top`) and between subgraphs (`Subgraph.map`). ## Implementation notes * Recall that subgraphs are not determined by their vertex sets, so `SetLike` does not apply to this kind of subobject. ## Todo * Images of graph homomorphisms as subgraphs. -/ universe u v namespace SimpleGraph /-- A subgraph of a `SimpleGraph` is a subset of vertices along with a restriction of the adjacency relation that is symmetric and is supported by the vertex subset. They also form a bounded lattice. Thinking of `V → V → Prop` as `Set (V × V)`, a set of darts (i.e., half-edges), then `Subgraph.adj_sub` is that the darts of a subgraph are a subset of the darts of `G`. -/ @[ext] structure Subgraph {V : Type u} (G : SimpleGraph V) where verts : Set V Adj : V → V → Prop adj_sub : ∀ {v w : V}, Adj v w → G.Adj v w edge_vert : ∀ {v w : V}, Adj v w → v ∈ verts symm : Symmetric Adj := by aesop_graph -- Porting note: Originally `by obviously` #align simple_graph.subgraph SimpleGraph.Subgraph initialize_simps_projections SimpleGraph.Subgraph (Adj → adj) variable {ι : Sort} {V : Type u} {W : Type v} /-- The one-vertex subgraph. -/ @[simps] protected def singletonSubgraph (G : SimpleGraph V) (v : V) : G.Subgraph where verts := {v} Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm _ _ := False.elim #align simple_graph.singleton_subgraph SimpleGraph.singletonSubgraph /-- The one-edge subgraph. -/ @[simps] def subgraphOfAdj (G : SimpleGraph V) {v w : V} (hvw : G.Adj v w) : G.Subgraph where verts := {v, w} Adj a b := ⟦(v, w)⟧ = ⟦(a, b)⟧ adj_sub h := by rw [← G.mem_edgeSet, ← h] exact hvw edge_vert {a b} h := by apply_fun fun e ↦ a ∈ e at h simp only [Sym2.mem_iff, true_or, eq_iff_iff, iff_true] at h exact h #align simple_graph.subgraph_of_adj SimpleGraph.subgraphOfAdj namespace Subgraph variable {G : SimpleGraph V} {G₁ G₂ : G.Subgraph} {a b : V} protected theorem loopless (G' : Subgraph G) : Irreflexive G'.Adj := fun v h ↦ G.loopless v (G'.adj_sub h) #align simple_graph.subgraph.loopless SimpleGraph.Subgraph.loopless theorem adj_comm (G' : Subgraph G) (v w : V) : G'.Adj v w ↔ G'.Adj w v := ⟨fun x ↦ G'.symm x, fun x ↦ G'.symm x⟩ #align simple_graph.subgraph.adj_comm SimpleGraph.Subgraph.adj_comm @[symm] theorem adj_symm (G' : Subgraph G) {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj_symm SimpleGraph.Subgraph.adj_symm protected theorem Adj.symm {G' : Subgraph G} {u v : V} (h : G'.Adj u v) : G'.Adj v u := G'.symm h #align simple_graph.subgraph.adj.symm SimpleGraph.Subgraph.Adj.symm protected theorem Adj.adj_sub {H : G.Subgraph} {u v : V} (h : H.Adj u v) : G.Adj u v := H.adj_sub h #align simple_graph.subgraph.adj.adj_sub SimpleGraph.Subgraph.Adj.adj_sub protected theorem Adj.fst_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ∈ H.verts := H.edge_vert h #align simple_graph.subgraph.adj.fst_mem SimpleGraph.Subgraph.Adj.fst_mem protected theorem Adj.snd_mem {H : G.Subgraph} {u v : V} (h : H.Adj u v) : v ∈ H.verts := h.symm.fst_mem #align simple_graph.subgraph.adj.snd_mem SimpleGraph.Subgraph.Adj.snd_mem protected theorem Adj.ne {H : G.Subgraph} {u v : V} (h : H.Adj u v) : u ≠ v := h.adj_sub.ne #align simple_graph.subgraph.adj.ne SimpleGraph.Subgraph.Adj.ne /-- Coercion from `G' : Subgraph G` to a `SimpleGraph G'.verts`. -/ @[simps] protected def coe (G' : Subgraph G) : SimpleGraph G'.verts where Adj v w := G'.Adj v w symm _ _ h := G'.symm h loopless v h := loopless G v (G'.adj_sub h) #align simple_graph.subgraph.coe SimpleGraph.Subgraph.coe @[simp] theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.coe_adj_sub SimpleGraph.Subgraph.coe_adj_sub -- Given `h : H.Adj u v`, then `h.coe : H.coe.Adj ⟨u, _⟩ ⟨v, _⟩`. protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) : H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h #align simple_graph.subgraph.adj.coe SimpleGraph.Subgraph.Adj.coe /-- A subgraph is called a spanning subgraph* if it contains all the vertices of `G`. -/ def IsSpanning (G' : Subgraph G) : Prop := ∀ v : V, v ∈ G'.verts #align simple_graph.subgraph.is_spanning SimpleGraph.Subgraph.IsSpanning theorem isSpanning_iff {G' : Subgraph G} : G'.IsSpanning ↔ G'.verts = Set.univ := Set.eq_univ_iff_forall.symm #align simple_graph.subgraph.is_spanning_iff SimpleGraph.Subgraph.isSpanning_iff /-- Coercion from `Subgraph G` to `SimpleGraph V`. If `G'` is a spanning subgraph, then `G'.spanningCoe` yields an isomorphic graph. In general, this adds in all vertices from `V` as isolated vertices. -/ @[simps] protected def spanningCoe (G' : Subgraph G) : SimpleGraph V where Adj := G'.Adj symm := G'.symm loopless v hv := G.loopless v (G'.adj_sub hv) #align simple_graph.subgraph.spanning_coe SimpleGraph.Subgraph.spanningCoe @[simp] theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCoe.Adj u v) : G.Adj u v := G'.adj_sub h #align simple_graph.subgraph.adj.of_spanning_coe SimpleGraph.Subgraph.Adj.of_spanningCoe theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by simp [Subgraph.spanningCoe] #align simple_graph.subgraph.spanning_coe_inj SimpleGraph.Subgraph.spanningCoe_inj /-- `spanningCoe` is equivalent to `coe` for a subgraph that `IsSpanning`. -/ @[simps] def spanningCoeEquivCoeOfSpanning (G' : Subgraph G) (h : G'.IsSpanning) : G'.spanningCoe ≃g G'.coe where toFun v := ⟨v, h v⟩ invFun v := v left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.spanning_coe_equiv_coe_of_spanning SimpleGraph.Subgraph.spanningCoeEquivCoeOfSpanning /-- A subgraph is called an induced subgraph if vertices of `G'` are adjacent if they are adjacent in `G`. -/ def IsInduced (G' : Subgraph G) : Prop := ∀ {v w : V}, v ∈ G'.verts → w ∈ G'.verts → G.Adj v w → G'.Adj v w #align simple_graph.subgraph.is_induced SimpleGraph.Subgraph.IsInduced /-- `H.support` is the set of vertices that form edges in the subgraph `H`. -/ def support (H : Subgraph G) : Set V := Rel.dom H.Adj #align simple_graph.subgraph.support SimpleGraph.Subgraph.support theorem mem_support (H : Subgraph G) {v : V} : v ∈ H.support ↔ ∃ w, H.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_support SimpleGraph.Subgraph.mem_support theorem support_subset_verts (H : Subgraph G) : H.support ⊆ H.verts := fun _ ⟨_, h⟩ ↦ H.edge_vert h #align simple_graph.subgraph.support_subset_verts SimpleGraph.Subgraph.support_subset_verts /-- `G'.neighborSet v` is the set of vertices adjacent to `v` in `G'`. -/ def neighborSet (G' : Subgraph G) (v : V) : Set V := {w \| G'.Adj v w} #align simple_graph.subgraph.neighbor_set SimpleGraph.Subgraph.neighborSet theorem neighborSet_subset (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G.neighborSet v := fun _ ↦ G'.adj_sub #align simple_graph.subgraph.neighbor_set_subset SimpleGraph.Subgraph.neighborSet_subset theorem neighborSet_subset_verts (G' : Subgraph G) (v : V) : G'.neighborSet v ⊆ G'.verts := fun _ h ↦ G'.edge_vert (adj_symm G' h) #align simple_graph.subgraph.neighbor_set_subset_verts SimpleGraph.Subgraph.neighborSet_subset_verts @[simp] theorem mem_neighborSet (G' : Subgraph G) (v w : V) : w ∈ G'.neighborSet v ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_neighbor_set SimpleGraph.Subgraph.mem_neighborSet /-- A subgraph as a graph has equivalent neighbor sets. -/ def coeNeighborSetEquiv {G' : Subgraph G} (v : G'.verts) : G'.coe.neighborSet v ≃ G'.neighborSet v where toFun w := ⟨w, w.2⟩ invFun w := ⟨⟨w, G'.edge_vert (G'.adj_symm w.2)⟩, w.2⟩ left_inv _ := rfl right_inv _ := rfl #align simple_graph.subgraph.coe_neighbor_set_equiv SimpleGraph.Subgraph.coeNeighborSetEquiv /-- The edge set of `G'` consists of a subset of edges of `G`. -/ def edgeSet (G' : Subgraph G) : Set (Sym2 V) := Sym2.fromRel G'.symm #align simple_graph.subgraph.edge_set SimpleGraph.Subgraph.edgeSet theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet := Sym2.ind (fun _ _ ↦ G'.adj_sub) #align simple_graph.subgraph.edge_set_subset SimpleGraph.Subgraph.edgeSet_subset @[simp] theorem mem_edgeSet {G' : Subgraph G} {v w : V} : ⟦(v, w)⟧ ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl #align simple_graph.subgraph.mem_edge_set SimpleGraph.Subgraph.mem_edgeSet theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet) (hv : v ∈ e) : v ∈ G'.verts := by revert hv refine' Sym2.ind (fun v w he ↦ _) e he intro hv rcases Sym2.mem_iff.mp hv with (rfl \| rfl) · exact G'.edge_vert he · exact G'.edge_vert (G'.symm he) #align simple_graph.subgraph.mem_verts_if_mem_edge SimpleGraph.Subgraph.mem_verts_if_mem_edge /-- The `incidenceSet` is the set of edges incident to a given vertex. -/ def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet \| v ∈ e} #align simple_graph.subgraph.incidence_set SimpleGraph.Subgraph.incidenceSet theorem incidenceSet_subset_incidenceSet (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G.incidenceSet v := fun _ h ↦ ⟨G'.edgeSet_subset h.1, h.2⟩ #align simple_graph.subgraph.incidence_set_subset_incidence_set SimpleGraph.Subgraph.incidenceSet_subset_incidenceSet theorem incidenceSet_subset (G' : Subgraph G) (v : V) : G'.incidenceSet v ⊆ G'.edgeSet := fun _ h ↦ h.1 #align simple_graph.subgraph.incidence_set_subset SimpleGraph.Subgraph.incidenceSet_subset /-- Give a vertex as an element of the subgraph's vertex type. -/ @[reducible] def vert (G' : Subgraph G) (v : V) (h : v ∈ G'.verts) : G'.verts := ⟨v, h⟩ #align simple_graph.subgraph.vert SimpleGraph.Subgraph.vert /-- Create an equal copy of a subgraph (see `copy_eq`) with possibly different definitional equalities. See Note [range copy pattern]. -/ def copy (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : Subgraph G where verts := V'' Adj := adj' adj_sub := hadj.symm ▸ G'.adj_sub edge_vert := hV.symm ▸ hadj.symm ▸ G'.edge_vert symm := hadj.symm ▸ G'.symm #align simple_graph.subgraph.copy SimpleGraph.Subgraph.copy theorem copy_eq (G' : Subgraph G) (V'' : Set V) (hV : V'' = G'.verts) (adj' : V → V → Prop) (hadj : adj' = G'.Adj) : G'.copy V'' hV adj' hadj = G' := Subgraph.ext _ _ hV hadj #align simple_graph.subgraph.copy_eq SimpleGraph.Subgraph.copy_eq /-- The union of two subgraphs. -/ instance : Sup G.Subgraph where sup G₁ G₂ := { verts := G₁.verts ∪ G₂.verts Adj := G₁.Adj ⊔ G₂.Adj adj_sub := fun hab => Or.elim hab (fun h => G₁.adj_sub h) fun h => G₂.adj_sub h edge_vert := Or.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => Or.imp G₁.adj_symm G₂.adj_symm } /-- The intersection of two subgraphs. -/ instance : Inf G.Subgraph where inf G₁ G₂ := { verts := G₁.verts ∩ G₂.verts Adj := G₁.Adj ⊓ G₂.Adj adj_sub := fun hab => G₁.adj_sub hab.1 edge_vert := And.imp (fun h => G₁.edge_vert h) fun h => G₂.edge_vert h symm := fun _ _ => And.imp G₁.adj_symm G₂.adj_symm } /-- The `top` subgraph is `G` as a subgraph of itself. -/ instance : Top G.Subgraph where top := { verts := Set.univ Adj := G.Adj adj_sub := id edge_vert := @fun v _ _ => Set.mem_univ v symm := G.symm } /-- The `bot` subgraph is the subgraph with no vertices or edges. -/ instance : Bot G.Subgraph where bot := { verts := ∅ Adj := ⊥ adj_sub := False.elim edge_vert := False.elim symm := fun _ _ => id } instance : SupSet G.Subgraph where sSup s := { verts := ⋃ G' ∈ s, verts G' Adj := fun a b => ∃ G' ∈ s, Adj G' a b adj_sub := by rintro a b ⟨G', -, hab⟩ exact G'.adj_sub hab edge_vert := by rintro a b ⟨G', hG', hab⟩ exact Set.mem_iUnion₂_of_mem hG' (G'.edge_vert hab) symm := fun a b h => by simpa [adj_comm] using h } instance : InfSet G.Subgraph where sInf s := { verts := ⋂ G' ∈ s, verts G' Adj := fun a b => (∀ ⦃G'⦄, G' ∈ s → Adj G' a b) ∧ G.Adj a b adj_sub := And.right edge_vert := fun hab => Set.mem_iInter₂_of_mem fun G' hG' => G'.edge_vert <\| hab.1 hG' symm := fun _ _ => And.imp (forall₂_imp fun _ _ => Adj.symm) G.adj_symm } @[simp] theorem sup_adj : (G₁ ⊔ G₂).Adj a b ↔ G₁.Adj a b ∨ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.sup_adj SimpleGraph.Subgraph.sup_adj @[simp] theorem inf_adj : (G₁ ⊓ G₂).Adj a b ↔ G₁.Adj a b ∧ G₂.Adj a b := Iff.rfl #align simple_graph.subgraph.inf_adj SimpleGraph.Subgraph.inf_adj @[simp] theorem top_adj : (⊤ : Subgraph G).Adj a b ↔ G.Adj a b := Iff.rfl #align simple_graph.subgraph.top_adj SimpleGraph.Subgraph.top_adj @[simp] theorem not_bot_adj : ¬ (⊥ : Subgraph G).Adj a b := not_false #align simple_graph.subgraph.not_bot_adj SimpleGraph.Subgraph.not_bot_adj @[simp] theorem verts_sup (G₁ G₂ : G.Subgraph) : (G₁ ⊔ G₂).verts = G₁.verts ∪ G₂.verts := rfl #align simple_graph.subgraph.verts_sup SimpleGraph.Subgraph.verts_sup @[simp] theorem verts_inf (G₁ G₂ : G.Subgraph) : (G₁ ⊓ G₂).verts = G₁.verts ∩ G₂.verts := rfl #align simple_graph.subgraph.verts_inf SimpleGraph.Subgraph.verts_inf @[simp] theorem verts_top : (⊤ : G.Subgraph).verts = Set.univ := rfl #align simple_graph.subgraph.verts_top SimpleGraph.Subgraph.verts_top @[simp] theorem verts_bot : (⊥ : G.Subgraph).verts = ∅ := rfl #align simple_graph.subgraph.verts_bot SimpleGraph.Subgraph.verts_bot @[simp] theorem sSup_adj {s : Set G.Subgraph} : (sSup s).Adj a b ↔ ∃ G ∈ s, Adj G a b := Iff.rfl #align simple_graph.subgraph.Sup_adj SimpleGraph.Subgraph.sSup_adj @[simp] theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b := Iff.rfl #align simple_graph.subgraph.Inf_adj SimpleGraph.Subgraph.sInf_adj @[simp] theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b := by simp [iSup] #align simple_graph.subgraph.supr_adj SimpleGraph.Subgraph.iSup_adj @[simp] theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b := by simp [iInf] #align simple_graph.subgraph.infi_adj SimpleGraph.Subgraph.iInf_adj theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) : (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b := sInf_adj.trans <\| and_iff_left_of_imp <\| by obtain ⟨G', hG'⟩ := hs exact fun h => G'.adj_sub (h _ hG') #align simple_graph.subgraph.Inf_adj_of_nonempty SimpleGraph.Subgraph.sInf_adj_of_nonempty theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b := by rw [iInf, sInf_adj_of_nonempty (Set.range_nonempty _)] simp #align simple_graph.subgraph.infi_adj_of_nonempty SimpleGraph.Subgraph.iInf_adj_of_nonempty @[simp] theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Sup SimpleGraph.Subgraph.verts_sSup @[simp] theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' := rfl #align simple_graph.subgraph.verts_Inf SimpleGraph.Subgraph.verts_sInf @[simp] theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts := by simp [iSup] #align simple_graph.subgraph.verts_supr SimpleGraph.Subgraph.verts_iSup @[simp] theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf] #align simple_graph.subgraph.verts_infi SimpleGraph.Subgraph.verts_iInf theorem verts_spanningCoe_injective : (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by intro G₁ G₂ h rw [Prod.ext_iff] at h exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2) /-- For subgraphs `G₁`, `G₂`, `G₁ ≤ G₂` iff `G₁.verts ⊆ G₂.verts` and `∀ a b, G₁.adj a b → G₂.adj a b`. -/ instance distribLattice : DistribLattice G.Subgraph := { show DistribLattice G.Subgraph from verts_spanningCoe_injective.distribLattice _ (fun _ _ => rfl) fun _ _ => rfl with le := fun x y => x.verts ⊆ y.verts ∧ ∀ ⦃v w : V⦄, x.Adj v w → y.Adj v w } instance : BoundedOrder (Subgraph G) where top := ⊤ bot := ⊥ le_top x := ⟨Set.subset_univ _, fun _ _ => x.adj_sub⟩ bot_le _ := ⟨Set.empty_subset _, fun _ _ => False.elim⟩ -- Note that subgraphs do not form a Boolean algebra, because of `verts`. instance : CompletelyDistribLattice G.Subgraph := { Subgraph.distribLattice with le := (· ≤ ·) sup := (· ⊔ ·) inf := (· ⊓ ·) top := ⊤ bot := ⊥ le_top := fun G' => ⟨Set.subset_univ _, fun a b => G'.adj_sub⟩ bot_le := fun G' => ⟨Set.empty_subset _, fun a b => False.elim⟩ sSup := sSup -- porting note: needed `apply` here to modify elaboration; previously the term itself was fine. le_sSup := fun s G' hG' => ⟨by apply Set.subset_iUnion₂ G' hG', fun a b hab => ⟨G', hG', hab⟩⟩ sSup_le := fun s G' hG' => ⟨Set.iUnion₂_subset fun H hH => (hG' _ hH).1, by rintro a b ⟨H, hH, hab⟩ exact (hG' _ hH).2 hab⟩ sInf := sInf sInf_le := fun s G' hG' => ⟨Set.iInter₂_subset G' hG', fun a b hab => hab.1 hG'⟩ le_sInf := fun s G' hG' => ⟨Set.subset_iInter₂ fun H hH => (hG' _ hH).1, fun a b hab => ⟨fun H hH => (hG' _ hH).2 hab, G'.adj_sub hab⟩⟩ iInf_iSup_eq := fun f => Subgraph.ext _ _ (by simpa using iInf_iSup_eq) (by ext; simp [Classical.skolem]) } @[simps] instance subgraphInhabited : Inhabited (Subgraph G) := ⟨⊥⟩ #align simple_graph.subgraph.subgraph_inhabited SimpleGraph.Subgraph.subgraphInhabited @[simp] theorem neighborSet_sup {H H' : G.Subgraph} (v : V) : (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_sup SimpleGraph.Subgraph.neighborSet_sup @[simp] theorem neighborSet_inf {H H' : G.Subgraph} (v : V) : (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_inf SimpleGraph.Subgraph.neighborSet_inf @[simp] theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v := rfl #align simple_graph.subgraph.neighbor_set_top SimpleGraph.Subgraph.neighborSet_top @[simp] theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ := rfl #align simple_graph.subgraph.neighbor_set_bot SimpleGraph.Subgraph.neighborSet_bot @[simp] theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) : (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v := by ext simp #align simple_graph.subgraph.neighbor_set_Sup SimpleGraph.Subgraph.neighborSet_sSup @[simp] theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) : (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by ext simp #align simple_graph.subgraph.neighbor_set_Inf SimpleGraph.Subgraph.neighborSet_sInf @[simp] theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) : (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v := by simp [iSup] #align simple_graph.subgraph.neighbor_set_supr SimpleGraph.Subgraph.neighborSet_iSup @[simp] theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) : (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v := by simp [iInf] #align simple_graph.subgraph.neighbor_set_infi SimpleGraph.Subgraph.neighborSet_iInf @[simp] theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet := rfl #align simple_graph.subgraph.edge_set_top SimpleGraph.Subgraph.edgeSet_top @[simp] theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_bot SimpleGraph.Subgraph.edgeSet_bot @[simp] theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_inf SimpleGraph.Subgraph.edgeSet_inf @[simp] theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet := Set.ext <\| Sym2.ind (by simp) #align simple_graph.subgraph.edge_set_sup SimpleGraph.Subgraph.edgeSet_sup @[simp] theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Sup SimpleGraph.Subgraph.edgeSet_sSup @[simp] theorem edgeSet_sInf (s : Set G.Subgraph) : (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet := by ext e induction e using Sym2.ind simp #align simple_graph.subgraph.edge_set_Inf SimpleGraph.Subgraph.edgeSet_sInf @[simp] theorem edgeSet_iSup (f : ι → G.Subgraph) : (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by simp [iSup] #align simple_graph.subgraph.edge_set_supr SimpleGraph.Subgraph.edgeSet_iSup @[simp] theorem edgeSet_iInf (f : ι → G.Subgraph) : (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet := by simp [iInf] #align simple_graph.subgraph.edge_set_infi SimpleGraph.Subgraph.edgeSet_iInf @[simp] theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl #align simple_graph.subgraph.spanning_coe_top SimpleGraph.Subgraph.spanningCoe_top @[simp] theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl #align simple_graph.subgraph.spanning_coe_bot SimpleGraph.Subgraph.spanningCoe_bot /-- Turn a subgraph of a `SimpleGraph` into a member of its subgraph type. -/ @[simps] def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph where verts := Set.univ Adj := H.Adj adj_sub e := h e edge_vert _ := Set.mem_univ _ symm := H.symm #align simple_graph.to_subgraph SimpleGraph.toSubgraph theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support := Rel.dom_mono h.2 #align simple_graph.subgraph.support_mono SimpleGraph.Subgraph.support_mono theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) : (toSubgraph H h).IsSpanning := Set.mem_univ #align simple_graph.to_subgraph.is_spanning SimpleGraph.toSubgraph.isSpanning theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe := h.2 #align simple_graph.subgraph.spanning_coe_le_of_le SimpleGraph.Subgraph.spanningCoe_le_of_le /-- The top of the `Subgraph G` lattice is equivalent to the graph itself. -/ def topEquiv : (⊤ : Subgraph G).coe ≃g G where toFun v := ↑v invFun v := ⟨v, trivial⟩ left_inv _ := rfl right_inv _ := rfl map_rel_iff' := Iff.rfl #align simple_graph.subgraph.top_equiv SimpleGraph.Subgraph.topEquiv /-- The bottom of the `Subgraph G` lattice is equivalent to the empty graph on the empty vertex type. -/ def botEquiv : (⊥ : Subgraph G).coe ≃g (⊥ : SimpleGraph Empty) where toFun v := v.property.elim invFun v := v.elim left_inv := fun ⟨_, h⟩ ↦ h.elim right_inv v := v.elim map_rel_iff' := Iff.rfl #align simple_graph.subgraph.bot_equiv SimpleGraph.Subgraph.botEquiv theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet := Sym2.ind h.2 #align simple_graph.subgraph.edge_set_mono SimpleGraph.Subgraph.edgeSet_mono theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) : Disjoint H₁.edgeSet H₂.edgeSet := disjoint_iff_inf_le.mpr <\| by simpa using edgeSet_mono h.le_bot #align disjoint.edge_set Disjoint.edgeSet /-- Graph homomorphisms induce a covariant function on subgraphs. -/ @[simps] protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where verts := f '' H.verts Adj := Relation.Map H.Adj f f adj_sub := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact f.map_rel (H.adj_sub h) edge_vert := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact Set.mem_image_of_mem _ (H.edge_vert h) symm := by rintro _ _ ⟨u, v, h, rfl, rfl⟩ exact ⟨v, u, H.symm h, rfl, rfl⟩ #align simple_graph.subgraph.map SimpleGraph.Subgraph.map theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by intro H H' h constructor · intro simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro v hv rfl exact ⟨_, h.1 hv, rfl⟩ · rintro _ _ ⟨u, v, ha, rfl, rfl⟩ exact ⟨_, _, h.2 ha, rfl, rfl⟩ #align simple_graph.subgraph.map_monotone SimpleGraph.Subgraph.map_monotone theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} : (H ⊔ H').map f = H.map f ⊔ H'.map f := by ext1 · simp only [Set.image_union, map_verts, verts_sup] · ext simp only [Relation.Map, map_adj, sup_adj] constructor · rintro ⟨a, b, h \| h, rfl, rfl⟩ · exact Or.inl ⟨_, _, h, rfl, rfl⟩ · exact Or.inr ⟨_, _, h, rfl, rfl⟩ · rintro (⟨a, b, h, rfl, rfl⟩ \| ⟨a, b, h, rfl, rfl⟩) · exact ⟨_, _, Or.inl h, rfl, rfl⟩ · exact ⟨_, _, Or.inr h, rfl, rfl⟩ #align simple_graph.subgraph.map_sup SimpleGraph.Subgraph.map_sup /-- Graph homomorphisms induce a contravariant function on subgraphs. -/ @[simps] protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph where verts := f ⁻¹' H.verts Adj u v := G.Adj u v ∧ H.Adj (f u) (f v) adj_sub h := h.1 edge_vert h := Set.mem_preimage.1 (H.edge_vert h.2) symm _ _ h := ⟨G.symm h.1, H.symm h.2⟩ #align simple_graph.subgraph.comap SimpleGraph.Subgraph.comap theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) := by intro H H' h constructor · intro simp only [comap_verts, Set.mem_preimage] apply h.1 · intro v w simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff] intro apply h.2 #align simple_graph.subgraph.comap_monotone SimpleGraph.Subgraph.comap_monotone theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) : H.map f ≤ H' ↔ H ≤ H'.comap f := by refine' ⟨fun h ↦ ⟨fun v hv ↦ _, fun v w hvw ↦ _⟩, fun h ↦ ⟨fun v ↦ _, fun v w ↦ _⟩⟩ · simp only [comap_verts, Set.mem_preimage] exact h.1 ⟨v, hv, rfl⟩ · simp only [H.adj_sub hvw, comap_adj, true_and_iff] exact h.2 ⟨v, w, hvw, rfl, rfl⟩ · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp] rintro w hw rfl exact h.1 hw · simp only [Relation.Map, map_adj, forall_exists_index, and_imp] rintro u u' hu rfl rfl exact (h.2 hu).2 #align simple_graph.subgraph.map_le_iff_le_comap SimpleGraph.Subgraph.map_le_iff_le_comap /-- Given two subgraphs, one a subgraph of the other, there is an induced injective homomorphism of the subgraphs as graphs. -/ @[simps] def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe where toFun v := ⟨↑v, And.left h v.property⟩ map_rel' hvw := h.2 hvw #align simple_graph.subgraph.inclusion SimpleGraph.Subgraph.inclusion theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by intro v w h rw [inclusion, FunLike.coe, Subtype.mk_eq_mk] at h exact Subtype.ext h #align simple_graph.subgraph.inclusion.injective SimpleGraph.Subgraph.inclusion.injective /-- There is an induced injective homomorphism of a subgraph of `G` into `G`. -/ @[simps] protected def hom (x : Subgraph G) : x.coe →g G where toFun v := v map_rel' := x.adj_sub #align simple_graph.subgraph.hom SimpleGraph.Subgraph.hom @[simp] lemma coe_hom (x : Subgraph G) : (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) := rfl theorem hom.injective {x : Subgraph G} : Function.Injective x.hom := fun _ _ ↦ Subtype.ext #align simple_graph.subgraph.hom.injective SimpleGraph.Subgraph.hom.injective /-- There is an induced injective homomorphism of a subgraph of `G` as a spanning subgraph into `G`. -/ @[simps] def spanningHom (x : Subgraph G) : x.spanningCoe →g G where toFun := id map_rel' := x.adj_sub #align simple_graph.subgraph.spanning_hom SimpleGraph.Subgraph.spanningHom theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom := fun _ _ ↦ id #align simple_graph.subgraph.spanning_hom.injective SimpleGraph.Subgraph.spanningHom.injective theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) : x.neighborSet v ⊆ y.neighborSet v := fun _ h' ↦ h.2 h' #align simple_graph.subgraph.neighbor_set_subset_of_subgraph SimpleGraph.Subgraph.neighborSet_subset_of_subgraph instance neighborSet.decidablePred (G' : Subgraph G) [h : DecidableRel G'.Adj] (v : V) : DecidablePred (· ∈ G'.neighborSet v) := h v #align simple_graph.subgraph.neighbor_set.decidable_pred SimpleGraph.Subgraph.neighborSet.decidablePred /-- If a graph is locally finite at a vertex, then so is a subgraph of that graph. -/ instance finiteAt {G' : Subgraph G} (v : G'.verts) [DecidableRel G'.Adj] [Fintype (G.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G.neighborSet v) (G'.neighborSet_subset v) #align simple_graph.subgraph.finite_at SimpleGraph.Subgraph.finiteAt /-- If a subgraph is locally finite at a vertex, then so are subgraphs of that subgraph. This is not an instance because `G''` cannot be inferred. -/ def finiteAtOfSubgraph {G' G'' : Subgraph G} [DecidableRel G'.Adj] (h : G' ≤ G'') (v : G'.verts) [Fintype (G''.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset (G''.neighborSet v) (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.finite_at_of_subgraph SimpleGraph.Subgraph.finiteAtOfSubgraph instance (G' : Subgraph G) [Fintype G'.verts] (v : V) [DecidablePred (· ∈ G'.neighborSet v)] : Fintype (G'.neighborSet v) := Set.fintypeSubset G'.verts (neighborSet_subset_verts G' v) instance coeFiniteAt {G' : Subgraph G} (v : G'.verts) [Fintype (G'.neighborSet v)] : Fintype (G'.coe.neighborSet v) := Fintype.ofEquiv _ (coeNeighborSetEquiv v).symm #align simple_graph.subgraph.coe_finite_at SimpleGraph.Subgraph.coeFiniteAt theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) : G'.verts.toFinset.card = Fintype.card V := by simp only [isSpanning_iff.1 h, Set.toFinset_univ] congr #align simple_graph.subgraph.is_spanning.card_verts SimpleGraph.Subgraph.IsSpanning.card_verts /-- The degree of a vertex in a subgraph. It's zero for vertices outside the subgraph. -/ def degree (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] : ℕ := Fintype.card (G'.neighborSet v) #align simple_graph.subgraph.degree SimpleGraph.Subgraph.degree theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : (G'.neighborSet v).toFinset.card = G'.degree v := by rw [degree, Set.toFinset_card] #align simple_graph.subgraph.finset_card_neighbor_set_eq_degree SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)] [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v := by rw [← card_neighborSet_eq_degree] exact Set.card_le_of_subset (G'.neighborSet_subset v) #align simple_graph.subgraph.degree_le SimpleGraph.Subgraph.degree_le theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)] [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v := Set.card_le_of_subset (neighborSet_subset_of_subgraph h v) #align simple_graph.subgraph.degree_le' SimpleGraph.Subgraph.degree_le' @[simp] theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)] [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree] exact Fintype.card_congr (coeNeighborSetEquiv v) #align simple_graph.subgraph.coe_degree SimpleGraph.Subgraph.coe_degree @[simp] theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)] [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v := by rw [← card_neighborSet_eq_degree, Subgraph.degree] congr! #align simple_graph.subgraph.degree_spanning_coe SimpleGraph.Subgraph.degree_spanningCoe theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] : G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w := by rw [← finset_card_neighborSet_eq_degree, Finset.card_eq_one, Finset.singleton_iff_unique_mem] simp only [Set.mem_toFinset, mem_neighborSet] #align simple_graph.subgraph.degree_eq_one_iff_unique_adj SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj end Subgraph section MkProperties /-! ### Properties of `singletonSubgraph` and `subgraphOfAdj` -/ variable {G : SimpleGraph V} {G' : SimpleGraph W} instance nonempty_singletonSubgraph_verts (v : V) : Nonempty (G.singletonSubgraph v).verts := ⟨⟨v, Set.mem_singleton v⟩⟩ #align simple_graph.nonempty_singleton_subgraph_verts SimpleGraph.nonempty_singletonSubgraph_verts @[simp] theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) : G.singletonSubgraph v ≤ H ↔ v ∈ H.verts := by refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩ intro h constructor · rwa [singletonSubgraph_verts, Set.singleton_subset_iff] · exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_le_iff SimpleGraph.singletonSubgraph_le_iff @[simp] theorem map_singletonSubgraph (f : G →g G') {v : V} : Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) := by ext <;> simp only [Relation.Map, Subgraph.map_adj, singletonSubgraph_adj, Pi.bot_apply, exists_and_left, and_iff_left_iff_imp, IsEmpty.forall_iff, Subgraph.map_verts, singletonSubgraph_verts, Set.image_singleton] exact False.elim #align simple_graph.map_singleton_subgraph SimpleGraph.map_singletonSubgraph @[simp] theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ := rfl #align simple_graph.neighbor_set_singleton_subgraph SimpleGraph.neighborSet_singletonSubgraph @[simp] theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ := Sym2.fromRel_bot #align simple_graph.edge_set_singleton_subgraph SimpleGraph.edgeSet_singletonSubgraph theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} : H = G.singletonSubgraph v ↔ H.verts = {v} := by refine' ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ _⟩ ext · rw [h, singletonSubgraph_verts] · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff] intro ha have ha1 := ha.fst_mem have ha2 := ha.snd_mem rw [h, Set.mem_singleton_iff] at ha1 ha2 subst_vars exact ha.ne rfl #align simple_graph.eq_singleton_subgraph_iff_verts_eq SimpleGraph.eq_singletonSubgraph_iff_verts_eq instance nonempty_subgraphOfAdj_verts {v w : V} (hvw : G.Adj v w) : Nonempty (G.subgraphOfAdj hvw).verts := ⟨⟨v, by simp⟩⟩ #align simple_graph.nonempty_subgraph_of_adj_verts SimpleGraph.nonempty_subgraphOfAdj_verts @[simp] theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).edgeSet = {⟦(v, w)⟧} := by ext e refine' e.ind _ simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff, forall₂_true_iff] #align simple_graph.edge_set_subgraph_of_adj SimpleGraph.edgeSet_subgraphOfAdj set_option autoImplicit true in lemma subgraphOfAdj_le_of_adj (H : G.Subgraph) (h : H.Adj v w) : G.subgraphOfAdj (H.adj_sub h) ≤ H := by constructor · intro x rintro (rfl \| rfl) <;> simp [H.edge_vert h, H.edge_vert h.symm] · simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] rintro _ _ (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) <;> simp [h, h.symm] theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw := by ext <;> simp [or_comm, and_comm] #align simple_graph.subgraph_of_adj_symm SimpleGraph.subgraphOfAdj_symm @[simp] theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) : Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) := by ext · simp only [Subgraph.map_verts, subgraphOfAdj_verts, Set.mem_image, Set.mem_insert_iff, Set.mem_singleton_iff] constructor · rintro ⟨u, rfl \| rfl, rfl⟩ <;> simp · rintro (rfl \| rfl) · use v simp · use w simp · simp only [Relation.Map, Subgraph.map_adj, subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] constructor · rintro ⟨a, b, ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩, rfl, rfl⟩ <;> simp · rintro (⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩) · use v, w simp · use w, v simp #align simple_graph.map_subgraph_of_adj SimpleGraph.map_subgraphOfAdj theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} := (G.subgraphOfAdj hvw).neighborSet_subset_verts _ #align simple_graph.neighbor_set_subgraph_of_adj_subset SimpleGraph.neighborSet_subgraphOfAdj_subset @[simp] theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet v = {w} := by ext u suffices w = u ↔ u = w by simpa [hvw.ne.symm] using this rw [eq_comm] #align simple_graph.neighbor_set_fst_subgraph_of_adj SimpleGraph.neighborSet_fst_subgraphOfAdj @[simp] theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet w = {v} := by rw [subgraphOfAdj_symm hvw.symm] exact neighborSet_fst_subgraphOfAdj hvw.symm #align simple_graph.neighbor_set_snd_subgraph_of_adj SimpleGraph.neighborSet_snd_subgraphOfAdj @[simp] theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ := by ext simp [hv.symm, hw.symm] #align simple_graph.neighbor_set_subgraph_of_adj_of_ne_of_ne SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) : (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ := by split_ifs <;> subst_vars <;> simp [*] #align simple_graph.neighbor_set_subgraph_of_adj SimpleGraph.neighborSet_subgraphOfAdj theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph u ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_fst_le_subgraph_of_adj SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} : G.singletonSubgraph v ≤ G.subgraphOfAdj h := by constructor <;> simp [-Set.bot_eq_empty] exact fun _ _ ↦ False.elim #align simple_graph.singleton_subgraph_snd_le_subgraph_of_adj SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj end MkProperties namespace Subgraph variable {G : SimpleGraph V} /-! ### Subgraphs of subgraphs -/ /-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/ @[reducible] protected def coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph := Subgraph.map G'.hom #align simple_graph.subgraph.coe_subgraph SimpleGraph.Subgraph.coeSubgraph /-- Given a subgraph of `G`, restrict it to being a subgraph of another subgraph `G'` by taking the portion of `G` that intersects `G'`. -/ @[reducible] protected def restrict {G' : G.Subgraph} : G.Subgraph → G'.coe.Subgraph := Subgraph.comap G'.hom #align simple_graph.subgraph.restrict SimpleGraph.Subgraph.restrict lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) : (G'.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ := by simp [Relation.Map] lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) : (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w := Iff.rfl theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) : Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' := by ext · simp · rw [restrict_adj, coeSubgraph_adj] simpa using G''.adj_sub #align simple_graph.subgraph.restrict_coe_subgraph SimpleGraph.Subgraph.restrict_coeSubgraph theorem coeSubgraph_injective (G' : G.Subgraph) : Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) := Function.LeftInverse.injective restrict_coeSubgraph #align simple_graph.subgraph.coe_subgraph_injective SimpleGraph.Subgraph.coeSubgraph_injective lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) : Subgraph.coeSubgraph H' ≤ H := by constructor · simp · rintro v w ⟨_, _, h, rfl, rfl⟩ exact H'.adj_sub h lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) : Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' := by ext · simp [and_comm] · simp_rw [coeSubgraph_adj, restrict_adj] simp only [exists_and_left, exists_prop, ge_iff_le, inf_adj, and_congr_right_iff] intro h simp [H.edge_vert h, H.edge_vert h.symm] /-! ### Edge deletion -/ /-- Given a subgraph `G'` and a set of vertex pairs, remove all of the corresponding edges from its edge set, if present. See also: `SimpleGraph.deleteEdges`. -/ def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where verts := G'.verts Adj := G'.Adj \ Sym2.ToRel s adj_sub h' := G'.adj_sub h'.1 edge_vert h' := G'.edge_vert h'.1 symm a b := by simp [G'.adj_comm, Sym2.eq_swap] #align simple_graph.subgraph.delete_edges SimpleGraph.Subgraph.deleteEdges section DeleteEdges variable {G' : G.Subgraph} (s : Set (Sym2 V)) @[simp] theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts := rfl #align simple_graph.subgraph.delete_edges_verts SimpleGraph.Subgraph.deleteEdges_verts @[simp] theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬⟦(v, w)⟧ ∈ s := Iff.rfl #align simple_graph.subgraph.delete_edges_adj SimpleGraph.Subgraph.deleteEdges_adj @[simp] theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) : (G'.deleteEdges s).deleteEdges s' = G'.deleteEdges (s ∪ s') := by ext <;> simp [and_assoc, not_or] #align simple_graph.subgraph.delete_edges_delete_edges SimpleGraph.Subgraph.deleteEdges_deleteEdges @[simp] theorem deleteEdges_empty_eq : G'.deleteEdges ∅ = G' := by ext <;> simp #align simple_graph.subgraph.delete_edges_empty_eq SimpleGraph.Subgraph.deleteEdges_empty_eq @[simp] theorem deleteEdges_spanningCoe_eq : G'.spanningCoe.deleteEdges s = (G'.deleteEdges s).spanningCoe := by ext simp #align simple_graph.subgraph.delete_edges_spanning_coe_eq SimpleGraph.Subgraph.deleteEdges_spanningCoe_eq theorem deleteEdges_coe_eq (s : Set (Sym2 G'.verts)) : G'.coe.deleteEdges s = (G'.deleteEdges (Sym2.map (↑) '' s)).coe := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists, not_and, and_congr_right_iff] intro constructor · intro hs refine' Sym2.ind _ rintro ⟨v', hv'⟩ ⟨w', hw'⟩ simp only [Sym2.map_pair_eq, Quotient.eq] contrapose! rintro (_ \| _) <;> simpa only [Sym2.eq_swap] · intro h' hs exact h' _ hs rfl #align simple_graph.subgraph.delete_edges_coe_eq SimpleGraph.Subgraph.deleteEdges_coe_eq theorem coe_deleteEdges_eq (s : Set (Sym2 V)) : (G'.deleteEdges s).coe = G'.coe.deleteEdges (Sym2.map (↑) ⁻¹' s) := by ext ⟨v, hv⟩ ⟨w, hw⟩ simp #align simple_graph.subgraph.coe_delete_edges_eq SimpleGraph.Subgraph.coe_deleteEdges_eq theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by constructor <;> simp (config := { contextual := true }) [subset_rfl] #align simple_graph.subgraph.delete_edges_le SimpleGraph.Subgraph.deleteEdges_le theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') : G'.deleteEdges s' ≤ G'.deleteEdges s := by constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj, true_and_iff, and_imp, subset_rfl] exact fun _ _ _ hs' hs ↦ hs' (h hs) #align simple_graph.subgraph.delete_edges_le_of_le SimpleGraph.Subgraph.deleteEdges_le_of_le @[simp] theorem deleteEdges_inter_edgeSet_left_eq : G'.deleteEdges (G'.edgeSet ∩ s) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_left_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_left_eq @[simp] theorem deleteEdges_inter_edgeSet_right_eq : G'.deleteEdges (s ∩ G'.edgeSet) = G'.deleteEdges s := by ext <;> simp (config := { contextual := true }) [imp_false] #align simple_graph.subgraph.delete_edges_inter_edge_set_right_eq SimpleGraph.Subgraph.deleteEdges_inter_edgeSet_right_eq theorem coe_deleteEdges_le : (G'.deleteEdges s).coe ≤ (G'.coe : SimpleGraph G'.verts) := by intro v w simp (config := { contextual := true }) #align simple_graph.subgraph.coe_delete_edges_le SimpleGraph.Subgraph.coe_deleteEdges_le theorem spanningCoe_deleteEdges_le (G' : G.Subgraph) (s : Set (Sym2 V)) : (G'.deleteEdges s).spanningCoe ≤ G'.spanningCoe := spanningCoe_le_of_le (deleteEdges_le s) #align simple_graph.subgraph.spanning_coe_delete_edges_le SimpleGraph.Subgraph.spanningCoe_deleteEdges_le end DeleteEdges /-! ### Induced subgraphs -/ /- Given a subgraph, we can change its vertex set while removing any invalid edges, which gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which, unlike for subgraphs, results in a graph with a different vertex type. -/ /-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual notion of an induced subgraph, but, in general, `s` is taken to be the new vertex set and edges are induced from the subgraph `G'`. -/ @[simps] def induce (G' : G.Subgraph) (s : Set V) : G.Subgraph where verts := s Adj u v := u ∈ s ∧ v ∈ s ∧ G'.Adj u v adj_sub h := G'.adj_sub h.2.2 edge_vert h := h.1 symm _ _ h := ⟨h.2.1, h.1, G'.symm h.2.2⟩ #align simple_graph.subgraph.induce SimpleGraph.Subgraph.induce theorem _root_.SimpleGraph.induce_eq_coe_induce_top (s : Set V) : G.induce s = ((⊤ : G.Subgraph).induce s).coe := by ext simp #align simple_graph.induce_eq_coe_induce_top SimpleGraph.induce_eq_coe_induce_top section Induce variable {G' G'' : G.Subgraph} {s s' : Set V} theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by constructor · simp [hs] · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp] intro v w hv hw ha exact ⟨hs hv, hs hw, hg.2 ha⟩ #align simple_graph.subgraph.induce_mono SimpleGraph.Subgraph.induce_mono @[mono] theorem induce_mono_left (hg : G' ≤ G'') : G'.induce s ≤ G''.induce s := induce_mono hg subset_rfl #align simple_graph.subgraph.induce_mono_left SimpleGraph.Subgraph.induce_mono_left @[mono] theorem induce_mono_right (hs : s ⊆ s') : G'.induce s ≤ G'.induce s' := induce_mono le_rfl hs #align simple_graph.subgraph.induce_mono_right SimpleGraph.Subgraph.induce_mono_right @[simp] theorem induce_empty : G'.induce ∅ = ⊥ := by ext <;> simp #align simple_graph.subgraph.induce_empty SimpleGraph.Subgraph.induce_empty @[simp] theorem induce_self_verts : G'.induce G'.verts = G' := by ext · simp · constructor <;> simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff] exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩ #align simple_graph.subgraph.induce_self_verts SimpleGraph.Subgraph.induce_self_verts lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts := calc G' = G'.induce G'.verts := Subgraph.induce_self_verts.symm _ ≤ (⊤ : G.Subgraph).induce G'.verts := Subgraph.induce_mono_left le_top lemma le_induce_union : G'.induce s ⊔ G'.induce s' ≤ G'.induce (s ∪ s') := by constructor · simp only [verts_sup, induce_verts, Set.Subset.rfl] · simp only [sup_adj, induce_adj, Set.mem_union] rintro v w (h \| h) <;> simp [h] lemma le_induce_union_left : G'.induce s ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).1 lemma le_induce_union_right : G'.induce s' ≤ G'.induce (s ∪ s') := by exact (sup_le_iff.mp le_induce_union).2 theorem singletonSubgraph_eq_induce {v : V} : G.singletonSubgraph v = (⊤ : G.Subgraph).induce {v} := by ext <;> simp (config := { contextual := true }) [-Set.bot_eq_empty, Prop.bot_eq_false] #align simple_graph.subgraph.singleton_subgraph_eq_induce SimpleGraph.Subgraph.singletonSubgraph_eq_induce theorem subgraphOfAdj_eq_induce {v w : V} (hvw : G.Adj v w) : G.subgraphOfAdj hvw = (⊤ : G.Subgraph).induce {v, w} := by ext · simp · constructor · intro h simp only [subgraphOfAdj_adj, Quotient.eq, Sym2.rel_iff] at h obtain ⟨rfl, rfl⟩ \| ⟨rfl, rfl⟩ := h <;> simp [hvw, hvw.symm] · intro h simp only [induce_adj, Set.mem_insert_iff, Set.mem_singleton_iff, top_adj] at h obtain ⟨rfl \| rfl, rfl \| rfl, ha⟩ := h <;> first \|exact (ha.ne rfl).elim\|simp #align simple_graph.subgraph.subgraph_of_adj_eq_induce SimpleGraph.Subgraph.subgraphOfAdj_eq_induce end Induce /-- Given a subgraph and a set of vertices, delete all the vertices from the subgraph, if present. Any edges incident to the deleted vertices are deleted as well. -/ @[reducible] def deleteVerts (G' : G.Subgraph) (s : Set V) : G.Subgraph := G'.induce (G'.verts \ s) #align simple_graph.subgraph.delete_verts SimpleGraph.Subgraph.deleteVerts section DeleteVerts variable {G' : G.Subgraph} {s : Set V} theorem deleteVerts_verts : (G'.deleteVerts s).verts = G'.verts \ s := rfl #align simple_graph.subgraph.delete_verts_verts SimpleGraph.Subgraph.deleteVerts_verts theorem deleteVerts_adj {u v : V} : (G'.deleteVerts s).Adj u v ↔ u ∈ G'.verts ∧ ¬u ∈ s ∧ v ∈ G'.verts ∧ ¬v ∈ s ∧ G'.Adj u v := by simp [and_assoc] #align simple_graph.subgraph.delete_verts_adj SimpleGraph.Subgraph.deleteVerts_adj @[simp] theorem deleteVerts_deleteVerts (s s' : Set V) : (G'.deleteVerts s).deleteVerts s' = G'.deleteVerts (s ∪ s') := by ext <;> simp (config := { contextual := true }) [not_or, and_assoc] #align simple_graph.subgraph.delete_verts_delete_verts SimpleGraph.Subgraph.deleteVerts_deleteVerts @[simp] theorem deleteVerts_empty : G'.deleteVerts ∅ = G' := by simp [deleteVerts] #align simple_graph.subgraph.delete_verts_empty SimpleGraph.Subgraph.deleteVerts_empty theorem deleteVerts_le : G'.deleteVerts s ≤ G' := by constructor <;> simp [Set.diff_subset] #align simple_graph.subgraph.delete_verts_le SimpleGraph.Subgraph.deleteVerts_le @[mono] theorem deleteVerts_mono {G' G'' : G.Subgraph} (h : G' ≤ G'') : G'.deleteVerts s ≤ G''.deleteVerts s := induce_mono h (Set.diff_subset_diff_left h.1) #align simple_graph.subgraph.delete_verts_mono SimpleGraph.Subgraph.deleteVerts_mono @[mono] theorem deleteVerts_anti {s s' : Set V} (h : s ⊆ s') : G'.deleteVerts s' ≤ G'.deleteVerts s := induce_mono (le_refl _) (Set.diff_subset_diff_right h) #align simple_graph.subgraph.delete_verts_anti SimpleGraph.Subgraph.deleteVerts_anti @[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by ext <;>	simp (config := { contextual := true }) [imp_false]	@[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s := by ext <;>	Mathlib.Combinatorics.SimpleGraph.Subgraph.1289_0.BlhiAiIDADcXv8t	@[simp] theorem deleteVerts_inter_verts_left_eq : G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s	Mathlib_Combinatorics_SimpleGraph_Subgraph

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