diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzasum" "b/data_all_eng_slimpj/shuffled/split2/finalzzasum"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzasum"
@@ -0,0 +1,5 @@
+{"text":"\\section*{Introduction}\n\nThe aim of this paper is to describe a simple method to compute the Manin black products with the operads $\\sA ss$, $\\sC om$ and $pre\\sL ie$. For instance, this allows the computation of $\\sA ss\\bullet\\sC om$ and $pre\\sL ie\\bullet pre\\sL ie$ (Examples \\ref{ex:ass} and \\ref{ex:prelie}), answering some questions posed by Loday \\cite{L}  (these had beeen answered already \\cite{bai2,GK}). While simple methods to compute $pre\\sL ie\\bullet-$ were already known \\cite{bai,bai3}, to our knowledge the results are new for the functors $\\sA ss\\bullet-$, $\\sC om\\bullet-$. We illustrate the method by several explicit computations: among these, we show that in the square diagram of operads introduced by Chapton \\cite{cha}   \n\\[\\xymatrix{ pre\\sL ie\\ar[r]&\\sD end\\ar[r]& \\sZ inb\\\\ \\sL ie\\ar[r]\\ar[u]&\\sA ss\\ar[r]\\ar[u]&\\sC om\\ar[u]\\\\\\sL eib\\ar[r]\\ar[u]&di\\sA ss\\ar[r]\\ar[u]&\\sP erm\\ar[u] }\\]\nwhere it is well known that the top row is the Manin black product of the middle one with $pre\\sL ie$ and the bottom row is the Manin white product of the middle one with $\\sP erm$, it is also true that the top row is the Manin \\emph{white} product of the middle one with $\\sZ inb$ and the bottom row is the Manin \\emph{black} product of the middle one with $\\sL eib$.\n\nWe explain the method by considering the case of $(\\sA ss,\\cup)$, where we denote by $\\cup$ the generating associative product. It is convenient to consider first the (Koszul) dual computation of the Manin white\nproduct $\\sA ss\\circ-$. Roughly, given an operad $\\Oh$, which will be always an operad in vector spaces over a field $\\mathbb{K}$ and moreover binary, quadratic and finitely generated by non-symmetric operations $\\cdot_i$, $i=1,\\ldots,p$, symmetric operations $\\bullet_j$, $j=1,\\ldots,q$, and anti-symmetric operations $[-,-]_k$, $k=1,\\ldots,r$, the operad $\\sA ss\\circ\\Oh$ is generated by the tensor product operations $\\cup\\otimes\\cdot_i$, $\\cup\\otimes\\cdot_i^{op}$, $\\cup\\otimes\\bullet_j$, $\\cup\\otimes[-,-]_k$ (all of which are non-symmetric, and where $\\cdot_i^{op}$ are the opposite products) together with the relations holding in the tensor product $A\\ten V$ of a generic $\\sA ss$-algebra $(A,\\cup)$ and a generic $\\Oh$-algebra $(V,\\cdot_i,\\bullet_j,\\ast_k)$. We define a functor $\\op{Ass}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$ by replacing the generic $\\sA ss$-algebra $(A,\\cup)$ in the above definition of $\\sA ss\\circ-$ with the dg associative algebra $(C^*(\\Delta_1;\\mathbb{K}),\\cup)$ of non-degenerate cochains on the $1$-simplex with the usual cup product, and show that $\\op{Ass}_\\circ(\\Oh)=\\sA ss\\circ\\Oh$, essentially because the only relations satisfied in $(C^*(\\Delta_1;\\mathbb{K}),\\cup)$ are the associativity relations. \n\nWe notice that besides the relation of $\\sA ss\\circ\\Oh$-algebra, the tensor product operations (which we will simply call the cup products in the body of the paper) on $C^*(\\Delta_1;V)=C^*(\\Delta_1;\\mathbb{K})\\otimes V$ satisfy the Leibniz identity with respect to the differential: moreover, for any $X\\subset\\Delta_1$ (that is, the boundary or one of the vertices) the subcomplex of relative cochains $C^*(\\Delta_1,X;V)\\subset C^*(\\Delta_1;V)$ is a dg $\\sA ss\\circ\\Oh$-ideal. We reassume the above properties by saying that $C^*(\\Delta_1;V)$ with the tensor product operations (cup products) is a local dg $\\sA ss\\circ\\Oh$-algebra. \n\nConversely, our trick to compute $\\sA ss\\bullet\\Oh$ consists in imposing a local dg $\\Oh$-algebra structure $(C^*(\\Delta_1;V),\\cdot_i,\\bullet_j,[-,-]_k)$ on the complex $C^*(\\Delta_1;V)$. Notice that the space $C^*(\\Delta_1;V)$ splits into the direct sum of three copies of $V$, we write $C^*(\\Delta_1;V)=V_0\\oplus V_1\\oplus V_{01}$, where $V_0$ (resp.: $V_1$) is the copy corresponding to the left (resp.: right) vertex and $V_{01}$ is the copy corresponding to the $1$-dimensional cell. The locality assumption and the Leibniz relation with respect to the differential imply that the whole $\\Oh$-algebra structure on $C^*(\\Delta_1;V)$ is determined by the products $\\cdot_i:V_0\\otimes V_{01}\\to V_{01}$, $\\cdot_i:V_{01}\\otimes V_0\\to V_{01}$, $\\bullet_j:V_0\\otimes V_{01}\\to V_{01}$, $[-,-]_k:V_0\\otimes V_{01}\\to V_{01}$, that is, by the datum of $2p+q+r$ non-symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $V$: then the relations of $\\Oh$-algebra on $C^*(\\Delta_1;V)$ are equivalent to certain relations on the products $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, and this defines a functor $\\op{Ass}_\\bullet(-):\\mathbf{Op}\\to\\mathbf{Op}$. We remark that the computation of this functor is completely mechanical. Finally, this is exactly the same as the functor $\\sA ss\\bullet-$: this will be proved at the very end of the paper, by showing that the functors $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$ form an adjoint pair (notice how the counit is obvious: given an $\\Oh$-algebra structure on $V$, there is a local dg $\\op{Ass}_\\circ(\\Oh)$-algebra structure on $C^*(\\Delta_1;V)$ via the tensor product operations, and by defninition this is the same as an $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$-algebra structure on $V$).\n\nThe trick to compute $\\sC om\\bullet-$ and $pre\\sL ie\\bullet-$ (and dually $\\sL ie\\circ-$ and $\\sP erm\\circ-$) is similar: we replace the dg associative algebra $(C^*(\\Delta_1;\\mathbb{K}),\\cup)$ in the above discussion with the dg Lie algebra $(C^*(\\Delta_1;\\mathbb{K}),[-,-]=\\cup-\\cup^{op})$ in the first case, and with the dg (right) permutative algebra $(C^*(\\Delta_1,v_l;\\mathbb{K}),\\cup)$ (where $v_l\\subset\\Delta_1$ is the left vertex) in the second case.\n\n\\begin{ack} The author is grateful to Domenico Fiorenza for some useful discussions.\n\\end{ack}\n\n\\bigskip\n\n\\subsection*{Preliminary remarks.} The author is not an actual expert on operads (and is afraid this might be painfully evident throughout the reading to anyone who is), accordingly, we will think of operads n\\\"aively, as defined by the corresponding type of algebras. Moreover, although the constructions seem to make sense in more general settings (for instance, dg operads), we limit ourselves to work in the category  $\\mathbf{Op}$ of finitely generated binary quadratic symmetric operads on vector spaces over a field $\\mathbb{K}$.\n\nGiven a vector space $V$, we denote by $V^{\\ten n}$, $V^{\\odot n}$, $V^{\\wedge n}$, $n\\geq0$, the tensor powers, symmetric powers and exterior powers of $V$ respectively. We shall always denote by $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ an operad in $\\mathbf{Op}$ generated by non-symmetric products $\\cdot_i$, $i=1,\\ldots,p$, commutative products~$\\bullet_j$, $j=1,\\ldots,q$, and anti-commutative brackets $[-,-]_k$, $k=1,\\ldots,r$: the relations shall be omitted from the notation. For the definitions of the Koszul duality functor $-^!:\\mathbf{Op}\\to\\mathbf{Op}$ and the Manin white and black products $-\\circ-,-\\bullet-:\\mathbf{Op}\\times\\mathbf{Op}\\to\\mathbf{Op}$ we refer to \\cite{V,LV}. For the definitions of the various operads we will consider, where not already specified, we refer to \\cite{LV,zinb}.\n\nWe denote by $\\Delta_1$ the standard $1$-simplex, which shall be represented as an arrow $\\to$, by $v_i\\subset\\Delta_1$, $i=l,r$, the left and right vertex respectively, and by $\\de\\Delta_1=v_l\\sqcup v_r\\subset\\Delta_1$ the boundary. Given a vector space $V$, we shall denote by $C^\\ast(\\Delta_1;V)$ (resp.: $C^*(\\Delta_1\\de\\Delta_1;V)$, $C^*(\\Delta_1,v_i;V)$, $i=l,r$) the usual complex of non-degenerate (resp.: relative) cochains with coefficients in $V$. We shall depict a $0$-cochain in $C^*(\\Delta_1;V)$ as $_x\\to_y$, $x,y\\in V$ (moreover, we write $_x\\to$ for $_x\\to_0$ and $\\to_y$ for $_0\\to_y$), and a $1$-cochain as $\\xrightarrow{x}$, similarly in the relative cases. The differential $d:C^0(\\Delta_1;V)\\to C^1(\\Delta_1;V)$ is the usual one $d(_x\\to_y)=\\xrightarrow{y-x}$.\n\\section{A trick to compute Manin black products}\n\nIn this section we shall introduce, and compute in several cases, endofunctors $\\mathbf{Op}\\to\\mathbf{Op}$ which we denote by  $\\op{Ass}_\\bullet(-)$, $\\op{Com}_\\bullet(-)$,  $\\op{preLie}_\\bullet(-)$, later we shall prove that these coincide with the functors  $\\sA ss\\bullet-$, $\\sC om\\bullet-$, $pre\\sL ie\\bullet-$ respectively.\n\n\n\n\\begin{definition}\\label{def:ManinBlack} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ in $\\mathbf{Op}$ and a vector space $V$, a structure of $\\op{Ass}_\\bullet(\\Oh)$-algebra on $V$ is the datum of operations \n\\begin{multline*} \\cdot'_i:C^\\ast(\\Delta_1;V)^{\\ten 2}\\to C^*(\\Delta_1;V),\\quad\\bullet'_j:C^\\ast(\\Delta_1;V)^{\\odot2}\\to C^*(\\Delta_1;V)\\\\\\mbox{and}\\quad[-,-]'_k:C^\\ast(\\Delta_1;V)^{\\wedge2}\\to C^*(\\Delta_1;V)\\quad\\mbox{such that}\\end{multline*}\n\\begin{enumerate}\n\\item\\label{item:dg} $(C^*(\\Delta_1;V),d,\\cdot'_i,\\bullet'_j,[-,-]'_k)$ is a dg $\\Oh$-algebra structure on $C^*(\\Delta_1;V)$, and moreover the following locality assumption holds:\\\\\n\n\\item\\label{item:locality} for all closed subsets $X\\subset\\Delta_1$ (i.e., $X=v_l,v_r,\\de\\Delta_1$, cf. the preliminary remarks) the complex of relative cochains $C^*(\\Delta_1,X;V)\\subset C^*(\\Delta_1;V)$ is a dg $\\Oh$-ideal.\n\\end{enumerate}\n\\end{definition}\nThe functor $\\op{preLie}_\\bullet(-)$ is defined similarly. \n\\begin{definition} A $\\op{preLie}_\\bullet(\\Oh)$-algebra structure on $V$ is the datum of a dg $\\Oh$-algebra structure $(C^*(\\Delta_1,v_l;V),d,\\cdot'_i,\\bullet'_j,[-,-]'_k)$ on $C^*(\\Delta_1,v_l;V)$ (we notice that in this case the locality assumption, that is, the fact that $C^*(\\Delta_1,\\de\\Delta_1;V)\\subset C^*(\\Delta_1,v_l;V)$ is a dg $\\Oh$-ideal, is satisfied for trivial degree reasons).\n\\end{definition}  \n\\begin{remark}\\label{rem:prelie}\n\tThis actually defines $\\op{preLie}_\\bullet(-)=\\op{preLie}_{r,\\bullet}(-)$, by replacing the vertex $v_l$ with the one $v_r$ in the previous definition we get a second endofunctor $\\op{preLie}_{l,\\bullet}(-)$ : more on this in Remark~\\ref{rem:leftcase}.\n\\end{remark} \n\\newcommand{\\w}{\\widetilde}\n\nIt is not immediately obvious that this defines endofunctors $\\mathbf{Op}\\to\\mathbf{Op}$: we consider first the case of $\\op{Ass}_\\bullet(-)$. Given the datum $(C^\\ast(\\Delta_1;V),d,\\cdot_i',\\bullet_j',[-,-]_k')$ of a local dg $\\Oh$-algebra structure on $(C^\\ast(\\Delta_1;V),d)$, we notice that the locality assumption implies $_x\\to\\cdot_i'\\to_y\\:=\\:\\to_x\\cdot_i'\\,\\,_y\\!\\to=0$, thus, applying the differential $d$ and Leibniz rule we see that $_x\\to\\cdot_i'\\xrightarrow{y}\\:=\\:\\xrightarrow{x}\\cdot_i'\\to_y$, and similarly $\\to_x\\cdot_i'\\xrightarrow{y}\\:=\\:\\xrightarrow{x}\\cdot_i'\\,\\,_y\\to$. We define non-symmetric products $\\prec_i,\\succ_i:V^{\\ten 2}\\to V$ on $V$ by \n\\[ _x\\to\\cdot_i'\\xrightarrow{y}\\,\\,=:\\,\\,\\xrightarrow{x\\prec_iy}\\,\\,:=\\,\\,\\xrightarrow{x}\\cdot_i'\\to_y,\\qquad\\to_x\\cdot_i'\\xrightarrow{y}\\,\\,=:\\,\\,-\\xrightarrow{y\\succ_i x}\\,\\,:=\\,\\,\\xrightarrow{x}\\cdot_i'\\,\\,_y\\to.\\]\nAlways by locality and Leibniz rule we have a product $\\cdot_i:V^{\\ten2}\\to V$ defined equivalently by $\\to_x\\cdot_i'\\to_y=\\to_{x\\cdot_iy}$ or $_x\\to\\cdot_i'\\,_y\\to=_{x\\cdot_i y}\\to$, and moreover $x\\cdot_i y= x\\prec_i y - y\\succ_i x$. In fact, \n\\[\\xrightarrow{x\\cdot_iy}=d\\left(\\to_x\\cdot_i'\\to_y\\right)=\\xrightarrow{x}\\cdot_i'\\to_y+\\to_x\\cdot_i'\\xrightarrow{y}=\\xrightarrow{x\\prec_iy-y\\succ_ix}=-d\\left(_x\\to\\cdot_i'\\,_y\\to\\right).\\]\nIn the same way, there are non-symmetric products $\\circ_j,\\ast_k:V^{\\ten2}\\to V$ on $V$, defined by the formulas \n\\begin{multline*} _x\\to\\bullet_j'\\xrightarrow{y}\\,\\,=\\,\\,\\xrightarrow{x}\\bullet_j'\\to_y\\,\\,=\\,\\,\\xrightarrow{y}\\bullet_j'\\,\\,_x\\to\\,\\,=\\,\\,\\to_y\\bullet_j'\\xrightarrow{x}\\,\\,=:\\,\\,\\xrightarrow{x\\circ_j y},\\\\ \\left[ _x\\to,\\xrightarrow{y}\\right]_k'\\,\\,=\\,\\, \\left[ \\xrightarrow{x},\\to_y\\right]_k' \\,\\,=\\,\\, -\\left[\\xrightarrow{y},\\,_x\\to\\right]_k' \\,\\,=\\,\\, -\\left[ \\to_y,\\xrightarrow{x}\\right]_k'\\,\\,=:\\,\\, \\xrightarrow{x\\ast_k y},\\end{multline*}\nproducts $\\bullet_j:V^{\\odot 2}\\to V$, $[-,-]_k:V^{\\wedge2}\\to V$ defined by \n\\[\\to_x\\bullet_j'\\to_y=\\to_{x\\bullet_j y},\\qquad_x\\to\\bullet_j'\\,_y\\to=_{x\\bullet_jy}\\to,\\qquad \\left[ \\to_x,\\to_y\\right]_k'=\\to_{[x,y]_k},\\qquad\\left[ _x\\to,\\,_y\\to \\right]_k'=_{[x,y]_k}\\to, \\]\nand moreover by Leibniz rule\n\\[ x\\bullet_j y = x\\circ_j y + y \\circ_j x, \\qquad [x,y]_k=x\\ast_k y - y\\ast_k x.\\]\n\n\\emph{Warning:} the previous formulas will be used extensively troughout the computations of this section without further mention.\n \nIn other words, the datum of an $\\op{Ass}_\\bullet(\\Oh)$-algebra structure on $V$ is the same as the one of operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k:V^{\\ten 2}\\to V$ on $V$, inducing operations $\\cdot_i',\\bullet_j',[-,-]_k'$ on $C^\\ast(\\Delta_1;V)$ via the previous formulas: the requirement that these make the latter into a (by construction, local dg) $\\Oh$-algebra translates into a finite set of terniary relations on the operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$. More precisely, for every relation $R(x,y,z)=0$ satisfied in an $\\Oh$-algebra, we get six (in general not independent) relations in the operad $\\op{Ass}_\\bullet(\\Oh)$ as in the next lemma.\n\n\\begin{lemma}\\label{lem:rel} Given operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $V$ inducing operations $\\cdot_i',\\bullet_j',[-,-]_k'$ on $C^\\ast(\\Delta_1;V)$ as above, then the latter is a local dg $\\Oh$-algebra if and only if for every relation $R(x,y,z)=0$ in the operad $\\Oh$ the six relations  \n\\begin{multline*}0=R(\\xrightarrow{x},\\to_y,\\,_z\\to)=R(\\xrightarrow{x},\\,_y\\to,\\to_z)=R(\\to_x,\\xrightarrow{y},\\,_z\\to)=\\\\=R(_x\\to,\\xrightarrow{y},\\to_z)=R(\\to_x,\\,_y\\to,\\xrightarrow{z})=R(_x\\to,\\to_y,\\xrightarrow{z})=0,\\end{multline*}\nare satisfied.\n\\end{lemma}\n\\begin{proof} The only if part is clear. For the if part we have to show that the given relations imply all the others. First of all, we notice that the only relations not trivially satisfied are those in total degree 0 or 1. Moreover, in total degree 0 all necessary relations follow by the locality assumption but the ones $R(_x\\to,\\,_y\\to,\\,_z\\to)=0=R(\\to_x,\\to_y,\\to_z)$. In total degree 1, all the necessary relations are satisfied by hypothesis but the ones\n\\begin{multline*}0=R(_x\\to,\\,_y\\to,\\xrightarrow{z})=R(\\to_x,\\to_y,\\xrightarrow{z})=R(_x\\to,\\xrightarrow{y},\\,_z\\to)=\\\\=R(\\to_x,\\xrightarrow{y},\\to_z)=R(\\xrightarrow{x},\\,_y\\to,\\,_z\\to)=R(\\xrightarrow{x},\\to_y,\\to_z)=0.\\end{multline*}\nFor instance, to prove the first one we apply $d$ to the relation $0=R(_x\\to,\\,_y\\to,\\to_z)$ (which follows by locality), and by Leibniz rule and the hypothesis of the lemma\n\\[0 = -R(\\xrightarrow{x},\\,_y\\to,\\to_z)-R(_x\\to,\\xrightarrow{y},\\to_z)+R(_x\\to,\\,_y\\to,\\xrightarrow{z})=R(_x\\to,\\,_y\\to,\\xrightarrow{z}). \\]\nThe others are proved similarly. Finally, if we denote by $R(x,y,z)$ the relation $R$ computed in the operations $\\cdot_i$, $\\bullet_j$, $[-,-]_k$ on $V$, we have $R(\\to_x,\\to_y,\\to_z)=\\to_{R(x,y,z)}$, and by Leibniz rule\n\\[\\xrightarrow{R(x,y,z)}=d\\left(\\to_{R(x,y,z)}\\right)=R(\\xrightarrow{x},\\to_y,\\to_z)+R(\\to_x,\\xrightarrow{y},\\to_z)+R(\\to_x,\\to_y,\\xrightarrow{z})=0,\\]\nhence $R(\\to_x,\\to_y,\\to_z)=0$, and $R(_x\\to,\\,_y\\to,\\,_z\\to)=0$ is proved similarly.\n\\end{proof}\n\\begin{example}\\label{ex:ass} We consider the operad $\\sL ie$ of Lie algebras. By the previous argument,  an $\\op{Ass}_\\bullet(\\sL ie)$-algebra structure on $V$ is the datum of $\\ast:V^{\\ten 2}\\to V$ such that\n\\begin{multline*} \\left[\\left[_x\\to,\\to_y\\right],\\xrightarrow{z}\\right]=0=\\left[ _x\\to,\\left[\\to_y,\\xrightarrow{z}\\right]\\right]+\\left[\\left[_x\\to,\\xrightarrow{z}\\right],\\to_y\\right]=\\\\=\\left[_x\\to,\\xrightarrow{-z\\ast y}\\right]+\\left[\\xrightarrow{x\\ast z},\\to_y\\right]=\\xrightarrow{-x\\ast(z\\ast y)+(x\\ast z)\\ast y},\\qquad\\forall x,y,z\\in V.\n\\end{multline*}\nIn other words, $\\ast$ is an associative product on $V$. The other five relations to be checked in the previous lemma follow from this one by symmetry of the Jacobi identity, thus, we see that $\\op{Ass}_\\bullet(\\sL ie)=\\sA ss$, the operad of associative algebras.  \\end{example}\n\nWe may repeat the previous considerations in the case of $\\op{preLie}_\\bullet(-)$, showing that a dg $\\Oh$-algebra structure $(C^\\ast(\\Delta_1,v_l;V),d,\\cdot_i'.\\bullet_j',[-,-]_k')$ on the complex $C^\\ast(\\Delta_1,v_l;V)$ is determined, via the previous formulas, by operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k:V^{\\ten 2}\\to V$ on $V$. The analog of Lemma \\ref{lem:rel} is the following\n\\begin{lemma}\\label{lem:relpl}\u25cb Given operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $V$ inducing operations $\\cdot_i',\\bullet_j',[-,-]_k'$ on $C^\\ast(\\Delta_1,v_l;V)$ as above, then the latter is a local dg $\\Oh$-algebra if and only if for every relation $R(x,y,z)=0$ in the operad $\\Oh$ the three relations  \n\\begin{equation*}R(\\xrightarrow{x},\\to_y,\\to_z)=R(\\to_x,\\xrightarrow{y},\\to_z)=R(\\to_x,\\to_y,\\xrightarrow{z})=0,\\end{equation*}\nare satisfied. \n\\end{lemma}\n\\begin{proof} It follows from the proof of Lemma \\ref{lem:rel}.\\end{proof}\n\\begin{example}\\label{ex:prelie} A $\\op{preLie}_\\bullet(\\sL ie)$-algebra structure on $V$ is the datum of $\\ast:V^{\\ten 2}\\to V$ such that, where as usual $[x,y]=x\\ast y - y \\ast x$,\n\\begin{multline*} \\left[\\xrightarrow{x},\\left[\\to_y,\\to_z\\right]\\right]=\\left[\\xrightarrow{x},\\to_{[y,z]} \\right]=\\xrightarrow{x\\ast[y,z]}=\\\\=\\left[ \\left[\\xrightarrow{x},\\to_y\\right],\\to_z\\right]+\\left[\\to_y,\\left[\\xrightarrow{x},\\to_z\\right]\\right]=\\left[\\xrightarrow{x\\ast y},\\to_z\\right]+\\left[\\to_y,\\xrightarrow{x\\ast z}\\right]=\\xrightarrow{(x\\ast y)\\ast z- (x\\ast z)\\ast y},\\qquad\\forall x,y,z\\in V.\n\\end{multline*}\nIn other words, $\\ast$ is a right pre-Lie product on $V$. Since the remaining two relations to be checked in the previous lemma follow from this one by symmetry of the Jacobi identity, we see that $\\op{preLie}_\\bullet(\\sL ie)=pre\\sL ie$, the operad of (right) pre-Lie algebras. \\end{example}\n\\begin{remark}\\label{rem:leftcase} As was said in Remark \\ref{rem:prelie} the above construction is actually the one of $\\op{preLie}_\\bullet(\\sL ie)=\\op{preLie}_{r,\\bullet}(\\sL ie)$: applying the functor $\\op{preLie}_{l,\\bullet}(-)$ introduced there we find by similar computations as in the above example that $\\op{preLie}_{l,\\bullet}(\\sL ie)$ is the operad of \\emph{left} pre-Lie algebras. More in general, it is easy to see that $\\op{preLie}_{l,\\bullet}(\\Oh)$ is always the opposite of the operad $\\op{preLie}_{r,\\bullet}(\\Oh)=\\op{preLie}_{\\bullet}(\\Oh)$ for any $\\Oh\\in\\mathbf{Op}$. We won't insist further on this point, and restrict our computations to the right case.\n\\end{remark}\n\nBy the proof of Lemma \\ref{lem:rel} we have morphisms of operads $\\Oh\\to \\op{Ass}_\\bullet(\\Oh)$, $\\Oh\\to \\op{preLie}_\\bullet(\\Oh)$, sending an $\\op{Ass}_\\bullet(\\Oh)$-algebra structure $(V,\\prec_i,\\succ_i,\\circ_j,\\ast_k)$ on $V$ to the associated $\\Oh$-algebra structure \\[(V\\:\\:,\\:\\:x\\cdot_iy=x\\prec_i y-y\\succ_ix\\:\\:,\\:\\:x\\bullet_j y=x\\circ_j y+y\\circ_j x\\:\\:,\\:\\:[x,y]_k=x\\ast_k y-y\\ast_k x),\\] \nand similarly in the other case.\n\n\\begin{definition} A $\\op{Com}_\\bullet(\\Oh)$-algebra is an $\\op{Ass}_\\bullet(\\Oh)$-algebra with trivial associated $\\Oh$-algebra. In other words, the operad $\\op{Com}_\\bullet(\\Oh)$ is generated by non-symmetric products $\\star_i$, anti-commutative brackets $\\{-,-\\}_j$ and commutative products $\\circledast_k$, together with the relations obtained from the ones of $\\op{Ass}_\\bullet(\\Oh)$-algebra by further imposing the identities \\[x\\prec_i y= y\\succ_i x=: x\\star_i y,\\qquad x\\circ_j y=-y\\circ_j x =:\\{x,y\\}_j,\\qquad x\\ast_k y= y\\ast_k x=: x\\circledast_k y.\\]\n\\end{definition}\n\n\\begin{example} We see immediately by Example \\ref{ex:ass} that $\\op{Com}_\\bullet(\\sL ie)=\\sC om$, the operad of commutative and associative algebras. \n\\end{example}\n\n\\begin{theorem}\\label{th:black} For any operad $\\Oh$ in $\\mathbf{Op}$, we have natural isomorphisms\n$\\op{Ass}_\\bullet(\\Oh)=\\sA ss\\bullet\\Oh$, $\\op{Com}_\\bullet(\\Oh)=\\sC om\\bullet\\Oh$, $\\op{preLie}_\\bullet(\\Oh)=pre\\sL ie\\bullet\\Oh$.\n\\end{theorem}\n\nThe proof is postponed to the next section (it follows from theorems \\ref{th:asswhite}, \\ref{th:permwhite}, \\ref{th:liewhite} and \\ref{th:adj}). In the remaining of this section we shall illustrate the result by several explicit computations. \n\n\\begin{remark}\\label{rem:curlyeqprec}\n\tIn some of the following computations it is conventient to choose a different basis for the operations of $\\op{Ass}_\\bullet(\\Oh)$ and $\\op{preLie}_\\bullet(\\Oh)$, by replacing the products $\\succ_i$ with the opposite products $\\curlyeqsucc_i:=-\\succ^{op}_i$: with the previous notations $\\to_x\\cdot'_i\\xrightarrow{y}=:\\xrightarrow{x\\curlyeqsucc_i y}:=\\xrightarrow{x}\\cdot'_i\\,_y\\to$.\n\\end{remark}\n\n\\begin{example} Given non-negative integers $p,q,r$, we denote by $\\sM ag_{p,q,r}$ the operad generated by $p$ magmatic non-symmetric operations, $q$ magmatic commutative operations and $r$ magmatic anti-commutative operations with no relations among them. The previous arguments show that $\\op{Ass}_\\bullet(\\sM ag_{p,q,r} )=\\op{preLie}_\\bullet (\\sM ag_{p,q,r})=\\sM ag_{2p+q+r,0,0}$, whereas $\\op{Com}_\\bullet(\\sM ag_{p,q,r})=\\sM ag_{p,r,q}$.\n\\end{example}\n\nWe introduce a further notation.\n\n\\begin{definition}\\label{def:+x} Given operads $\\mathcal{O},\\mathcal{P}\\in\\mathbf{Op}$, we denote by $\\mathcal{O}+\\mathcal{P}$ the operad defined by saying that an $(\\mathcal{O}+\\mathcal{P})$-algebra structure on $V$ is the datum of both an $\\Oh$-algebra structure and a $\\mathcal{P}$-algebra structure on $V$ with no relations among them, while we denote by $\\Oh\\times\\mathcal{P}$ the operad defined in the same way and by further imposing that every triple product involving an operation from $\\Oh$ and one from $\\mathcal{P}$ vanishes. It is easy to see that the functors $-+-,-\\times-:\\mathbf{Op}\\to\\mathbf{Op}$ are Koszul dual in the sense that $(\\Oh+\\sP)^!=\\Oh^!\\times\\sP^!$ for any pair of operads $\\Oh,\\sP\\in\\mathbf{Op}$. \\end{definition}\n\n\\begin{example} We consider the operad $(\\sA ss,\\cdot)$ of associative algebras. Given an $\\op{Ass}_\\bullet(\\sA ss)$-algebra structure $(V,\\prec,\\succ)$ on a vector space, we shall denote the associator of the associated product $\\cdot'$ on $C^\\ast(\\Delta_1;V)$ by $A_{\\cdot'}(-,-,-)$. Straightforward computations show that\n\\[ 0=A_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})\\:=\\: \\left(_x\\to\\cdot'\\to_y\\right)\\cdot'\\xrightarrow{z}-\\,_x\\to\\cdot'(\\to_y\\cdot'\\xrightarrow{z}) \\:=\\: 0+\\,_x\\to\\cdot'\\xrightarrow{z\\succ y}\\:=\\:\\xrightarrow{x\\prec(z\\succ y)},\\]\n\\begin{eqnarray} \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,{}_y\\xrightarrow{},\\xrightarrow{z} \\right) &=& \\xrightarrow{(y\\prec z)\\succ x}, \\\\\n\\nonumber 0=A_{\\cdot'}\\left({}_x\\xrightarrow{},\\xrightarrow{y},\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-x\\prec(y\\prec z)}, \\\\\n\\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},{}_z\\xrightarrow{} \\right) &=& \\xrightarrow{z\\succ(y\\succ x)-(z\\succ y)\\succ x}, \\\\\n\\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{x},{}_y\\xrightarrow{},\\xrightarrow{}_z \\right) &=& \\xrightarrow{-(y\\succ x)\\prec z},\\\\ \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{x},\\xrightarrow{}_y,{}_z\\xrightarrow{} \\right) &=& \\xrightarrow{-z\\succ(x\\prec y)}. \\end{eqnarray}\nConversely, according to Lemma \\ref{lem:rel}, the above six relations on $\\prec,\\succ$ imply the vanishing of $A_{\\cdot'}(-,-,-)$ on $C^*(\\Delta_1;V)$. Hence, we see that an $\\op{Ass}_\\bullet( \\sA ss)$-algebra structure on $V$ is the datum of two non-symmetric products $\\prec,\\succ:V^{\\ten 2}\\to V$ which are associative and such that moreover\n\\[x\\prec(y\\succ z)=(x\\prec y)\\succ z =x\\succ(y\\prec z)=(x\\succ y)\\prec z =0,\\qquad\\forall x,y,z\\in V.\\]\nWith the notations from the previous definition we found that that $\\op{Ass}_\\bullet( \\sA ss)=\\sA ss\\times\\sA ss$. By further imposing $x\\prec y-y\\succ x =: x\\cdot y= 0$, that is, $x \\prec y = y\\succ x=:x\\star y$, we see that the operad $\\op{Com}_\\bullet(\\sA ss)$ is generated by a non symmetric product $\\star$ such that $(x\\star y)\\star z=x\\star (y\\star z)=0$: in other words $\\op{Com}_\\bullet(\\sA ss)= nil\\sA ss$, the operad of two step nilpotent associative algebras, in accord with Theorem \\ref{th:black} and the computations in \\cite{GK}. To compute the relations of $\\op{preLie}_\\bullet(\\sA ss)$ it is convenient to replace the generating set of operations $\\prec,\\succ$ with the one $\\prec,\\curlyeqsucc=-\\succ^{op}$ as explained in Remark \\ref{rem:curlyeqprec}, then we get the following relations according to Lemma \\ref{lem:relpl} (notice that $x\\cdot y:=x\\prec y-y\\succ x=x\\prec y + x\\curlyeqsucc y$)\n\\begin{eqnarray} \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{x},\\xrightarrow{}_y,\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-x\\prec(y\\cdot z)}, \\\\ \n\\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},\\xrightarrow{}_z \\right) & = & \\xrightarrow{(x\\curlyeqsucc y)\\prec z-x\\curlyeqsucc(y\\prec z)}, \\\\ \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{}_y,\\xrightarrow{z} \\right) &=& \\xrightarrow{(x\\cdot y)\\curlyeqsucc z - x\\curlyeqsucc(y\\curlyeqsucc z)}, \\end{eqnarray}\nwhich are exactly the dendirform relations for the products $\\prec,\\curlyeqsucc$, thus $\\op{preLie}_\\bullet(\\sA ss)=\\sD end$, the operad of dendriform algebras, which as well known is also $\\sD end=pre\\sL ie\\bullet\\sA ss$.\n\\end{example}\n\\begin{example} Next we consider the operad $(\\sC om,\\bullet)$ of commutative and associative algebras. Given an $\\op{Ass}_\\bullet(\\sC om)$ algebra structure $(V,\\circ)$ on a vector space, together with the asociated commutative product $\\bullet'$ on $C^*(\\Delta_1;V)$, similar computations as in the previous example show that the vanishing of the associator $A_{\\bullet'}(-,-,-)$ is equivalent to the identities $(x\\circ y)\\circ z = x\\circ(y\\circ z)=0$, thus recovering once again $\\op{Com}_\\bullet(\\sA ss)= \\sC om\\bullet \\sA ss = nil\\sA ss$. If we further impose that the commutative product $x\\bullet y = x\\circ y + y\\circ x$ on $V$ vanishes, we see that $\\op{Com}_\\bullet(\\sC om)$ is the operad generated by an anti-commutative bracket $\\{x,y\\}:=x\\circ y=-y\\circ x$ such that $\\{ \\{ x,y \\}, z \\}=0$, or in other words $\\op{Com}_\\bullet(\\sC om)=nil\\sL ie$, the operad of two step nilpotent Lie algebras. Finally, a $\\op{preLie}_\\bullet(\\sC om)$-algebra structure on $V$ is the datum of a non-symmetric product $\\circ:V^{\\ten 2}\\to V$ such that, where as usual we denote by $x\\bullet y=x\\circ y + y\\circ x$, \\[ 0=A_{\\bullet'}(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{x\\circ(y\\bullet z)-(x\\circ y) \\circ z},\\qquad\\forall x,y,z\\in V, \\] \nand a second relation coming from Lemma \\ref{lem:relpl}, namely, $(x\\circ y)\\circ z= (x\\circ z)\\circ y$, already follows from this one. We find that $\\op{preLie}_\\bullet(\\sC om)=\\sZ inb$, the operad of (right) Zinbiel algebra, according to the well known fact $pre\\sL ie\\bullet \\sC om=\\sZ inb$ \\cite{V}.\n\\end{example}\n\n\\begin{example} We consider the operad $pre\\sL ie$ of (right) pre-Lie algebras. Given an $\\op{Ass}_\\bullet(pre\\sL ie)$-algebra structure $(V,\\prec,\\succ)$ on $V$, together with the associated dg pre-Lie algebra $(C^\\ast(\\Delta_1;V),d,\\cdot')$, the associator $A_{\\cdot'}(-,-,-)$ on $C^\\ast(\\Delta_1;V)$ is computed as in Example \\ref{ex:ass}. Writing the generating relation in $pre\\sL ie$ as $0=R(\\alpha,\\beta,\\gamma)=A_{\\cdot'}(\\alpha,\\beta,\\gamma)-A_{\\cdot'}(\\alpha,\\gamma,\\beta)$, $\\forall \\alpha,\\beta,\\gamma\\in C^*(\\Delta_1;V)$, we find that according to Lemma \\ref{lem:rel} this is equivalent to the following relations on $\\prec,\\succ$ (the remaining three follow from these ones by symmetry of $R$) \n\t\\[ 0 = R(_x\\to,\\to_y,\\xrightarrow{z})=\\xrightarrow{x\\prec(z\\succ y)-(x\\prec z)\\prec y+x\\prec(z\\prec y)}\\]\n\t\\[ 0 = R(\\to_x,\\,_y\\to,\\xrightarrow{z})=\\xrightarrow{(y \\prec z)\\succ x -y\\succ(z\\succ x)+(y\\succ z)\\succ x}\\]\n\t\\[ 0 = R(\\xrightarrow{x},\\,_y\\to,\\to_z)=\\xrightarrow{-(y\\succ x)\\prec z+ y\\succ(x\\prec z)}\\]\n\t\t\nThese are precisely the relations of dendriform algebra, thus once again $\\op{Ass}_\\bullet(pre\\sL ie)=\\sD end$. If we impose that the right pre-Lie product $x \\cdot y = x\\prec y - y\\succ x$ on $V$ vanishes, we get as well known \\cite{LV} the Zinbiel relation for $x\\star y:=x\\prec y=y\\succ x$, that is, once again $\\op{Com}_\\bullet(pre\\sL ie)=\\sZ inb$. Finally, in a $\\op{preLie}_\\bullet(pre\\sL ie)$-algebra we get the relations (the remaining one follows from the first one and simmetry of $R$)\n\\[0 = R(\\to_x,\\to_y,\\xrightarrow{z})=\\xrightarrow{-z\\succ (x\\cdot y)-(z\\succ y)\\succ x+(z\\succ x)\\prec y-(z\\prec y)\\succ x}\\]\n\\[0 = R(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{(x\\prec y)\\prec z -(x\\prec z)\\prec y- x\\prec(y\\cdot z -z\\cdot y)}\\]\nThe reader will recognize the relations of $L$-dendriform algebra \\cite{bai2,V}. We recall the proof of the following fact, giving the two forgetful functors from the category of $(pre\\sL ie\\bullet pre\\sL ie)$-algebras to the one of $pre\\sL ie$-algebras. \n\\begin{proposition} Let $V$ be a vector space, together with non symmetric products $\\prec,\\succ:V^{\\ten2}\\to V$ such that the following relations, where we put $x\\cdot y:= x\\prec y-y\\succ x$, $x\\triangleleft y := x\\prec y + x\\succ y$, $[x,y]:=x\\cdot y-y\\cdot x=x\\triangleleft y-y\\triangleleft x$, are verified\n\t\\[ x\\prec [y,z]=(x\\prec y)\\prec z - (x\\prec z)\\prec y,\\qquad (x\\succ y)\\prec z=x\\succ(y\\cdot z)+(x\\triangleleft z)\\succ y,\\qquad\\forall x,y,z\\in V. \\]\nThen the products $\\triangleleft,\\cdot$ are right pre-Lie products on $V$ and $[-,-]$ is a Lie bracket.\n\\end{proposition}\n\t\n\\begin{proof} The relations in the claim of the lemma imply the pre-Lie relation for $\\cdot$ according to the previous computations and the proof of Lemma \\ref{lem:rel}. The fact that $[-,-]$ is a Lie bracket follows. Finally, we may write the second relation as $A_{\\succ}(x,z,y)=(x\\succ y)\\prec z - (x\\prec z)\\succ y -x\\succ(y\\prec z)$: substituting in \n\t\t\\[ A_{\\triangleleft}(x,y,z)=A_{\\prec}(x,y,z)+A_{\\succ}(x,y,z)+(x\\prec y)\\succ z-x\\prec(y\\succ z)+(x\\succ y)\\prec z-x\\succ(y\\prec z)\\]\n\t\twe find that \n\t\t\\begin{multline*}\n\t\tA_{\\triangleleft}(x,y,z)= A_\\prec(x,y,z)-x\\prec(y\\succ z)+\\\\+(x\\succ z)\\prec y-(x\\prec y)\\succ z-x\\succ(z\\prec y)+(x\\prec y)\\succ z+(x\\succ y)\\prec z-x\\succ(y\\prec z).\n\t\t\\end{multline*}\n\t\tThe bottom row is clearly symmetric in $y$ and $z$, while the same statement for the right hand side of the top row is just another way of writing the first relation in the claim of the lemma.\\end{proof}\n\\end{example}\n\n\\begin{example}\\label{ex:leib} An interesting example is the operad $\\sL eib$ of (right) Leibniz algebras: this is the operad generated by a single non symmetric product $\\cdot$ and the relation $0=R_\\cdot(x,y,z):=(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y$. From Lemma \\ref{lem:rel} we get the following five indipendent relations on $\\prec,\\succ$\n\t\\begin{multline*} 0=R_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})\\:=\\: \\left(_x\\to\\cdot'\\to_y\\right)\\cdot'\\xrightarrow{z}-\\,_x\\to\\cdot'(\\to_y\\cdot'\\xrightarrow{z})-(_x\\to\\cdot'\\xrightarrow{z})\\cdot'\\to_y \\:= \\\\ 0+\\,_x\\to\\cdot'\\xrightarrow{z\\succ y}-\\xrightarrow{x\\prec z}\\cdot'\\to_y\\:=\\:\\xrightarrow{x\\prec(z\\succ y)-(x\\prec z)\\prec y},\\end{multline*}\n\t\n\t\\begin{eqnarray} \\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,{}_y\\xrightarrow{},\\xrightarrow{z} \\right) &=& \\xrightarrow{(y\\prec z)\\succ x-y\\succ(z\\succ x)}, \\\\\n\t\\nonumber 0=R_{\\cdot'}\\left({}_x\\xrightarrow{},\\xrightarrow{y},\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-x\\prec(y\\prec z)}, \\\\\n\t\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},{}_z\\xrightarrow{} \\right) &=& \\xrightarrow{z\\succ(y\\succ x)-(z\\succ y)\\succ x}, \\\\\n\t\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{x},{}_y\\xrightarrow{},\\xrightarrow{}_z \\right) &=& \\xrightarrow{-(y\\succ x)\\prec z+y\\succ(x\\prec z)},\n\t\\end{eqnarray}\n\tand the remaining one is equivalent to the last one.\nThese are the precisely the relations of diassociative algebra on $(V,\\prec,\\succ)$, in other words we found that $\\op{Ass}_\\bullet(\\sL eib)=di\\sA ss$, the operad of diassociative algebras. If we further impose $0=x\\cdot y=x\\prec y -y\\succ x$ and we put as usual $x\\star y :=x\\prec y=y\\succ x$, the previous relations reduce to the two independent ones $(x\\star y)\\star z=x\\star(y\\star z)$, $(x\\star y)\\star z=x\\star(z\\star y)$: in other words, we found that $\\op{Com}_\\bullet(\\sL eib)=\\sP erm$, the operad of (right) permutative algebras. Finally,  we consider the operad $\\op{preLie}_\\bullet(\\sL eib)$: this is generated by non-symmetric products $\\prec,\\succ$ and the relations, where as usual $x\\cdot y= x\\prec y- y\\succ x$, \n\\begin{eqnarray} \\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{x},\\xrightarrow{}_y,\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-(x\\prec z)\\prec y-x\\prec(y\\cdot z)}, \\\\\n\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},\\xrightarrow{}_z \\right) &=& \\xrightarrow{-(y\\succ x)\\prec z+(y\\prec z)\\succ x+y\\succ(x\\cdot z)}, \\\\\n\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{}_y,\\xrightarrow{z} \\right) &=& \\xrightarrow{-z\\succ(x\\cdot y)-(z\\succ y)\\succ x+(z\\succ x)\\prec y}.\n\\end{eqnarray}\nThese are equivalent to the following relations for $\\prec,\\succ$\n\\begin{multline*} x\\prec(y\\cdot z)=(x\\prec y)\\prec z -(x\\prec z)\\prec y,\\qquad (x\\succ y)\\succ z=(x\\prec y)\\succ z, \\\\ x\\succ(y\\cdot z)=(x\\succ y)\\prec z-(x\\prec z)\\succ y. \\end{multline*}\n\n\\end{example}\n\n\\begin{example}\\label{ex:pois} We consider the operad $(\\sP ois,\\bullet, [-,-])$ of commutative Poisson algebras. By the previous computations, an $\\operatorname{Ass}_\\bullet(\\sP ois)$-algebra structure on $V$ is the datum $(V,\\circ,\\ast)$ of an $\\op{Ass}_\\bullet(\\sC om)=nil\\sA ss$-algebra structure $(V,\\circ)$ and an $\\op{Ass}_\\bullet(\\sL ie)=\\sA ss$-algebra structure $(V,\\ast)$ satisfying the additional relations induced, as in Lemma \\ref{lem:rel}, by the Poisson identity $0=[x,y\\bullet z]-[x,y]\\bullet z - y\\bullet[x,z]$ in the operad $\\sP ois$. These additional relations are easily computed, for instance\n\t\\begin{multline*}  0=\\left[\\xrightarrow{x},\\,_y\\to\\bullet'\\to_z\\right]' - \\left[\\xrightarrow{x},\\,_y\\to\\right]'\\bullet'\\to_z-\\,_y\\to\\bullet'\\left[\\xrightarrow{x},\\to_z\\right]' = \\\\=\\,0\\,-\\xrightarrow{-y\\ast x}\\cdot'\\to_z -\\,_y\\to\\cdot'\\xrightarrow{x\\ast z}\\:=\\:\\xrightarrow{(y\\ast x)\\circ z - y\\circ (x\\ast z)}.\n\t\\end{multline*} \n\tProceeding in this way we find the following relations for $\\circ,\\ast$\n\t\\[ (x\\ast y)\\circ z= x\\ast(y\\circ z)=x\\circ(y\\ast z)= (x\\circ y)\\ast z.\\] \n\tSimilarly, a $\\operatorname{Com}_\\bullet(\\sP ois)$-algebra structure on $V$ is the datum $(V,\\{-,-\\},\\circledast)$ of a $\\operatorname{Com}_\\bullet(\\sC  om)=nil\\sL ie$-algebra structure $(V,\\{-,-\\})$ and a $\\operatorname{Com}_\\bullet(\\sL ie)=\\sC om$-algebra structure $(V,\\circledast)$ such that moreover the above identities hold. Consider the one $\\{x,y\\}\\circledast z=\\{x\\circledast y,z \\} $: as the left and right hand side are respectively symmetric and anti-symmetric in $x$ and $y$ both vanish, thus we have the relations\n\t\\[ \\{x\\circledast y,z \\}=x\\circledast\\{y,z\\}=0,\\qquad\\forall x,y,z\\in V.\\]\n\tIn other words, we found $\\operatorname{Com}_\\bullet(\\sP ois)=nil\\sL ie\\times\\sC om$. Finally, a $\\op{preLie}_\\bullet(\\sP ois)$-algebra structure on $V$ is the datum $(V,\\circ,\\ast)$ of a $\\op{preLie}_\\bullet(\\sC om)=\\sZ inb$-algebra structure $(V,\\circ)$ and a $\\op{preLie}_\\bullet(\\sL ie)=pre\\sL ie$-algebra structure $(V,\\ast)$, such that moreover, where as usual we denote by $x\\bullet y =x\\circ y+ y\\circ x$ and $[x,y]=x\\ast y-y\\ast x$ the associated $\\sC om$- and $\\sL ie$-algebra structures on $V$ respectively,\n\t\\begin{multline*}  0=\\left[\\xrightarrow{x},\\to_y\\bullet'\\to_z\\right]' - \\left[\\xrightarrow{x},\\,\\to_y\\right]'\\bullet'\\to_z-\\to_y\\bullet'\\left[\\xrightarrow{x},\\to_z\\right]' = \\\\=\\left[\\xrightarrow{x},\\to_{y\\bullet z}\\right]'-\\xrightarrow{x\\ast y}\\bullet'\\to_z -\\to_y\\bullet'\\xrightarrow{x\\ast z}=\\xrightarrow{x\\ast(y\\bullet z) -(x\\ast y)\\circ z - (x\\ast z)\\circ y},\n\t\\end{multline*} \n\t\\begin{multline*}  0=\\left[\\to_x,\\xrightarrow{y}\\bullet'\\to_z\\right]' - \\left[\\to_x,\\xrightarrow{y}\\right]'\\bullet'\\to_z-\\xrightarrow{y}\\bullet'\\large[\\to_x,\\to_z\\large]' = \\\\=\\left[\\to_x,\\xrightarrow{y\\circ z}\\right]'-\\xrightarrow{-y\\ast x}\\bullet'\\to_z -\\xrightarrow{y}\\bullet'\\to_{[x,z]}=\\xrightarrow{-(y\\circ z)\\ast x+(y\\ast x)\\circ z - y\\circ [x,z]}.\n\t\\end{multline*} \n\tWe conclude that $\\op{preLie}_\\bullet(\\sP ois)=pre\\sP ois$, Aguiar's operad of right pre-Poisson algebras \\cite{A}, in accord with the fact \\cite{U} that $pre\\sL ie\\bullet\\sP ois = pre\\sP ois$.\n\\end{example}\n\n\n\\begin{example}\\label{ex:perm} We consider the operad $\\sP erm$: recall that a $\\sP erm$-algebra structure, or (right) permutative algebra structure, on $V$ is the datum of an associative product $\\cdot$ which is commutative on the right hand side whenever there are three or more variables, or, in other words, satisfies the relations $(x\\cdot y)\\cdot z=x\\cdot(y\\cdot z)=x\\cdot(z\\cdot y)$. Thus, an $\\op{Ass}_\\bullet(\\sP erm)$-algebra structure on $V$ is the datum of an $\\op{Ass}_\\bullet(\\sA ss)=\\sA ss\\times\\sA ss$ algebra structure $(V,\\prec,\\prec)$, as in Example \\ref{ex:ass}, such that moreover\n\\[ 0 \\:=\\: _x\\to\\cdot'\\left( \\to_y\\cdot'\\xrightarrow{z}\\right)\\,-\\,_x\\to\\cdot'\\left( \\xrightarrow{z}\\cdot'\\to_y\\right) =\\xrightarrow{-x\\prec(z\\prec y)-x\\prec(z\\prec y)}=\\xrightarrow{-x\\prec(y\\prec z)}\\]\n\\[ 0 \\:=\\: \\to_x\\cdot'\\left( _y\\to\\cdot'\\xrightarrow{z}\\right)-\\to_x\\cdot'\\left( \\xrightarrow{z}\\cdot'\\,_y\\to \\right) =\\xrightarrow{-(y\\prec z)\\prec x-(y\\prec z)\\prec x}=\\xrightarrow{-(y\\prec z)\\prec x}.\\]\nIn other words, $\\op{Ass}_\\bullet(\\sP erm)=nil\\sA ss\\times nil\\sA ss$. Further imposing $0=x\\prec y-y\\prec x=x\\cdot y$, we find that $\\op{Com}_\\bullet(\\sP erm)=nil\\sA ss$. Finally, as in Remark \\ref{rem:curlyeqprec} and Example \\ref{ex:ass}, it is convenient in the compuation of $\\op{preLie}_\\bullet(\\sP erm)$ to replace the product $\\prec$ by the opposite one $\\curlyeqsucc:=-\\prec^{op}$, satisfying the formulas $\\to_x\\cdot'\\xrightarrow{y}=\\xrightarrow{x}\\cdot'\\,_y\\to=:\\xrightarrow{x\\curlyeqsucc y}$. In fact, as we saw in \\ref{ex:ass} the relations of $\\op{preLie}_\\bullet(\\sP erm)$-algebra translate in the relations of $\\op{preLie}_\\bullet(\\sA ss)=\\sD end$-algebra for the products $\\prec,\\curlyeqsucc$, and we get the additional relations (where as usual $x\\cdot y=x\\prec y-y\\prec x=x\\prec y+ x\\curlyeqsucc y$)\n\\[ 0= \\xrightarrow{x}\\cdot'\\left( \\to_y\\cdot'\\to_z\\right)-\\xrightarrow{x}\\cdot'\\left(\\to_z\\cdot'\\to_y\\right)=\\xrightarrow{x\\prec(y\\cdot z) - x\\prec(z\\cdot y)}\\]\n\\[ 0= \\to_x\\cdot'(\\xrightarrow{y}\\cdot'\\to_z)-\\to_x\\cdot'\\left( \\to_z\\cdot'\\xrightarrow{y}\\right)= \\xrightarrow{x\\curlyeqsucc(y\\prec z)-x\\curlyeqsucc(z\\curlyeqsucc y)}.\\]\nThis is in accord with \\cite[Theorem 21]{V}.\n\\end{example}\n\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$, an $\\Oh_{adm}$-algebra structure on $V$ is the datum of non-symmetric operations $\\prec_i,\\succ_i,\\circ_j.\\ast_k:V^{\\ten 2}\\to V$ such that the induced operations $x\\cdot_i y:=x\\prec_i y+x\\succ_iy$, $x\\bullet_j y= x\\circ_j y+y\\circ_j x$, $[x,y]_k=x\\ast_k y-y\\ast_k x$ define an $\\Oh$-algebra structure on $(V,\\cdot_i,\\bullet_j,[-,-]_k)$. This defines the operad $(\\Oh_{adm},\\prec_i,\\succ_i,\\circ_j,\\ast_k)$ of $\\Oh$ admissible algebras.\n\\end{definition}\n\n\\begin{example}\\label{ex:lieadm} We consider the operad $\\sL ie_{adm}$ of Lie admissible algebras, that is, the operad generated by a non-symmetric operation $\\cdot$ such that the commutator is a Lie bracket. We shall write the only relation in $\\sL ie_{adm}$ as $0=R_\\cdot(x_1,x_2,x_3)=\\sum_{\\sigma\\in S_3}\\varepsilon(\\sigma)A_\\cdot(x_{\\sigma(1)},x_{\\sigma(2)},x_{\\sigma(3)})$, where $\\varepsilon(\\sigma)$ is the sign of the permutation $\\sigma$ and as usual $A_\\cdot(-,-,-)$ is the asociator of $\\cdot$. Given an $\\operatorname{Ass}_\\bullet(\\sL ie_{adm})$-algebra structure $(V,\\prec,\\succ)$ on $V$, together with the associated $\\sL ie_{adm}$-algebra structure on $C^*(\\Delta_1;V)$, the associator $A_{\\cdot'}(-,-,-)$ is computed as in Example \\ref{ex:ass}. By Lemma \\ref{lem:rel} the vanishing of $R_\\cdot'(-,-,-)$ is equivalent to the following relation on $\\prec,\\succ$ (the others follow from this one by symmetry of $R$) \n\\begin{multline*} 0= R_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})=A_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})+ A_{\\cdot'}(\\to_y,\\xrightarrow{z},\\,_x\\to)+A_{\\cdot'}(\\xrightarrow{z},\\,_x\\to,\\to_y)-\\\\-A_{\\cdot'}(\\to_y,\\,_x\\to,\\xrightarrow{z})-A_{\\cdot'}(_x\\to,\\xrightarrow{z},\\to_y)-A_{\\cdot'}(\\xrightarrow{z},\\to_y,\\,_x\\to)=\n\\end{multline*} \n\\[=\\xrightarrow{x\\prec(z\\succ y) + x\\succ(z\\succ y)-(x\\succ z)\\succ y - (x\\succ z)\\prec y-(x\\prec z)\\succ y-(x\\prec z)\\prec y -x\\prec(z\\prec y)+x\\succ(z\\prec y)}\\]\nThis is exactly the relation saying that the product $x\\cup y:=x\\prec y + x\\succ y$ on $V$ (so denoted to distinguish it from the Lie admissible product $x\\cdot y=x\\prec y-y\\succ x$) is associative. Hence, we found $\\op{Ass}_\\bullet(\\sL ie_{adm})=\\sA ss_{adm}$. Further imposing $x\\prec y=y\\succ x=:x\\star y$, we see that $x\\cup y= x\\star y + y\\star x$ is an associative and commutative product, and thus that $\\op{Com}_\\bullet(\\sL ie_{adm})=\\sC om_{adm}$. For the operad $(\\operatorname{preLie}_\\bullet(\\sL ie_{adm}),\\prec,\\succ)$ we find the relation (and the others follow by symmetry of $R$), where as usual $x\\cdot y=x\\prec y-y\\succ x$,\n\\begin{multline*} 0= R_{\\cdot'}(\\to_x,\\to_y,\\xrightarrow{z})=A_{\\cdot'}(\\to_x,\\to_y,\\xrightarrow{z})+ A_{\\cdot'}(\\to_y,\\xrightarrow{z},\\to_x)+A_{\\cdot'}(\\xrightarrow{z},\\to_x,\\to_y)-\\\\-A_{\\cdot'}(\\to_y,\\to_x,\\xrightarrow{z})-A_{\\cdot'}(\\to_x,\\xrightarrow{z},\\to_y)-A_{\\cdot'}(\\xrightarrow{z},\\to_y,\\to_x)= \n\\end{multline*} \n\\[=\\xrightarrow{-z\\succ(x\\cdot y) - (z\\succ y)\\succ x - (z\\succ y)\\prec x + (z\\prec x)\\succ y + (z\\prec x)\\prec y-z\\prec(x\\cdot y)+z\\succ (y\\cdot x) + (z\\succ x)\\succ y + (z\\succ x)\\prec y -(z\\prec y)\\succ x -(z\\prec y)\\prec x+ z\\prec(y\\cdot x)}\\]\nPutting $x\\cup y = x\\prec y + x\\succ y$, the reader will check that the above relation can be rewritten as $0=A_\\cup(z,x,y)-A_\\cup(z,y,x)$, and thus we found $\\op{preLie}_\\bullet(\\sL ie_{adm})=pre\\sL ie_{adm}$.\\end{example} \n\n\\begin{remark}\n\tThe latter example suggests $\\Oh\\bullet\\sL ie_{adm}=\\Oh_{adm}$ for every operad $\\Oh$ in $\\mathbf{Op}$.\t\n\\end{remark}\n\n\\begin{example}\\label{ex:postlie} We consider the operad $(post\\sL ie,\\cdot,[-,-])$ of post-Lie algebras: this is the operad generated by a non-symmetric product $\\cdot$ and a Lie bracket $[-,-]$ satisfying the relations $0=R_1(x,y,z)=[x,y]\\cdot z-[x,y\\cdot z]-[x\\cdot z,y]$ and $0=R_2(x,y,z)=x\\cdot(y\\cdot z +z\\cdot y + [z,y])-(x\\cdot y)\\cdot z+(x\\cdot z)\\cdot y$. In the operad $(\\op{Ass}_\\bullet(\\Oh),\\prec,\\succ,\\ast)$ we get the associativity relation for the product $\\ast$ and the six independent relations \n\\[ 0=R_1(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{(x\\ast y)\\prec z-x\\ast(y\\prec z)}, \\]\n\\[ 0=R_1(\\to_x,\\xrightarrow{y},_z\\to)=\\xrightarrow{z\\succ(y\\ast x)-(z\\succ y)\\ast x},\\]\n\\[ 0=R_1(_x\\to,\\to_y,\\xrightarrow{z})=\\xrightarrow{x\\ast(z\\succ y)-(x\\prec z)\\ast y},\\]\n\\[ 0=R_2(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{x\\prec(y\\prec z+z\\prec y+y\\ast z)-(x\\prec y)\\prec z},\\]\n\\[ 0=R_2(\\to_x,\\xrightarrow{y},_z\\to)= \\xrightarrow{(z\\succ y+ y\\prec z +z\\star y)\\succ x-z\\succ(y\\succ x)},\\]\n\\[ 0=R_2( \\xrightarrow{x},_y\\to,\\to_z)=\\xrightarrow{(y\\succ x)\\prec z-y\\succ(x\\prec z)}.\\]\nThese are the relations of dendriform trialgebra \\cite{LR}, hence we found $\\op{Ass}_\\bullet(post\\sL ie)=\\mathcal{T}ridend$. If we further impose $x\\prec y= y\\succ x=: y\\star x$, $x\\star y=y\\star x=:x\\circledast y$, we find that the operad $(\\op{Com}_\\bullet(post\\sL ie),\\star,\\circledast)$ is defined by the following relations\n\\[ (x\\circledast y)\\circledast z=x\\circledast(y\\circledast z),\\quad(x\\circledast y)\\star z=x\\circledast(y\\star z),\\quad(x\\star y)\\star z =x\\star(y\\star z+z\\star y +y\\circledast z). \\]\nWe shall denote this operad by $post\\sC om:=\\op{Com}_\\bullet(post\\sL ie)$. We leave to the interested reader the computation of $\\op{preLie}_\\bullet(post\\sL ie)$. We have morphism of operads  \n\\[ (\\sL ie,\\{-,-\\})\\to(post\\sL ie,\\cdot,[-,-]):\\{-,-\\}\\:\\:\\longrightarrow\\:\\:\\cdot\\:-\\:\\cdot^{op}\\:+\\:[-,-],\\]\n\\[ (\\sA ss,\\cup)\\to(\\mathcal{T}ridend,\\prec,\\succ,\\ast):\\cup\\:\\:\\longrightarrow\\:\\:\\prec+\\succ+\\ast,\\]\n\\[(\\sC om,\\bullet)\\to(post\\sC om,\\star,\\circledast):\\bullet\\:\\:\\longrightarrow\\:\\: \\star+\\star^{op}+\\circledast, \\] \nin the first (as well as the second) case this is well known, the other two follow by functoriality. \n\\end{example}\n\n\\begin{example} We consider the operad generated by a Lie bracket $[-,-]$ and a right Leibniz product $\\cdot$ satisfying the additional relations \n\t\\[ [x,y]\\cdot z=[x,y\\cdot z]+[x\\cdot z,y],\\qquad x\\cdot[y,z]=x\\cdot(y\\cdot z). \\]\n\tIt can be checked that this is the Koszul dual $(post\\sC om^!, \\cdot,[-,-])$ of the operad $post\\sC om:=\\op{Com}_\\bullet(post\\sL ie)$ from the previous example. To aid in the computations, we notice that this is the same, up to changing the sign of the operation $\\cdot$, as the operad we obtain from $(post\\sL ie,\\cdot,[-,-])$ by further imposing the right Leibniz identity for $\\cdot$. By the previous example and Example \\ref{ex:leib}, we see that the operad $\\op{Ass}_\\bullet(post\\sC om^!)$ is the same as the one we obtain from the operad $\\mathcal{T}ridend$ of dendriform trialgebras by further imposing the diassociativity relations for $\\prec,\\succ$, and then changing the signs of $\\prec,\\succ$: the reader will readily verify that this is the operad $\\op{Ass}_\\bullet(post\\sC om^!)=\\mathcal{T}riass$ of triassociative algebras by Loday and Ronco \\cite{LR}. Likewise, $\\op{Com}_\\bullet(post\\sC om^!)$ is the operad we obtain from $(post\\sC om,\\star,\\circledast)$, defined as in the previous example, by further imposing that $\\star$ is a right permutative product, and then changing the sign of $\\star$: the reader will readily verify that this is the same as the operad $\\sC omtrias$ of commutative trialgebras \\cite{Vpos}.   \n\\end{example}\n\\newcommand{\\sR}{\\mathcal{R}}\n\n\\section{A Koszul dual trick to compute Manin white products}\n\nWe denote by $-\\circ-:\\mathbf{Op}\\times\\mathbf{Op}\\to\\mathbf{Op}$ the Manin white product of operads \\cite{V}, it gives $\\mathbf{Op}$ a structure of symmetric monoidal category, and by $-^!:\\mathbf{Op}\\to\\mathbf{Op}$ the Koszul duality functor. Recall that the Manin white and black products are Koszul dual, in the sense that $(\\Oh\\bullet\\sP)^!=\\Oh^!\\circ\\sP^!$ for every pair of operads $\\Oh$ and $\\sP$ in $\\mathbf{Op}$, and moreover for every operad $\\Oh$ the functors $\\xymatrix{\\Oh\\bullet-:\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\Oh^!\\circ-\\ar@<2pt>[l]}$ form an adjoint pair. recall that $\\sA ss^!=\\sA ss$, $\\sC om^!=\\sL ie$ and $pre\\sL ie^!=\\sP erm$. In the previous section we introduced easily computable functors $\\op{Ass}_\\bullet(-)$, $\\op{Com}_\\bullet(-),\\op{preLie}_\\bullet(-):\\mathbf{Op}\\to\\mathbf{Op}$ and claimed that they coincide with the respective Manin black products $\\sA ss\\bullet-,\\sC om\\bullet-, pre\\sL ie\\bullet-$. The aim of this section is to introduce the respective right adjoint functors $\\op{Ass}_\\circ(-),\\op{Lie}_\\circ(-),\\op{Perm}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$ (the adjointness relation will be shown at the very end of the section) and prove that they in fact coincide with the Manin white products $\\sA ss\\circ-,\\sL ie\\circ-,\\sP erm\\circ-$:\nthis will also complete the proof of Theorem \\ref{th:black}.\n\nWe consider first the case of $\\operatorname{Ass}_\\circ(-)$. Given an (as usual, binary, quadratic and finitely generated) operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$, the operad $\\op{Ass}_\\circ(\\Oh)$ is generated by non symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$.  We want a morphism of operads $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ corresponding to the counit of the adjunction  $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$: in other words, by definition of $\\op{Ass}_\\bullet(-)$, given an $\\Oh$-algebra structure $(V,\\cdot_i,\\bullet_j,[-,-]_k)$ on a vector space $V$ we want an induced local dg $\\op{Ass}_\\circ(\\Oh)$-algebra structure on the complex $C^*(\\Delta_1;V)=C^*(\\Delta_1;\\mathbb{K})\\otimes V$. Denoting by $\\cup$ the usual cup product of cochains on $C^*(\\Delta_1;\\mathbb{K})$, the $\\op{Ass}_\\circ(\\Oh)$-algebra structure on $C^*(\\Delta_1;V)$ will be given by the tensor product operations $\\prec_i=\\cup\\otimes\\cdot_i$, $\\succ_i=\\cup\\otimes\\cdot_i^{op}$, $\\circ_j=\\cup\\ten\\bullet_j$, $\\ast_k=\\cup\\ten[-,-]_k$: explicitly, with the same notations as in the previous section, \n\\[ \\to_x\\prec_i\\to_y\\:=\\:\\to_{x\\cdot_i y}\\:=\\:\\to_y\\succ_i\\to_x,\\qquad \\to_x\\circ_j\\to_y\\:=\\:\\to_{x\\bullet_j y},\\qquad\\to_x\\ast_k\\to_y\\:=\\: \\to_{[x,y]_k}, \\] \\[ _x\\to\\prec_i\\,_y\\to\\:=\\:_{x\\cdot_i y}\\to\\:=\\:_y\\to\\succ_i\\,_x\\to,\\qquad _x\\to\\circ_j\\,_y\\to\\:=\\:_{x\\bullet_j y}\\to,\\qquad_x\\to\\ast_k\\,_y\\to\\:=\\: _{[x,y]_k}\\to, \\] \\[ _x\\to\\prec_i\\xrightarrow{y}=_y\\to\\succ_i\\xrightarrow{x}\\:=\\:\\xrightarrow{x\\cdot_i y}\\:=\\:\\xrightarrow{x}\\prec_i\\to_y= \\xrightarrow{y}\\succ_i\\to_{x},\\] \\[ _x\\to\\circ_j\\xrightarrow{y}\\:=\\:\\xrightarrow{x\\bullet_j y}\\:=\\:\\xrightarrow{x}\\circ_j\\to_y,\\qquad  _x\\to\\ast_k\\xrightarrow{y}\\:=\\:\\xrightarrow{[x,y]_k}\\:=\\:\\xrightarrow{x}\\ast_k\\to_y, \\] \nand the remaining products vanish. It is immediately seen that these operations satisfy the Leibniz identity with respect to $d$ and the locality assumption from Definition \\ref{def:ManinBlack}. We will generically call the operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $C^*(\\Delta_1;V)$ the cup products. \n\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\op{Ass}_\\circ(\\Oh)$ is generated by non-symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, together with the larger set of relations making $C^*(\\Delta_1;V)$ with the cup products a local dg $\\op{Ass}_\\circ(\\Oh)$-algebra for every $\\Oh$-algebra $V$.\\end{definition}\n\nNext we want to describe a generating set of relations of $\\op{Ass}_\\circ(\\Oh)$. Given a terniary operation $R(-,-,-):C^*(\\Delta_1;V)^{\\ten3}\\to C^*(\\Delta_1;V)$ in the cup products, we write $R=\\sum_{\\sigma\\in S_3}R_\\sigma$, where $S_3$ is the symmetric group and $R_\\sigma$ is spanned by triple products permuting the variables according to $\\sigma$. As in the proof of Lemma \\ref{lem:rel}, we see that the relation $R(-,-,-)=0$ holds in $C^*(\\Delta_1;V)$ if and only if\n\\begin{multline*}0=R(\\xrightarrow{x},\\to_y,\\,_z\\to)=R(\\xrightarrow{x},\\,_y\\to,\\to_z)=R(\\to_x,\\xrightarrow{y},\\,_z\\to)=\\\\=R(_x\\to,\\xrightarrow{y},\\to_z)=R(\\to_x,\\,_y\\to,\\xrightarrow{z})=R(_x\\to,\\to_y,\\xrightarrow{z})=0\\end{multline*}\nfor all $x,y,z\\in V$. We claim that this is true if and only if $R_\\sigma(-,-,-)=0$ holds in $C^*(\\Delta_1;V)$ for every $\\sigma\\in S_3$, where the \\vr if'' implication is obvious. To fix the ideas, we consider the identical permutation $\\id\\in S_3$ and prove $R(-,-,-)=0\\solose R_{\\id}(-,-,-)=0$: the remaining implications are proved similarly. We obtain the desired conclusion again by Lemma \\ref{lem:rel}, the point is that the only non-vanishing triple products that we can form out of the cochains $_1\\to,\\to_1,\\xrightarrow{1}\\in C^*(\\Delta_1;\\mathbb{K})$ are $(_1\\to\\cup\\xrightarrow{1})\\cup\\to_1\\:=\\:\\xrightarrow{1}\\:=\\:_1\\to\\cup(\\xrightarrow{1}\\cup\\to_1)$: this shows at once that $R_{\\id}(\\xrightarrow{x},\\to_y,\\,_z\\to)=R_{\\id}(\\xrightarrow{x},\\,_y\\to,\\to_z)=R_{\\id}(\\to_x,\\xrightarrow{y},\\,_z\\to)=R_{\\id}(\\to_x,\\,_y\\to,\\xrightarrow{z})=R_{\\id}(_x\\to,\\to_y,\\xrightarrow{z})=0$, whereas $R_{\\id}(_x\\to,\\xrightarrow{y},\\to_z)=R(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{R'(x,y,z)}=0$ by hypothesis, where $R'(-,-,-):V^{\\ten3}\\to V$ is a terniary operation in $\\Oh$; finally, if this holds for a generic $\\Oh$-algebra $V$, we conclude that $R'(-,-,-)=0$ is a relation in the operad $\\Oh$. We sum up the previous discussion in the following lemma.\n\\begin{lemma}\\label{lem:relwhite} The operad $\\op{Ass}_\\circ(\\Oh)$ has a generating set of relations $R(-,-,-)=0$ which are non-symmetric (where a relation is non-symmetric if $R=R_{\\id}$), consisting of all the possible splittings, as in the following \n\t\\[ R(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{R'(x,y,z)}=0,\\]\nof a relation $R'(-,-,-)=0$ in $\\Oh$ via the cup products on $C^*(\\Delta_1;V)$ (where $V$ is a generic $\\Oh$-algebra).\\end{lemma}\nIt is best to illustrate the previous lemma by some examples.\n\\begin{example} We consider the operad $(\\sC om,\\bullet)$: in this case $R'$ as in the previous lemma has to be non-symmetric itself, and since the space of non-symmetric relators of $\\sC om$ is one-dimensional, spanned by the associativity relation, we only get the splitting\n\t\\[ (_x\\to \\circ \\xrightarrow{y})\\circ\\to_y - _x\\to\\circ(\\xrightarrow{y}\\circ\\to_z) =\\xrightarrow{(x\\bullet y)\\bullet z-x\\bullet (y\\bullet z)}=0,\\]\n\ttelling us that the cup product $\\circ$ is associative: thus $\\op{Ass}_\\circ(\\sC om)=\\sA ss$ as expected. In the case of the operad $(\\sL ie,[-,-])$, again $R'$ as in the previous Lemma has to be non-symmetric, and since the space of relators of the operad $\\sL ie$ is one-dimensional generated by the Jacobi identity, and this can't be written in a non-symmetric form, we find $\\op{Ass}_\\circ(\\sL ie)=\\sM ag_{1,0,0}$, in accord with our computation of the Koszul dual $\\op{Ass}_\\bullet(\\sC om)=nil\\sA ss$. \n\t\\end{example}\n\n\\begin{remark} Whereas the computation of the functor $\\op{Ass}_\\bullet(-)$ is completely mechanical, as we apply Lemma \\ref{lem:rel} to a generating set of relations of $\\Oh$ to get a generating set of relations of $\\op{Ass}_\\bullet(\\Oh)$, the previous lemma is not as effective in the compution of $\\op{Ass}_\\circ(\\Oh)$: we illustrate this fact by considering the operad $(\\sP ois,\\bullet,[-,-])$. By the previous example, the operad $\\op{Ass}_\\circ(\\sP ois)$ is generated by an associative product $\\circ$ and a magmatic product $\\ast$. Since there is no way to split the Poisson identity $[x\\bullet y, z] -x\\bullet[y,z] - [x,z]\\bullet y=0$ as in the previous lemma, as this can't be written in a non-symmetric form, we may be tempted to conclude that $\\op{Ass}_\\circ(\\sP ois)=\\sA ss+\\sM ag_{1,0,0}$, but this is in contrast with our computation of the Koszul dual $\\op{Ass}_\\bullet(\\sP ois)$ in Example \\ref{ex:pois}. In fact, looking closely we find the splitting\n\\begin{multline*} (_x\\to\\circ \\xrightarrow{y})\\ast\\to_z-_x\\to\\circ(\\xrightarrow{y}\\ast\\to_z) + (_x\\to\\ast\\xrightarrow{y})\\circ\\to_z-_x\\to\\ast(\\xrightarrow{y}\\circ\\to_z) = \\\\ = \\xrightarrow{[x\\bullet y,z ]-x\\bullet[y,z]+[x,y]\\bullet z-[x,y\\bullet z]}=\\xrightarrow{[x,z]\\bullet y - y\\bullet[x,z]}=0.\n\\end{multline*}\nSo we find the relation $(\\alpha\\circ\\beta)\\ast\\gamma-\\alpha\\circ(\\beta\\ast\\gamma)+(\\alpha\\ast\\beta)\\circ\\gamma-\\alpha\\ast(\\beta\\circ\\gamma)=0$, $\\forall \\alpha,\\beta,\\gamma\\in C^*(\\Delta_1;V)$, in the operad $\\op{Ass}_\\circ(\\sP ois)$. Together with the associativity relation for $\\circ$, these generate the relations of $\\op{Ass}_\\circ(\\sP ois)$: the easiest way to see this is to check that the operad $\\op{Ass}_\\circ(\\sP ois)$ defined in this way is in fact the Koszul dual of $\\op{Ass}_\\bullet(\\sP ois)$ from \\ref{ex:pois}. Returning to the initial remark, the difficulty in applying Lemma \\ref{lem:relwhite} is that we have to look for the splittings of any relation $R'$ in $\\Oh$, and we can't limit ourselves to let $R'$ vary in a generating set of these relations, so there is no telling us in general if we are missing some of the possible splittings. As in the previous computation, the easiest way around this, and what we will do in practice in the following examples, is to take advantage of the computation of the Koszul dual $\\op{Ass}_\\bullet(\\Oh^!)$ from the previous section. Of course, this depends on the yet to be given proofs of theorems \\ref{th:black}, \\ref{th:asswhite},\\ref{th:permwhite} and \\ref{th:liewhite}.\n\\end{remark}\n\\begin{example} We consider the operad $(\\sA ss,\\cdot)$: in this case we know from Example \\ref{ex:ass} that $\\op{Ass}_\\circ(\\sA ss)=(\\sA ss\\bullet\\sA ss)^!=(\\sA ss\\times\\sA ss)^!=\\sA ss+\\sA ss$ (with the functors $-\\times-$ and $-+-$ as in Definition \\ref{def:+x}), and in fact we find the relations\n\t\\[ (_x\\to\\prec\\xrightarrow{y})\\prec\\to_z-_x\\to\\prec(\\xrightarrow{y}\\prec \\to_z)=\\xrightarrow{(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)}=0,  \\]\n\t\\[ (_x\\to\\succ\\xrightarrow{y})\\succ\\to_z-_x\\to\\succ(\\xrightarrow{y}\\succ \\to_z)=\\xrightarrow{z\\cdot(y\\cdot x)-(z\\cdot y)\\cdot x}=0,  \\] \n\tin the operad $(\\op{Ass}_\\circ(\\sA ss),\\prec,\\succ)$.\n\t\n\tWe consider the operad $(pre\\sL ie, \\cdot)$. The right pre-Lie relation $(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y+x\\cdot(z\\cdot y)=0$ can't be splitted as in Lemma \\ref{lem:relwhite}, for instance because there is no argument remaining inside every parenthesis. As the previous remark illustrates, this alone wouldn't be enough to conclude $\\op{Ass}_\\circ(pre\\sL ie)=\\sM ag_{2,0,0}$: on the other hand, this is true since it agrees with the computation of the Koszul dual $\\op{Ass}_\\bullet(\\sP erm)=nil\\sA ss\\times nil\\sA ss$ in Example \\ref{ex:perm}.\n\t\n\tWe consider the operad $(\\sL eib,\\cdot)$. Again, the right Leibniz relation  $(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y=0$ can't be splitted as in Lemma \\ref{lem:relwhite}, on the other hand, this imply the relation $x\\cdot(y\\cdot z+z\\cdot y) = (x\\cdot y)\\cdot z-(x\\cdot z)\\cdot y +(x\\cdot z)\\cdot y-(x\\cdot y)\\cdot z=0$ in $\\sL eib$: for the latter, we find the splittings\n\t\\[ _x\\to\\prec (\\xrightarrow{y}\\prec\\to_z + \\xrightarrow{y}\\succ\\to_z ) =\\xrightarrow{x\\cdot(y\\cdot z+z\\cdot y)}=0,\\]\n\t\\[ (_x\\to\\prec\\xrightarrow{y} + \\,_x\\to\\succ\\xrightarrow{y})\\succ\\to_z = \\xrightarrow{z\\cdot (x\\cdot y + y\\cdot x)}=0, \\]\n\thence the relations $\\alpha\\prec(\\beta\\prec\\gamma+\\beta\\succ\\gamma)=0=(\\alpha\\prec\\beta +\\alpha\\succ\\beta)\\succ\\gamma$ in $(\\op{Ass}_\\circ(\\sL eib),\\prec,\\succ)$. We leave to the reader to check that this is a generating set of relations, for instance by computing the Koszul dual $\\op{Ass}_\\bullet(\\sZ inb)$ with the method of the previous section.\n\n    As a final example, we consider $(\\sZ inb,\\cdot)$. In this case, we know form Example \\ref{ex:leib} that $\\op{Ass}_\\circ(\\sZ inb)=(\\sA ss\\bullet\\sL eib)^!=di\\sA ss^!=\\sD end$: in fact, we get the dendriform relations on the operad $(\\op{Ass}_\\circ(\\sZ inb),\\prec,\\succ)$, corresponding to the splittings as in Lemma \\ref{lem:relwhite} (notice that the relation $(x\\cdot y)\\cdot z-(x\\cdot z)\\cdot y=0$ holds in $\\sZ inb$)\n    \\[ _x\\to\\prec( \\xrightarrow{y}\\prec\\to_z+\\xrightarrow{y}\\succ\\to_z)-(_x\\to\\prec\\xrightarrow{y})\\prec\\to_z=\\xrightarrow{x\\cdot(y\\cdot z+ z\\cdot y)-(x\\cdot y)\\cdot z}=0\\]\n    \\[ (_x\\to\\prec\\xrightarrow{y}+\\,_x\\to\\succ\\xrightarrow{y})\\succ\\to_z -\\,_x\\to\\succ(\\xrightarrow{y}\\succ\\to_z)=\\xrightarrow{z\\cdot(x\\cdot y+y\\cdot x)-(z\\cdot y)\\cdot x}=0\\]\n    \\[ (_x\\to\\succ\\xrightarrow{y})\\prec\\to_z -\\, _x\\to\\succ(\\xrightarrow{y}\\prec\\to_z)=\\xrightarrow{(y\\cdot x)\\cdot z-(y\\cdot z)\\cdot x}=0.\\]\n\\end{example}\n\n\\begin{theorem}\\label{th:asswhite} There is a natural isomorphism $\\op{Ass}_\\circ(-)\\xrightarrow{\\cong}\\sA ss\\circ-$ of functors $\\mathbf{Op}\\to\\mathbf{Op}$.\n\\end{theorem}\n\\begin{proof} We denote by $(\\sA ss,\\cup)$ the associative operad with its generating product. Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\sA ss\\circ\\Oh$ is generated by the tensor product operations (which are non-symmetric) $\\cup\\otimes\\cdot_i, \\cup\\otimes\\cdot_i^{op}$, $\\cup\\otimes\\bullet_j$, $\\cup\\otimes[-,-]_k$, together with the larger set of relations holding in the tensor product $A\\otimes V$ of a generic $\\sA ss$-algebra $(A,\\cup)$ and a generic $\\Oh$-algebra $(V,\\cdot_i,\\bullet_j,[-,-]_k)$. The isomorphism of operads $\\op{Ass}_\\circ(\\Oh)\\leftrightarrow \\sA ss\\circ\\Oh$ is given by $\\prec_i\\leftrightarrow \\cup\\otimes\\cdot_i$, $\\succ_i\\leftrightarrow\\cup\\otimes\\cdot_i^{op}$, $\\circ_j\\leftrightarrow\\cup\\otimes\\bullet_j$ and finally $\\ast_k\\leftrightarrow\\cup\\otimes[-,-]_k$. This defines a morphism of operads $\\sA ss\\circ\\Oh\\to\\op{Ass}_\\circ(\\Oh)$: in fact, by construction of the cup products $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, these satisfy the relations of $\\sA ss\\circ\\Oh$-algebra in the tensor product $C^*(\\Delta_1;V)$ of the $\\sA ss$-algebra $(C^*(\\Delta_1;\\mathbb{K}),\\cup)$ (where we may forget the gradation) and the generic $\\Oh$-algebra $V$. Conversely, we must show that if we evaluate a generating relation $R(-,-,-)=0$ of $\\op{Ass}_\\circ(\\Oh)$, given as in Lemma \\ref{lem:relwhite}, in the operations $\\cup\\otimes\\cdot_i, \\cup\\otimes\\cdot_i^{op}$, $\\cup\\otimes\\bullet_j$, $\\cup\\otimes[-,-]_k$, we get a relation of the operad $\\sA ss\\circ\\Oh$: but this is also clear, since given a generic $\\sA ss$-algebra $(A,\\cup)$ we find $R(a\\otimes x,b\\otimes y,c\\otimes z)= (a\\cup b\\cup c)\\otimes R'(x,y,z)=0$, $\\forall a\\otimes x,b\\otimes y,c\\otimes z\\in A\\otimes V$, where $a\\cup b\\cup c:=(a \\cup b)\\cup c=a\\cup(b\\cup c)$.\n\\end{proof}\n\nThe construction of the functor $\\op{Perm}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$ is similar. Given a generic $\\Oh$-algebra $V$, the cup products on the complex $C^*(\\Delta_1,v_l;V)$ are defined by the same formulas as in the associative case.\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\op{Perm}_\\circ(\\Oh)$ is generated by non-symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, together with the larger set of relations making $C^*(\\Delta_1,v_l;V)$ with the cup products a local dg $\\op{Perm}_\\circ(\\Oh)$-algebra for every $\\Oh$-algebra $V$.\\end{definition}\n\nWe want to describe a generating set of relations of $\\op{Perm}_\\circ(\\Oh)$ as in the associative case. We denote the elements of the symmetric group $S_3=\\{\\id,(123),(132),(12),(13),(23)\\}$ according to their cycle decomposition. Given a terniary operation $R(-,-,-):C^*(\\Delta_1,v_l;V)^{\\otimes 3}\\to C^*(\\Delta_1,v_l;V)$ in the cup products, we write $R=R_1+R_2+R_3$, where $R_1:=R_{\\id}+R_{(23)}$, $R_2:=R_{(123)}+ R_{(12)}$ and $R_3:=R_{(132)}+R_{(13)}$. By Lemma \\ref{lem:relpl}, the relation $R(-,-,-)=0$ holds in $C^*(\\Delta_1,v_l;V)$ if and only if \n\\[ R(\\xrightarrow{x},\\to_y,\\to_z)=R(\\to_x,\\xrightarrow{y},\\to_z)=R(\\to_x,\\to_y,\\xrightarrow{z})=0\\]\nfor all $x,y,z\\in V$. We claim that this is true if and only if so is $R_i(-,-,-)=0$ for $i=1,2,3$: again, we limit ourselves to show $R(-,-,-)=0\\solose R_1(-,-,-)=0$. This time, the point is that the only way to form a non-vanishing triple product out of $\\xrightarrow{1},\\to_1,\\to_1\\in (C^*(\\Delta_1,v_l;\\mathbb{K}),\\cup)$ is by putting the degree one cochain $\\xrightarrow{1}$ on the left: this implies $R_1(\\to_x,\\xrightarrow{y},\\to_z)=R_1(\\to_x,\\to_y,\\xrightarrow{z})=0$, whereas $R_1(\\xrightarrow{x},\\to_y,\\to_z)=R(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{R'(x,y,z)}=0$ by hypothesis, where $R'(-,-,-)=0$ is a relation of $\\Oh$ since $V$ is generic, so we may conclude again by Lemma \\ref{lem:relpl}. To sum up\n\\begin{lemma}\\label{lem:relpm} The operad $\\op{Perm}_\\circ(\\Oh)$ has a generating set of relations $R(-,-,-)=0$ of the form $R=R_{\\id}+R_{(23)}$, consisting of all the possible splittings, as in the following \n\t\\[ R(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{R'(x,y,z)}=0,\\]\n\tof a relation $R'(-,-,-)=0$ in $\\Oh$ via the cup products on $C^*(\\Delta_1,v_l;V)$.\\end{lemma}\n\n\\begin{theorem}\\label{th:permwhite} There is a natural isomorphism $\\op{Perm}_\\circ(-)\\xrightarrow{\\cong}\\sP erm\\circ-$ of functors $\\mathbf{Op}\\to\\mathbf{Op}$.\n\\end{theorem}\n\\begin{proof}\n\tThis is done as in Theorem \\ref{th:asswhite} by noticing that the relations of $\\sP erm\\circ\\Oh$-algebra are satisfied by the cup products on the tensor product $C^*(\\Delta_1,v_l;V)$ of the right permutative algebra $(C^*(\\Delta_1,v_l;\\mathbb{K}),\\cup)$ and a generic $\\Oh$-algebra $V$. Conversely, given a generating relation $R(-,-,-)=0$ of $\\op{Perm}_\\circ(\\Oh)$ as in the previous lemma, it is straightforward to check that this holds more in general in the tensor product $A\\otimes V$ of a generic $\\sP erm$-algebra $A$ and a generic $\\Oh$-algebra $V$. \n\\end{proof}\n\\begin{example} We consider the operad $(\\sL ie,[-,-])$: by the previous theorem $\\op{Perm}_\\circ(\\sL ie)=\\sP erm\\circ\\sL ie=\\sL eib$, where the right Leibniz relation corresponds to the splitting as in Lemma \\ref{lem:relpm}\n\t\\[ (\\xrightarrow{x}\\ast \\to_y)\\ast\\to_z-\\xrightarrow{x}\\ast(\\to_y\\ast\\to_z)-(\\xrightarrow{x}\\ast\\to_z)\\ast\\to_y=\\xrightarrow{[[x,y],z]-[x,[y,z]]-[[x,z],y]}=0.\\]\n\nWe consider the operad $(\\sP ois,\\bullet,[-,-])$. This is generated by a right permutative product $\\circ$ and a right Leibniz product $\\ast$: we have the following splittings of the Poisson identity\n\\begin{multline*}\n0=\\xrightarrow{[x\\bullet y,z]-x\\bullet [y,z]-[x,z]\\bullet y} = \\left( \\xrightarrow{x}\\circ\\to_y\\right)\\ast\\to_z -\\xrightarrow{x}\\circ(\\to_y\\ast\\to_z)-\\left( \\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to{y}=\\\\= \\left( \\xrightarrow{x}\\circ\\to_y\\right)\\ast\\to_z +\\xrightarrow{x}\\circ(\\to_z\\ast\\to_y)-\\left( \\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to{y}, \\end{multline*}\n\\begin{multline*} 0=\\xrightarrow{[x,y\\bullet z]-[x,y]\\bullet z-[x,z]\\bullet y}= \\xrightarrow{x}\\ast\\left( \\to_y\\circ\\to_z\\right) -\\left(\\xrightarrow{x}\\ast\\to_y\\right)\\circ\\to_z-\\left(\\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to_y =\\\\ = \\xrightarrow{x}\\ast\\left( \\to_z\\circ\\to_y\\right) -\\left(\\xrightarrow{x}\\ast\\to_y\\right)\\circ\\to_z-\\left(\\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to_y,\\end{multline*}\ncorresponding to the three independent relations\n\\[ (\\alpha\\circ \\beta)\\ast\\gamma=\\alpha\\circ(\\beta\\ast\\gamma)+(\\alpha\\ast\\gamma)\\circ\\beta,\\quad\\alpha\\ast(\\beta\\circ\\gamma)=(\\alpha\\ast\\beta)\\circ\\gamma+(\\alpha\\ast\\gamma)\\circ\\beta,\\quad \\alpha\\circ(\\beta\\ast\\gamma)=-\\alpha\\circ(\\gamma\\ast\\beta), \\]\nin the operad $\\op{Perm}_\\circ(\\sP ois)$. In fact, these are the relations defining Aguiar's operad $\\sP erm\\circ\\sP ois=(pre\\sL ie\\bullet\\sP ois)^!$ of dual pre-Poisson algebras \\cite{A,U}.\n\nWe consider the operad $(\\sA ss,\\cdot)$: then we know $\\sP erm\\circ\\sA ss=di\\sA ss$. In this case, the relations we find as in Lemma \\ref{lem:relpm} turn out to be the diassociative relations for the generating products $\\prec$, $\\curlyeqsucc:=\\succ^{op}$ of $\\op{Ass}_\\circ(\\Oh)$: in fact, we find\n\\begin{equation*} 0=\\xrightarrow{(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)}=(\\xrightarrow{x}\\prec\\to_y)\\prec\\to_z -\\xrightarrow{x}\\prec(\\to_y\\prec\\to_z)=(\\xrightarrow{x}\\prec\\to_y)\\prec\\to_z -\\xrightarrow{x}\\prec(\\to_z\\succ\\to_y), \\end{equation*}\n\\begin{equation*} 0=\\xrightarrow{z\\cdot(y\\cdot x)-(z\\cdot y)\\cdot x}=(\\xrightarrow{x}\\succ\\to_y)\\succ\\to_z -\\xrightarrow{x}\\succ(\\to_y\\succ\\to_z)=\\\\=(\\xrightarrow{x}\\succ\\to_y)\\succ\\to_z -\\xrightarrow{x}\\succ(\\to_z\\prec\\to_y), \\end{equation*}\n\\[ 0=\\xrightarrow{(y\\cdot x)\\cdot z-y\\cdot(x\\cdot z)} =(\\xrightarrow{x}\\succ\\to_y)\\prec\\to_x-(\\xrightarrow{x}\\prec\\to_z)\\succ\\to_y.\\]\nThe reader may compare this with the computation of $\\op{preLie}_\\bullet(\\sA ss)$ in Example \\ref{ex:ass}, as well as what we said in Remark \\ref{rem:curlyeqprec}.\n\nWe consider the operad $(pre\\sL ie,\\cdot)$. We split the right pre-Lie identity as \n\\[0=\\xrightarrow{(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y +x\\cdot(z\\cdot y)}= (\\xrightarrow{x}\\prec\\to_y)\\prec\\to_z-\\xrightarrow{x}\\prec\\to_{y\\cdot z}-(\\xrightarrow{x}\\prec\\to_z)\\prec\\to_y+\\xrightarrow{x}\\prec\\to_{z\\cdot y},\\]\nSince we may split $\\to_{y\\cdot z}$ both as $\\to_y\\prec\\to_z=\\to_{y\\cdot z}=\\to_z\\succ\\to_y$, and similarly for $\\to_{z\\cdot y}$, we get the two independent relations  $(\\alpha\\prec\\beta)\\prec\\gamma-\\alpha\\prec(\\beta\\prec\\gamma)=(\\alpha\\prec\\gamma)\\prec\\beta-\\alpha\\prec(\\gamma\\prec\\beta)$ and $\\alpha\\prec(\\beta\\prec\\gamma)=\\alpha\\prec(\\gamma\\succ\\beta)$ in the operad $\\op{Perm}_\\circ(pre\\sL ie)$. We also have the splitting\n\\[0=\\xrightarrow{z\\cdot (y\\cdot x)-(z\\cdot y)\\cdot x-z\\cdot(x\\cdot y) +(z\\cdot x)\\cdot y}= (\\xrightarrow{x}\\succ\\to_y)\\succ\\to_z-\\xrightarrow{x}\\succ\\to_{z\\cdot y}-(\\xrightarrow{x}\\prec\\to_y)\\succ\\to_z+(\\xrightarrow{x}\\succ\\to_{z})\\prec\\to_y,\\]\ngiving the relations $(\\alpha\\succ\\beta)\\succ\\gamma-\\alpha\\succ(\\beta\\succ\\gamma)=(\\alpha\\prec\\beta)\\succ\\gamma-(\\alpha\\succ\\gamma)\\prec\\beta$ and $\\alpha\\succ(\\beta\\succ\\gamma)=\\alpha\\succ(\\gamma\\prec\\beta)$. We leave to the reader to check that this is a generating set of relations, and in fact the operad $(\\op{Perm}_\\circ(pre\\sL ie),\\prec,\\curlyeqsucc:=\\succ^{op})$ with the relations\n\\begin{eqnarray} \\nonumber\n(\\alpha\\prec\\beta)\\prec\\gamma-\\alpha\\prec(\\beta\\prec\\gamma)\\:=\\:(\\alpha\\prec\\gamma)\\prec\\beta-\\alpha\\prec(\\gamma\\prec\\beta),  & \\alpha\\prec(\\beta\\prec\\gamma)\\:=\\:\\alpha\\prec(\\beta\\curlyeqsucc\\gamma),  \\\\ \\nonumber (\\alpha\\curlyeqsucc\\beta)\\curlyeqsucc\\gamma-\\alpha\\curlyeqsucc(\\beta\\curlyeqsucc\\gamma)\\:=\\:(\\alpha\\curlyeqsucc\\gamma)\\prec\\beta-\\alpha\\curlyeqsucc(\\gamma\\prec\\beta), & (\\alpha\\curlyeqsucc\\beta)\\curlyeqsucc\\gamma \\:=\\:(\\alpha\\prec\\beta)\\curlyeqsucc\\gamma,\n\\end{eqnarray}\nis the Koszul dual of the operad $\\op{preLie}_\\bullet(\\sP erm)$ from Example \\ref{ex:perm}.\n\\end{example}\n\nWe finally come to the definition of $\\op{Lie}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$. In this case, we consider the complex $C^*(\\Delta_1;\\mathbb{K})$ as a $\\sL ie$-algebra via the bracket $[-,-]=\\cup-\\cup^{op}$, then given a generic $\\Oh$-algebra $(V,\\cdot_i,\\bullet_j,\\ast_k)$ we shall call the tensor product operations $\\star_i := [-,-]\\otimes\\cdot_i=(\\cup-\\cup^{op})\\otimes\\cdot_i=\\prec_i-\\succ_i^{op}$, $\\{-,-\\}_j:=[-,-]\\otimes\\bullet_j=\\circ_j-\\circ_j^{op}$, $\\circledast_k:=[-,-]\\otimes[-,-]_k=\\ast_k+\\ast_k^{op}$ the cup brackets on $C^*(\\Delta_1;V)$.\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\op{Lie}_\\circ(\\Oh)$ is generated by non-symmetric operations $\\star_i$, anti-symmetric operations $\\{-,-\\}_j$ and symmetric operations $\\circledast_k$ together with the larger set of relations making $C^*(\\Delta_1;V)$ with the cup brackets a local dg $\\op{Lie}_\\circ(\\Oh)$-algebra for every $\\Oh$-algebra $V$.\\end{definition}\n\\begin{theorem}\\label{th:liewhite} There is a natural isomorphism $\\op{Lie}_\\circ(-)\\xrightarrow{\\cong}\\sL ie\\circ-$ of functors $\\mathbf{Op}\\to\\mathbf{Op}$.\n\\end{theorem}\n\\begin{proof} This follows from Theorem \\ref{th:asswhite}: in fact, by construction $\\op{Lie}_\\circ(\\Oh)$ is the suboperad of $\\op{Ass}_\\circ(\\Oh)$ generated by the operations $\\star_i=\\prec_i-\\succ_i^{op}$, $\\{-,-\\}_j=\\circ_j-\\circ_j^{op}$, $\\circledast_k=\\ast_k+\\ast_k^{op}$, and similarly $\\sL ie\\circ\\Oh$ is the suboperad of $\\sA ss\\circ\\Oh$ generated by the operations $(\\cup-\\cup^{op})\\otimes\\cdot_i$, $(\\cup-\\cup^{op})\\otimes\\bullet_j$, $(\\cup-\\cup^{op})\\otimes[-,-]_k$.\n\\end{proof}\n\n \n\\begin{example}\\label{ex:liewhite} The computations from the previous section, together with the above theorem and the fact that $\\sL ie\\circ\\Oh=(\\sC om\\bullet\\Oh^!)^!$, imply for instance $\\op{Lie}_\\circ(\\sA ss)=\\op{Lie}_\\circ(pre\\sL ie)=\\sM ag_{1,0,0}$, $\\op{Lie}_\\circ(\\sL ie)=\\sM ag_{0,1,0}$, $\\op{Lie}_\\circ(\\sP ois)=\\sL ie+\\sM ag_{0,1,0}$, $\\op{Lie}_\\circ(\\sP erm)=\\sL eib$, $\\op{Lie}_\\circ(\\sZ inb)=pre\\sL ie$. To illustrate the latter, we show that the (right) pre-Lie relation holds in the tensor product $L\\otimes V$ of a generic Lie algebra $(L,[-,-])$ and a generic $\\sZ inb$-algebra $(V,\\cdot)$, equipped with the tensor product operation $\\star:=[-,-]\\otimes\\cdot$. The associator of $\\star$ is given by $A_\\star(l\\ten x,m\\ten y,n\\ten z)=[[l,m],n]\\ten (x\\cdot y)\\cdot z\\,-\\,[l,[m,n]]\\ten x\\cdot(y\\cdot z)$: to show that this is graded symmetric in the last two arguments, we compute\n\t\\[ [[l,m],n]\\ten (x\\cdot y)\\cdot z\\,-\\,[l,[m,n]]\\ten x\\cdot(y\\cdot z)\\,-\\,[[l,n],m]\\ten (x\\cdot z)\\cdot y\\,+\\,[l,[n,m]]\\ten x\\cdot(z\\cdot y)  = \\]\n\t\\[ =[l,[m,n]]\\ten\\left( (x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-x\\cdot(z\\cdot y) \\right)\\,+\\,[[l,n],m]\\ten\\left( (x\\cdot y)\\cdot z -(x\\cdot z)\\cdot y \\right)\\:=\\:0. \\] \n\t\n\n\t\n\t\n\\end{example}\n\nWe are finally ready to complete the proof of Theorem \\ref{th:black}: this will follow from the following theorem and theorems \\ref{th:asswhite}, \\ref{th:permwhite}, \\ref{th:liewhite}.\n\\begin{theorem}\\label{th:adj}  The following $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$, $\\xymatrix{\\op{Com}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Lie}_\\circ(-)\\ar@<2pt>[l]}$, $\\xymatrix{\\op{preLie}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Perm}_\\circ(-)\\ar@<2pt>[l]}$ are pairs of adjoint functors.\n\t\\end{theorem}\n\n\\begin{proof} We consider the case of $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$ in detail.\n\t\nSo far we used overlapping notations for the generating sets of operations of $\\op{Ass}_\\bullet(\\Oh)$ and $\\op{Ass}_\\circ(\\Oh)$: this isn't practical to prove adjointness, thus, only for this proof, we will have to change notations; moreover, we will use slightly different generating sets than the ones we used before. We consider first the case of a non-symmetric operation $\\cdot_i$ of $\\Oh$. Corresponding to $\\cdot_i$ there are two generating operations of $\\op{Ass}_\\bullet(\\Oh)$ which we denote by $\\underline{\\cdot_i}$ and $\\overline{\\cdot_i}$ respectively: we use the generating set of operations from Remark \\ref{rem:curlyeqprec}, explicitly, $\\underline{\\cdot_i}$ and $\\overline{\\cdot_i}$ are defined by the formulas $_x\\to\\cdot_i\\xrightarrow{y}\\:=\\:\\xrightarrow{x\\underline{\\cdot_i} y}\\:=\\:\\xrightarrow{x}\\cdot_i\\to_y$ and $\\to_x\\cdot_i\\xrightarrow{y}\\:=\\:\\xrightarrow{x\\overline{\\cdot_i}y}\\:=\\:\\xrightarrow{x}\\cdot_i\\,_y\\to$ (with the previous notations from remark \\ref{rem:curlyeqprec} we have $\\underline{\\cdot_i}=\\prec_i$ and $\\overline{\\cdot_i}=\\curlyeqsucc_i=-\\succ_i^{op}$). Likewise, corresponding to $\\cdot_i$ there are two generating operations of $\\op{Ass}_\\circ(\\Oh)$ which we denote by $\\cup\\otimes\\cdot_i$ and $\\cup^{op}\\otimes\\cdot_i$ respectively: explicitly, these are defined by the formulas $_x\\to(\\cup\\otimes\\cdot_i)\\xrightarrow{y}=\\xrightarrow{x\\cdot_i y}=\\xrightarrow{x}(\\cup\\otimes\\cdot_i)\\to_y$ and $\\to_x(\\cup^{op}\\otimes\\cdot_i)\\xrightarrow{y}=\\xrightarrow{x\\cdot_i y}=\\xrightarrow{x}(\\cup^{op}\\otimes\\cdot_i)\\,_y\\to$ (with the previous notations from this section we have $\\cup\\otimes\\cdot_i=\\prec_i$ and $\\cup^{op}\\otimes\\cdot_i=\\succ_i^{op}$). Finally, corresponding to $\\cdot_i$ there are four generating operations of the operad $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$, which we denote by $\\cup\\otimes\\underline{\\cdot_i}$, $\\cup^{op}\\otimes\\underline{\\cdot_i}$, $\\cup\\otimes\\overline{\\cdot_i}$ and $\\cup^{op}\\otimes\\overline{\\cdot_i}$ respectively, and four generating operations of $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$, which we denote by $\\underline{\\cup\\otimes\\cdot_i}$, $\\overline{\\cup\\otimes\\cdot_i}$, $\\underline{\\cup^{op}\\otimes\\cdot_i}$ and $\\overline{\\cup^{op}\\otimes\\cdot_i}$ respectively. Similarly, to a generating symmetric operation $\\bullet_j$ of $\\Oh$ correspond a generating (non-symmetric) operation $\\underline{\\bullet_j}$ of $\\op{Ass}_\\bullet(\\Oh)$ and a generating (non-symmetric) operation $\\cup\\otimes\\bullet_j$ of $\\op{Ass}_\\circ(\\Oh)$, as well as two generating (non-symmetric) operations $\\cup\\otimes\\underline{\\bullet_j}$, $\\cup^{op}\\otimes\\underline{\\bullet_j}$ of $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$ and two generating (non-symmetric) operations $\\underline{\\cup\\otimes\\bullet_j}$, $\\overline{\\cup\\otimes\\bullet_j}$ of $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$. Finally, to a generating antisymmetric operation $[-,-]_k$ of $\\Oh$ correspond generating (non-symmetric) operations $\\underline{[-,-]_k}$ and $\\cup\\otimes[-,-]_k$ of $\\op{Ass}_\\bullet(\\Oh)$ and $\\op{Ass}_\\circ(\\Oh)$ respectively, as well as generating operations $\\cup\\otimes\\underline{[-,-]_k}$, $\\cup^{op}\\otimes\\underline{[-,-]_k}$ of $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$ and $\\underline{\\cup\\otimes[-,-]_k}$, $\\overline{\\cup\\otimes[-,-]_k}$ of $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$ respectively.\n\t\nHaving established the previous notations, we will prove the theorem by explicitly exihibiting the unit $\\varepsilon_\\Oh:\\Oh\\to\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$ and the counit $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ of the adjunction. The construction of the counit is implicit in the very definition of the functors $\\op{Ass}_\\circ(-)$ and $\\op{Ass}_\\bullet(-)$: given an $\\Oh$-algebra $V$, the tensor product operations induce a local dg $\\op{Ass}_\\circ(\\Oh)$-algebra structure on the complex of $V$-valued cochains $C^*(\\Delta_1;V)=C^*(\\Delta_1;\\mathbb{K})\\otimes V$, but by definition this is the same as an $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$-algebra structure on the space $V$. Unraveling the definitions, we find that the counit $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ is explicitly given by \n\\[ \\underline{\\cup\\otimes\\cdot_i},\\:\\overline{\\cup^{op}\\otimes\\cdot_i}\\to\\cdot_i,\\qquad\\overline{\\cup\\otimes\\cdot_i},\\: \\underline{\\cup^{op}\\otimes\\cdot_i}\\to0, \\] \n\\[ \\underline{\\cup\\otimes\\bullet_j}\\to\\bullet_j,\\qquad\\overline{\\cup\\otimes\\bullet_j}\\to0,\\qquad\\underline{\\cup\\otimes[-,-]_k}\\to[-,-]_k,\\qquad\\overline{\\cup\\otimes[-,-]_k}\\to0. \\]\nAs already remarked, this defines a morphism of operads $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ by construction of the functors $\\op{Ass}_\\circ(-)$ and $\\op{Ass}_\\bullet(-)$. It remains to define the unit $\\varepsilon_\\Oh:\\Oh\\to\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$, this is explicitly given by\n\\[\\cdot_i\\to \\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i}, \\qquad\\bullet_j\\to\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op},\\qquad[-,-]_k\\to\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op}. \\]\nWe have to show that this is a morphism of operads. Given a generic $\\op{Ass}_\\bullet(\\Oh)$-algebra $(V,\\underline{\\cdot_i},\\overline{\\cdot_i},\\underline{\\bullet_j},\\underline{[-,-]_k})$, the complex $C^*(\\Delta_1;V)$ carries both an $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$-algebra structure via the tensor product operations and an $\\Oh$-algebra structure $(C^*(\\Delta_1;V),\\cdot_i,\\bullet_j,[-,-]_k)$ by definition of $\\op{Ass}_\\bullet(-)$. An easy verification shows $\\alpha(\\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i})\\beta=\\alpha\\cdot_i\\beta$ for all $\\alpha,\\beta\\in C^*(\\Delta_1;V)$. For instance,\n\\[ \\to_x (\\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i}) \\xrightarrow{y}=\\to_x(\\cup^{op}\\otimes\\overline{\\cdot_i})\\xrightarrow{y}=\\xrightarrow{x\\overline{\\cdot_i} y}=\\to_x\\cdot_i\\xrightarrow{y}.\\]\nSimilarly, one verifies $\\alpha(\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op})\\beta=\\alpha\\bullet_j\\beta$ and $\\alpha(\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op})\\beta=[\\alpha,\\beta]_k$ for all $\\alpha,\\beta\\in C^*(\\Delta_1;V)$. Finally, given a relation $R(-,-,-)=0$ of $\\Oh$, this induces a relation $R'(-,-,-)=0$ of $\\op{Ass}_\\bullet(\\Oh)$ as in Lemma \\ref{lem:rel}\n\\[  R(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{R'(x,y,z)}=0.\\]\nOn the other hand, by the previous considerations $R(-,-,-)$ in the left hand side can be computed equivalently either in the products $\\cdot_i,\\bullet_j,[-,-]_k$ of the $\\Oh$-algebra structure or in the products $\\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i}, \\:\\:\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op},\\:\\:\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op}$ of the $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$-algebra structure, which shows that $\\varepsilon_\\Oh$ sends an $\\Oh$-algebra relation to an $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$-algebra relation, and is thus a well defined morphism of operads.\n\nTo complete the proof, it remains to show that $\\varepsilon_{-}$ and $\\mu_{-}$ satisfy the conditions to be the unit and the counit of an adjunction. More precisely, we have to show that the compositions\n\\[ \\op{Ass}_\\bullet(\\Oh)\\xrightarrow{\\op{Ass}_\\bullet(\\varepsilon_\\Oh)}\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh)))\\xrightarrow{\\mu_{\\op{Ass}_\\bullet(\\Oh)}}\\op{Ass}_\\bullet(\\Oh),  \\]\n\\[ \\op{Ass}_\\circ(\\Oh)\\xrightarrow{\\varepsilon_{\\op{Ass}_\\circ(\\Oh)}}\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh)))\\xrightarrow{\\op{Ass}_\\circ(\\mu_\\Oh)}\\op{Ass}_\\circ(\\Oh),\\]\nare the respective identities. We consider the first one: notice that given a morphism $f:\\Oh\\to\\sP$ of operads, the morhism $\\op{Ass}_\\bullet(f)$ is defined by $\\op{Ass}_\\bullet(f)(\\underline{\\cdot_i})=\\underline{f(\\cdot_i)}$, $\\op{Ass}_\\bullet(f)(\\overline{\\cdot_i})=\\overline{f(\\cdot_i)}$, $\\op{Ass}_\\bullet(f)(\\underline{\\bullet_j})=\\underline{f(\\bullet_j)}$ and $\\op{Ass}_\\bullet(f)(\\underline{[-,-]_k})=\\underline{f([-,-]_k)}$. Now it is easy to compute the first composition using the previous formulas (for a non-symmetric operation $\\#$ we notice that $\\underline{(\\#^{op})}=(\\overline{\\#})^{op}$, in the following computation we apply this for $\\#=\\cup\\otimes\\underline{\\bullet_j}$ and $\\#=\\cup\\otimes\\underline{[-,-]_k}$)\n\\[ \\underline{\\cdot_i}\\to\\underline{\\cup\\otimes\\underline{\\cdot_i}}+\\underline{\\cup^{op}\\otimes\\overline{\\cdot_i}}\\to\\underline{\\cdot_i}+0,\\qquad\\overline{\\cdot_i}\\to\\overline{\\cup\\otimes\\underline{\\cdot_i}}+\\overline{\\cup^{op}\\otimes\\overline{\\cdot_i}}\\to0+\\overline{\\cdot_i},\\]\\[ \\underline{\\bullet_j}\\to\\underline{\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op}}=\\underline{\\cup\\otimes\\underline{\\bullet_j}}+(\\overline{\\cup\\otimes\\underline{\\bullet_j}})^{op}\\to\\underline{\\bullet_j}+0,  \\]\\[ \\underline{[-,-]_k}\\to\\underline{\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op}}=\\underline{\\cup\\otimes\\underline{[-,-]_k}}-(\\overline{\\cup\\otimes\\underline{[-,-]_k}})^{op}\\to\\underline{[-,-]_k}-0.  \\]\nNext we consider the second composition. We have $\\op{Ass}_\\circ(f)(\\cup\\otimes\\cdot_i)=\\cup\\otimes f(\\cdot_i)$, $\\op{Ass}_\\circ(f)(\\cup^{op}\\otimes \\cdot_i)=\\cup^{op}\\otimes f(\\cdot_i)$, and similarly for the other cases. As desired, the second composition is\n\\[\\cup\\otimes\\cdot_i\\to\\cup\\otimes\\underline{\\cup\\otimes\\cdot_i} +\\cup^{op}\\otimes\\overline{\\cup\\otimes\\cdot_i}\\to\\cup\\otimes\\cdot_i+0,\\]\\[ \\cup^{op}\\otimes\\cdot_i\\to\\cup\\otimes\\underline{\\cup^{op}\\otimes\\cdot_i} +\\cup^{op}\\otimes\\overline{\\cup^{op}\\otimes\\cdot_i}\\to0+\\cup^{op}\\otimes\\cdot_i,      \\]\n\\[\\cup\\otimes\\bullet_j\\to\\cup\\otimes\\underline{\\cup\\otimes\\bullet_j} +\\cup^{op}\\otimes\\overline{\\cup\\otimes\\bullet_j}\\to\\cup\\otimes\\bullet_j+0,\\]\n\\[ \\cup\\otimes[-,-]_k\\to\\cup\\otimes\\underline{\\cup\\otimes[-,-]_k} +\\cup^{op}\\otimes\\overline{\\cup\\otimes[-,-]_k}\\to\\cup\\otimes[-,-]_k+0.\\] \n\nThe remaining cases of $\\xymatrix{\\op{Com}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Lie}_\\circ(-)\\ar@<2pt>[l]}$ and  $\\xymatrix{\\op{preLie}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Perm}_\\circ(-)\\ar@<2pt>[l]}$ are proved by the same argument: in the former, we notice that we may identify $\\op{Com}_\\bullet(\\op{Lie}_\\circ(\\Oh))$, $\\Oh$ and $\\op{Lie}_\\circ(\\op{Com}_\\bullet(\\Oh))$ with quotients of the same free operad $(\\sM ag_{p,q,r},\\cdot_i,\\bullet_j,[-.-]_k)$ in such a way that the unit and the counit are the identities on the generating operations.\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe work over the complex number field $\\C$. A K3 surface is a compact simply connected, in the classical topology, smooth complex surface with nowhere vanishing global holomorphic $2$-form. An Enriques surface is a smooth complex surface which is isomorphic to a non-trivial \\'etale quotient of a K3 surface. The quotient map is necessarily of degree two and every Enriques surface is projective.\n\nOur main theorem is the following:\n\n\\begin{theorem}\\label{thm1}\n\nThere is a smooth projective surface $Y$ birational to some Enriques surface such that ${\\rm Aut}\\, (Y)$ is not finitely generated.\n\n\\end{theorem}\n\n\\begin{remark}\\label{rem1} Let $Y$ be a smooth projective surface birational to an Enriques surface $S$ and let $\\tilde{S}$ be the universal covering K3 surface of $S$.\n\\begin{enumerate}\n\\item ${\\rm Aut}^0(S) = \\{\\id_S\\}$, i.e., ${\\rm Aut}\\,(S)$ is discrete. This is because $H^0(S, T_{S}) = 0$ by $H^0(\\tilde{S}, T_{\\tilde{S}}) = 0$. On the other hand, ${\\rm Aut}\\, (S)$ itself is finitely generated. This is because, \"up to finite kernel and cokernel\", ${\\rm Aut}\\, (S)$ is isomorphic to the quotient group ${\\rm O}({\\rm NS}\\, (S)\/{\\rm torsion})\/W(S)$ of the arithmetic subgroup ${\\rm O}({\\rm NS}\\, (S)\/{\\rm torsion})$ by the Weyl group $W(S)$ generated by the reflections corresponding to the smooth rational curves on $S$ (see \\cite[Theorem]{Do84} for a more precise statement) and ${\\rm O}({\\rm NS}\\, (S)\/{\\rm torsion})$ is finitely generated by a general result on arithmetic subgroups of linear algebraic groups \\cite[Theorem 6.12]{BH62} (See also Theorem \\ref{thm11}). So, $S$ itself is not a candidate surface in Theorem \\ref{thm1}.\n\\item $S$ is the unique minimal model of $Y$ up to isomorphisms. So, we have a birational morphism $\\nu : Y \\to S$, which is a finite composition of blowings up at points. Therefore, we have $H^0(Y, T_Y) = 0$ and also an injective group homomorphism\n$${\\rm Aut}\\, (Y) \\subset {\\rm Bir}\\, (S) = {\\rm Aut}\\, (S)\\,\\, ; \\,\\, f \\mapsto \\nu \\circ f \\circ \\nu^{-1}\\,\\, ,$$\nvia $\\nu$. Note that a subgroup of a finitely generated group is not necessarily finitely generated (cf. Theorem \\ref{thm11}).\n\\end{enumerate}\n\\end{remark}\n\nWe show Theorem \\ref{thm1} by constructing $Y$ explicitly. Our construction is inspired by \\cite{Le18} for 6-dimensional examples, \\cite{DO19} and \\cite{Og19} for exmaples birational to K3 surfaces and also \\cite{Mu10} for his new construction of an Enriques surface with a numerically trivial involution, which is missed in an earlier paper \\cite{MN84}. As usual in study of Enriques surfaces, our construction is more involved than \\cite{DO19} and \\cite{Og19} for K3 surfaces, whereas the basic strategy of the construction is essentially the same.\n\nAs in \\cite{DO19} and \\cite{Og19}, the following purely group theoretical theorem (see eg. \\cite{Su82}) will be frequently used in this paper.\n\n\\begin{thm}\\label{thm11}\nLet $G$ be a group and $H \\subset G$ a subgroup of $G$. Assume that $H$ is of finite index, i.e., $[G :H] < \\infty$. Then, the group $H$\n is finitely generated if and only if $G$ is finitely generated.\n\\end{thm}\n\nIn this paper, for a variety $V$ we denote\nthe group of biregular automorphisms of $V$ and the group of birational automorphisms of $V$ by\n$${\\rm Aut}\\, (V),\\,\\, \\,\\, {\\rm Bir}\\, (V)$$\nrespectively, and for closed subsets  $W_1$, $W_2$, $\\ldots$, $W_n$ of $V$ the decomposition group and the inertia group by\n$${\\rm Dec}\\, (W_1, \\ldots, W_n) := {\\rm Dec}\\, (V, W_1, \\ldots, W_n) := \\{ f \\in {\\rm Aut}\\,(V)\\,\\, |\\,\\, f(W_i) = W_i (\\forall i)\\}\\,\\, ,$$\n$${\\rm Ine}\\, (W_1, \\ldots, W_n) := {\\rm Ine}\\, (V, W_1, \\ldots, W_n) := \\{ f \\in {\\rm Dec}\\, (V, W_1, \\ldots, W_n)\\, |\\, f_{W_i} = \\id_{W_i} (\\forall i)\\}.$$\nFor basic properties of surfaces, we refer to \\cite{BHPV04} and \\cite{CD89}.\n\nWe believe that large part of our construction should work also in positive characteristic $\\ge 3$ if the based field is carefully chosen (see e.g. for some sensitive aspect of the base field in positive characteristic \\cite{Og19}). We leave it to the readers who are interested in this generalization.\n\n\\medskip\\noindent\n{\\bf Acknowledgements.} We would like to thank Professor Jun-Muk Hwang for organizing one day workshop at KIAS,\nwhich made our collaboration possible, and Professor Yuya Matsumoto for very kind help concerning Figure \\ref{fig1}.\n\n\\section{Preliminaries}\\label{sect2}\n\nIn this section, first we fix some basic notation concerning a Kummer surface ${\\rm Km}\\, (E \\times F)$ of the product of two non-isogenous elliptic curves. Our notation follows \\cite{DO19} and \\cite{Og19}. Then we recall Mukai's construction of Enriques surfaces with a numerically trivial involution of odd type \\cite{Mu10} arising from ${\\rm Km}\\, (E \\times F)$. His construction is very crucial in our construction in Section \\ref{sect3}.\n\n\\subsection{Kummer surfaces of product type}\\label{sub1}\n\nLet $E$ be the elliptic curve defined by the Weierstrass equation\n$$y^2 = x(x-1)(x-t)\\,\\, ,$$\nand $F$ be the elliptic curve defined by the Weierstrass equation\n$$v^2 = u(u-1)(u-s)\\,\\, .$$\nNote that $E\/\\langle -1_E \\rangle = \\P^1$, the associated quotient map $E \\to \\P^1$ is given by $(x,y)\\mapsto x$ and the points $0$, $1$, $t$ and $\\infty$ of $\\P^1$ are exactly the branch points of this quotient map. The same holds for $F$ if we replace $t$ by $s$.\n\nThroughout this paper, we make the following assumption:\n\n\\begin{assumption}\\label{ass21} $t$ and $s$ are transcendental over $\\Q$ and the two elliptic curves\n$E$ and $F$ are not isogenous.\n\\end{assumption}\nAssumption \\ref{ass21} is satisfied if $s \\in \\C$ is generic with respect to a transcendental number $t \\in \\C$.\n\nLet\n$$X := {\\rm Km} (E \\times F)$$\nbe the Kummer K3 surface accociated to the product abelian surface $E \\times F$, that is, the minimal resolution of the quotient surface $E \\times F\/\\langle -1_{E\\times F} \\rangle$. We write $H^0(X, \\Omega_X^2) = \\C \\omega_X$. Since $E$ and $F$ are not isogenous, the Picard number $\\rho(X)$ of $X$ is $18$ (See eg.\\cite[Prop. 1 and Appendix]{Sh75}).\n\nLet $\\{a_i\\}_{i=1}^{4}$ and $\\{b_i\\}_{i=1}^{4}$ be the $2$-torsion subgroups of $F$ and $E$ respectively. Then $X$ contains 24 smooth rational curves which form the so called double Kummer pencil on $X$, as in Figure \\ref{fig1}. Here smooth rational curves $E_i$, $F_i$ ($1 \\le i \\le 4$) are arising from the elliptic curves $E \\times \\{a_i\\}$, $\\{b_i\\} \\times F$ on $E \\times F$. Smooth rational curves $C_{ij}$ ($1\\le i,j \\le 4$) are the exceptional curves over the $A_1$-singular points of the quotient surface $E \\times F\/\\langle -1_{E\\times F} \\rangle$. Throughout this paper, we will freely use the names of curves in Figure \\ref{fig1}.\n\n\\begin{figure}\n\n\\unitlength 0.1in\n\\begin{picture}(25.000000,24.000000)(-1.000000,-23.500000)\n\\put(4.500000, -22.000000){\\makebox(0,0)[rb]{$F_1$}\n\\put(9.500000, -22.000000){\\makebox(0,0)[rb]{$F_2$}\n\\put(14.500000, -22.000000){\\makebox(0,0)[rb]{$F_3$}\n\\put(19.500000, -22.000000){\\makebox(0,0)[rb]{$F_4$}\n\\put(0.250000, -18.500000){\\makebox(0,0)[lb]{$E_1$}\n\\put(0.250000, -13.500000){\\makebox(0,0)[lb]{$E_2$}\n\\put(0.250000, -8.500000){\\makebox(0,0)[lb]{$E_3$}\n\\put(0.250000, -3.500000){\\makebox(0,0)[lb]{$E_4$}\n\\put(6.000000, -16.000000){\\makebox(0,0)[lt]{$C_{11}$}\n\\put(6.000000, -11.000000){\\makebox(0,0)[lt]{$C_{12}$}\n\\put(6.000000, -6.000000){\\makebox(0,0)[lt]{$C_{13}$}\n\\put(6.000000, -1.000000){\\makebox(0,0)[lt]{$C_{14}$}\n\\put(11.000000, -16.000000){\\makebox(0,0)[lt]{$C_{21}$}\n\\put(11.000000, -11.000000){\\makebox(0,0)[lt]{$C_{22}$}\n\\put(11.000000, -6.000000){\\makebox(0,0)[lt]{$C_{23}$}\n\\put(11.000000, -1.000000){\\makebox(0,0)[lt]{$C_{24}$}\n\\put(16.000000, -16.000000){\\makebox(0,0)[lt]{$C_{31}$}\n\\put(16.000000, -11.000000){\\makebox(0,0)[lt]{$C_{32}$}\n\\put(16.000000, -6.000000){\\makebox(0,0)[lt]{$C_{33}$}\n\\put(16.000000, -1.000000){\\makebox(0,0)[lt]{$C_{34}$}\n\\put(21.000000, -16.000000){\\makebox(0,0)[lt]{$C_{41}$}\n\\put(21.000000, -11.000000){\\makebox(0,0)[lt]{$C_{42}$}\n\\put(21.000000, -6.000000){\\makebox(0,0)[lt]{$C_{43}$}\n\\put(21.000000, -1.000000){\\makebox(0,0)[lt]{$C_{44}$}\n\\special{pa 500 2200\n\\special{pa 500 0\n\\special{fp\n\\special{pa 1000 2200\n\\special{pa 1000 0\n\\special{fp\n\\special{pa 1500 2200\n\\special{pa 1500 0\n\\special{fp\n\\special{pa 2000 2200\n\\special{pa 2000 0\n\\special{fp\n\\special{pa 0 1900\n\\special{pa 450 1900\n\\special{fp\n\\special{pa 550 1900\n\\special{pa 950 1900\n\\special{fp\n\\special{pa 1050 1900\n\\special{pa 1450 1900\n\\special{fp\n\\special{pa 1550 1900\n\\special{pa 1950 1900\n\\special{fp\n\\special{pa 0 1400\n\\special{pa 450 1400\n\\special{fp\n\\special{pa 550 1400\n\\special{pa 950 1400\n\\special{fp\n\\special{pa 1050 1400\n\\special{pa 1450 1400\n\\special{fp\n\\special{pa 1550 1400\n\\special{pa 1950 1400\n\\special{fp\n\\special{pa 0 900\n\\special{pa 450 900\n\\special{fp\n\\special{pa 550 900\n\\special{pa 950 900\n\\special{fp\n\\special{pa 1050 900\n\\special{pa 1450 900\n\\special{fp\n\\special{pa 1550 900\n\\special{pa 1950 900\n\\special{fp\n\\special{pa 0 400\n\\special{pa 450 400\n\\special{fp\n\\special{pa 550 400\n\\special{pa 950 400\n\\special{fp\n\\special{pa 1050 400\n\\special{pa 1450 400\n\\special{fp\n\\special{pa 1550 400\n\\special{pa 1950 400\n\\special{fp\n\\special{pa 200 2000\n\\special{pa 600 1600\n\\special{fp\n\\special{pa 200 1500\n\\special{pa 600 1100\n\\special{fp\n\\special{pa 200 1000\n\\special{pa 600 600\n\\special{fp\n\\special{pa 200 500\n\\special{pa 600 100\n\\special{fp\n\\special{pa 700 2000\n\\special{pa 1100 1600\n\\special{fp\n\\special{pa 700 1500\n\\special{pa 1100 1100\n\\special{fp\n\\special{pa 700 1000\n\\special{pa 1100 600\n\\special{fp\n\\special{pa 700 500\n\\special{pa 1100 100\n\\special{fp\n\\special{pa 1200 2000\n\\special{pa 1600 1600\n\\special{fp\n\\special{pa 1200 1500\n\\special{pa 1600 1100\n\\special{fp\n\\special{pa 1200 1000\n\\special{pa 1600 600\n\\special{fp\n\\special{pa 1200 500\n\\special{pa 1600 100\n\\special{fp\n\\special{pa 1700 2000\n\\special{pa 2100 1600\n\\special{fp\n\\special{pa 1700 1500\n\\special{pa 2100 1100\n\\special{fp\n\\special{pa 1700 1000\n\\special{pa 2100 600\n\\special{fp\n\\special{pa 1700 500\n\\special{pa 2100 100\n\\special{fp\n\\end{picture\n \\caption{Curves $E_i$, $F_j$ and $C_{ij}$}\n \\label{fig1}\n\\end{figure}\n\nWe denote the unique point $E_j \\cap C_{ij}$ by $P_{ij}$ and the unique point $F_i \\cap C_{ij}$ by $P_{ij}'$. We may and do adapt $x$ (resp. $u$) the affine coordinate of $E_j$ and $F_i$ so that\n$$P_{1j} = 1\\,\\, ,\\,\\, P_{2j} = t\\,\\, ,\\,\\, P_{3j} = \\infty\\,\\, ,\\,\\, P_{4j} = 0$$\non $E_j$ with respect to the coordinate $x$ and\n$$P_{i1}' = 1\\,\\, ,\\,\\, P_{i2}' = s\\,\\, ,\\,\\, P_{i3}' = \\infty\\,\\, ,\\,\\,\nP_{i4}' = 0$$\non $F_i$ with respect to the coordinate $u$.\n\nSet\n$$\\theta := [(1_E, -1_F)] = [(-1_E, 1_F)] \\in {\\rm Aut}\\, (X)\\,\\, .$$\nThen $\\theta$ is an involution of $X$, i.e., an automorphism of $X$ of order $2$. The following lemma was proved in \\cite[Lemmas (1.3), (1.4)]{Og89} (See also \\cite{Og19}).\n\n\\begin{lemma}\\label{lem21}\n\\begin{enumerate}\n\\item $\\theta^* = \\id$ on ${\\rm Pic}\\, (X)$ and $\\theta^* \\omega_X = -\\omega_X$.\\item $f \\circ \\theta = \\theta \\circ f$ for all $f \\in {\\rm Aut}\\,(X)$.\n\\item Let $X^{\\theta}$ be the fixed locus of $\\theta$. Then $X^{\\theta} = \\cup_{i=1}^{4} (E_i \\cup F_i)$.\n\\item ${\\rm Aut}\\,(X) = {\\rm Dec}\\, (X, \\cup_{i=1}^{4} (E_i \\cup F_i))$.\n\\end{enumerate}\n\\end{lemma}\n\n\\subsection{Enriques surfaces with a numerically trivial involution of odd type}\\label{sub2}\n\nWe employ the same notation as in Subsection \\ref{sub1}. By Assumption \\ref{ass21}, the two ordered sets\n$$\\{P_{i1}', P_{i2}', P_{i3}', P_{i4}'\\}\\subset F_i\\cong \\P^1\\,\\, ,\\,\\, \\{P_{1j}, P_{2j}, P_{3j}, P_{4j}\\} \\subset E_j\\cong \\P^1$$\nare not projectively equivalent, i.e., not in the same orbit of the action of ${\\rm Aut}\\, (\\P^1) = {\\rm PGL}\\, (2, \\C)$ on $\\P^1$.\n\nWe recall the construction of\n Mukai \\cite{Mu10} for our $X = {\\rm Km}\\, (E \\times F)$.\n\nLet $$T := X\/\\langle \\theta \\rangle$$ be the quotient surface and $$q : X \\to T$$ be the quotient morphism. Then $T$ is a smooth projective surface such that $q(C_{ij})$ ($1 \\le i, j \\le 4$) is a $(-1)$-curve, i.e., a smooth rational curve with self interesection number $-1$. Then $T$ is obtained by the blowings up of $\\P^1 \\times \\P^1$ at the 16 points $p_{ij}$ ($1 \\le i, j \\le 4$) of $\\P^1 \\times \\P^1$. We may assume that $p_{ij}$ is the image of $C_{ij}$ under the composite morphism\n$$X \\to T \\to \\P^1 \\times \\P^1\\,\\, .$$\nLet us consider the Segre embedding\n$$\\P^1 \\times \\P^1 \\subset \\P^3\\,\\, ,$$\nand identify $\\P^1 \\times \\P^1$ with a smooth quadric surface\n$Q$ in $\\P^3$. Since the four points $p_{11}, p_{22}, p_{33}, p_{44} \\in Q$  are not coplaner in $\\P^3$, we may adjust coordinates $[x_1 : x_2 : x_3 : x_4]$ of $\\P^3$ so that the $4$ points are\n$$p_{11}=[1:0:0:0],\\,\\, \\, p_{22}= [0:1:0:0],\\,\\, \\, p_{33}=[0:0:1:0],\\,\\, \\, p_{44}=[0:0:0:1]\\,\\, .$$\nThen the equation of $Q$ is of the form\n$$\\alpha_1x_2x_3 + \\alpha_2x_1x_3 + \\alpha_3x_1x_2 + (x_1+x_2+x_3)x_4 = 0$$\nfor some complex numbers $\\alpha_i$ satisfying non-degeneracy condition.\nThen the Cremona involution of $\\P^3$\n$$\\tilde{\\tau}' : [x_1 :x_2 : x_3 : x_4] \\mapsto [\\frac{\\alpha_1}{x_1} : \\frac{\\alpha_2}{x_2} : \\frac{\\alpha_3}{x_3} : \\frac{\\alpha_1\\alpha_2\\alpha_3}{x_4}]$$ satisfies\n $\\tilde{\\tau}'(Q) = Q$, hence induces a birational automorphism of $Q$\n$$\\tau' := \\tilde{\\tau}'|_{Q} \\in {\\rm Bir}\\, (Q)\\,\\, .$$\nLet $I(\\tau')$ be the indeterminacy locus of $\\tau'$.\nBy the definition of $\\tau'$, we readily check the following (\\cite[Section 2]{Mu10}):\n\\begin{lemma}\\label{lem22}\n\\begin{enumerate}\n\\item $I(\\tau') = \\{p_{ii}\\}_{i=1}^{4}$ and $\\tau'$ contracts the (smooth) conic curve $C'_{i} := Q \\cap (x_i= 0)$ to $p_{ii}$.\n\\item $\\tau'$ interchanges the two lines through $p_{ii}$ for each $i=1$, $2$, $3$, $4$.\n\\item $\\mu^{-1}\\circ \\tau' \\circ \\mu \\in {\\rm Aut}\\, (B)$, where $\\mu : B \\to \\P^1 \\times \\P^1$ is the blowing up at the four points $p_{ii}$ ($1 \\le i \\le 4$).\n\\end{enumerate}\n\\end{lemma}\n\nBy the property (2), $\\tau'(p_{ij}) = p_{ji}$ if $1 \\le i \\not= j \\le 4$. Therefore $\\tau'$ lifts to $$\\tau \\in {\\rm Aut}\\, (T).$$ Since $q : X \\to T$ is the finite double cover branched along the unique anti-bicanonical divisor $$\\sum_{i=1}^{4} (q(E_i)+q(F_i))\\in |-2K_T|,$$ it follows that $\\tau$ lifts to an involution $$\\epsilon \\in {\\rm Aut}\\, (X).$$ Apriori, there are exactly the two choices of the lifting $\\epsilon$; if we denote one lifting by $\\epsilon_0$ then the other is $\\epsilon_0 \\circ \\theta$. Recall that $\\theta^*\\omega_X = -\\omega_X$. Thus, we may and do choose the unique lift $\\epsilon$ with $\\epsilon^* \\omega_X = -\\omega_X$.\nSet\n$$Z := X\/\\langle \\epsilon \\rangle\\,\\, .$$\nand denote the quotient morphism by\n$$\\pi : X \\to Z\\,\\, .$$\nThe following discovery due to\nMukai \\cite[Proposition 2]{Mu10} is also crucial for us:\n\\begin{proposition}\\label{prop21}\nThe involution $\\epsilon$ acts on $X$ freely. In particular,\n$Z$ is an Enriques surface with a numerically trivial involution $\\theta_Z \\in {\\rm Aut}\\, (Z)$ induced from $\\theta \\in {\\rm Aut}\\, (X)$.\n\\end{proposition}\nSet $$C_i := \\epsilon (C_{ii})\\,\\, (i = 1,\\,2,\\,3,\\, 4).$$\nThen, $C_i$ is the proper transform of the curve $C_i'$ in Lemma \\ref{lem22} under the morphism\n$$X \\to T \\to B \\to \\P^1 \\times \\P^1 = Q\\,\\, .$$\n\\begin{corollary}\\label{cor21}\n\\begin{enumerate}\n\\item $\\epsilon (E_i) = F_i$, $\\epsilon(F_i) = E_{i}$ for all $i =1$, $2$, $3$, $4$.\n\\item $\\epsilon(C_{ij}) = C_{ji}$ for all $i$, $j$ such that $i \\not= j$.\n\\item $(C_i, E_i) = (C_i, F_i) = 1$, $(C_i, C_{ii}) = 0$, $(C_i, C_{kj}) = 0$\nfor all $i$, $j$, $k$ such that $k \\not= j$.\n\\item $(C_i, E_j) = (C_i, F_j) = 0$ for all $i$, $j$ such that $j \\not= i$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof} The assertions (1) and (2) follow from the description of $\\tau$. Then the assertions (3) and (4) follow from $\\epsilon(C_{ii}) = C_{i}$ and the assertions (1) and (2), except possibly $(C_i, C_{ii}) = 0$. The latter follows from the fact that the conic curve $C'_{i} \\subset Q$ that is contracted to $p_{ii}$ by $\\tau'$ does not pass through $p_{ii}$ (See Lemma \\ref{lem22} (1)).\n\n\\end{proof}\n\n\\section{Construction and proof of Theorem \\ref{thm1}}\\label{sect3}\n\nWe employ the same notation and the assumption (Assumption \\ref{ass21}) as in Section \\ref{sect2}. For instance,\n$$X = {\\rm Km}\\, (E \\times F)\\,\\, ,\\,\\, Z = X\/\\langle \\epsilon \\rangle\\,\\, ,\\,\\, \\pi : X \\to Z\\,\\, .$$\nWe also use the following notation for curves and points on the Enriques surface $Z$:\n$$H_j := \\pi(E_j)\\,\\, ,\n\\,\\, D_{ij} := \\pi(C_{ij})\\,\\, , \\,\\, Q_{ij} := \\pi(P_{ij})\\,\\, ,$$\nand via the isomorphism $\\pi|_{E_j} : E_j \\to H_j$, we also regard $x$ as the affine coordinate of $H_j$. Then $Q_{ij} \\in H_j$ and\n$$x(Q_{1j}) = 1\\,\\, ,\\,\\, x(Q_{2j}) = t\\,\\, ,\\,\\, x(Q_{3j}) = \\infty\\,\\, ,\\,\\, x(Q_{4j}) = 0\\,\\, .$$\nBy Corollary \\ref{cor21}, we have\n$$\\pi^{-1}(H_j) = E_j \\cup F_j$$\nfor each $j =1$, $2$, $3$, $4$ and\n$$\\pi^{-1}(D_{ij}) = C_{ij} \\cup C_{ji}\\,\\, ,\\,\\, \\pi^{-1}(Q_{ij}) = \\{P_{ij},\nP_{ji}'\\}$$\nif $i \\not= j$, while\n$$\\pi^{-1}(D_{ii}) = C_{ii} \\cup C_i\\,\\, ,\\,\\, \\pi^{-1}(Q_{ii}) = \\{P_{ii} \\cup P_{i}\\}\\,\\, ,$$\nagain for each $i$.\nHere $P_{i}$ is the unique intersection point of $C_i \\cap F_i$.\n\nLet $\\mu_1 : Z_1 \\to Z$ be the blowing up at the point $Q_{32} \\in H_2$, i.e., the blowing up at $\\infty$ under the coordinate $x$ of $H_2$. Let\n$$E_{\\infty} := \\P(T_{Z,Q_{32}}) \\simeq \\P^1$$\nbe the exceptional divisor of $\\mu_1$. We then choose three mutually different points on $\\P(T_{Z,Q_{32}})$, say $Q_{32k}$ ($k=1$, $2$, $3$). Let $\\mu_2 : Z_2 \\to Z_1$ be the blowings up of $Z_1$ at the three points $Q_{32k}$.\n\nOur main theorem is Theorem \\ref{thm31} below. Clearly, Theorem \\ref{thm1} follows from Theorem \\ref{thm31} by taking $Y = Z_2$:\n\\begin{theorem}\\label{thm31}\n${\\rm Aut}\\, (Z_2)$ is not finitely generated.\n\\end{theorem}\n\nIn the rest of this section, we prove Theorem \\ref{thm31}.\n\nWe denote\n$$\\mu := \\mu_1 \\circ \\mu_2 : Z_2 \\to Z_1 \\to Z\\,\\, .$$\nBy $E_{32k}$, we denote the exceptional curve over $Q_{32k}$ under $\\mu_2$ and\nby $E_{\\infty}'$ the proper transform of $E_{\\infty}$ under $\\mu_2$.\n\nFirst we reduce the proof to $Z$. For this, we recall that\n$${\\rm Aut}\\, (Z_2) \\subset {\\rm Aut}\\, (Z)$$\nvia $\\mu$ (See Remark \\ref{rem1}).\nWe define\n$${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}}) := \\{ f \\in {\\rm Dec}\\,(Z, Q_{32})\\,|\\, df|_{T_{Z, Q_{32}}} = \\id_{T_{Z, Q_{32}}}\\}\\,\\, .$$\n\n\\begin{proposition}\\label{prop31}\n\\begin{enumerate}\n\\item There is a subgroup $K$ of\n${\\rm Aut}\\, (Z_2)$ such that $[{\\rm Aut}\\, (Z_2) : K] < \\infty$, ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}}) \\subset K$ via $\\mu$ and $[K : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})] < \\infty$.\n\n\\item If ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated, then ${\\rm Aut}\\, (Z_2)$ is not finitely generated.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} First we show (1). By the canonical bundle formula, we have\n$$|2K_{Z_2}| = \\{2E_{\\infty}' + 4(E_{321} + E_{322} + E_{323})\\}\\,\\, .$$\nSince ${\\rm Aut}\\, (Z_2)$ preserves $|2K_{Z_2}|$, it follows that\n$${\\rm Aut}\\, (Z_2) = {\\rm Dec}\\,(Z_2, E_{\\infty}', E_{321} \\cup E_{322} \\cup E_{323})\\,\\, .$$\nTherefore, via $\\tau_2$, we have\n$${\\rm Aut}\\, (Z_2) = {\\rm Dec}\\,(Z_1, E_{\\infty}, \\{Q_{321}, Q_{322}, Q_{323}\\}) \\subset {\\rm Aut}\\, (Z_1)\\,\\, .$$\nThus, the group\n$$K :=\n{\\rm Dec}\\,(Z_1, E_{\\infty}, \\{Q_{321}\\}, \\{Q_{322}\\}, \\{Q_{323}\\})$$\nis a subgroup of ${\\rm Aut}\\, (Z_2)$ with $[{\\rm Aut}\\, (Z_2) : K] \\le 6 = |{\\rm Aut}_{{\\rm set}}\\,(\\{Q_{321}, Q_{322}, Q_{323}\\})|$.\n\nWe will show that $K$ satisfies the requirement.\n\nSince only $\\id_{\\P^1}$ is the automorphism of $\\P^1$ pointwisely fixes three points, it follows that\n$$K = {\\rm Ine}\\,(Z_1, E_{\\infty})\\,\\, .$$\nSince $E_{\\infty} = \\P(T_{Z, Q_{32}})$, we deduce that\n$$K = \\{f \\in {\\rm Dec}\\, (Z, Q_{32})\\,|\\, df|_{T_{Z, Q_{32}}} = \\alpha(f)\n\\id_{T_{Z, Q_{32}}}\\,\\, (\\exists\\alpha(f) \\in \\C^{\\times})\\} \\subset {\\rm Dec}\\, (Z, Q_{32})\\,\\, .$$\nObserve that if $df|_{T_{Z, Q_{32}}} = \\alpha(f) \\id_{T_{Z, Q_{32}}}$ for $f \\in K$, then\n$$(df \\wedge df)^{\\otimes 2}|_{(\\wedge^2 T_{Z, Q_{32}})^{\\otimes 2}} = \\alpha(f)^4 \\id_{(\\wedge^2 T_{Z, Q_{32}})^{\\otimes 2}}\\,\\, .$$\nSince the line bundle $(\\Omega_Z^2)^{\\otimes 2}$ admits a nowhere vanishing global section, it follows that $\\alpha(f)^4$ is in the image ${\\rm Im}\\, r_2$ of the bicanonical representation\n$$r_2 : {\\rm Aut}\\, (Z) \\to {\\rm GL}(H^0(Z, (\\Omega_Z^2)^{\\otimes 2})) \\simeq \\C^{\\times}$$\nof ${\\rm Aut}\\, (Z)$ (\\cite[Section 14]{Ue75}). Since ${\\rm Im}\\, r_2$\nis finite by \\cite[Theorem 14.10]{Ue75}, it follows that\n$\\{\\alpha(f)\\, |\\, f \\in K\\}$ is also finite.\nHence $[K : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})] < \\infty$ as well.\n\nLet us show (2). Recall Theorem \\ref{thm11}. Then, if ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated, then $K$\nis not finitely generated by $[K : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})] < \\infty$. Hence ${\\rm Aut}\\, (Z_2)$ is not finitely generated, again by $[{\\rm Aut}\\, (Z_2) : K] < \\infty$.\n\\end{proof}\n\nIn what follows, we will show that ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated. This is a problem on the Enriques surface $Z$.\n\n\\begin{lemma}\\label{lem31}\n\\begin{enumerate}\n\\item Let $f \\in {\\rm Dec}\\, (Z, Q_{32})$. Then $f(H_2) = H_2$, i.e., $f \\in {\\rm Dec}\\, (Z, H_2)$.\n\n\\item The differential maps $df|_{T_{Z, Q_{32}}}$ for all $f \\in {\\rm Dec}\\, (Z, Q_{32})$ are simultaneously diagonalizable.\n\n\\item Let $f \\in {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$. Then $f \\in {\\rm Dec}\\, (Z, H_2)$ by (1)\nand\n$$d(f|_{H_2})|_{T_{H_2, Q_{32}}} = \\id_{T_{H_2, Q_{32}}}$$\nfor the induced action.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $f \\in {\\rm Dec}\\, (Z, Q_{32})$. Then, the one of the two lifts of $f$, say $\\tilde{f}$, satisfies $\\tilde{f}(P_{32}) = P_{32}$. Therefore the result follows from the corresponding result on $X$ (see eg. \\cite{DO19}).\n\nFor the convenience of the readers, we recall the proof here from \\cite{DO19}. Since $\\tilde{f} \\in {\\rm Dec}(X, \\cup_{j=1}^{4} (E_j \\cup F_j))$ by Lemma \\ref{lem21} (4) and $E_{2}$ is the unique component of $\\cup_{j=1}^{4} (E_j \\cup F_j)$, containinig $P_{32}$, it follows that $\\tilde{f} \\in {\\rm Dec}\\,(X, E_2)$. This shows (1).\n\nBy Lemma \\ref{lem21} (1), (3), one has $\\theta(R) = R$ for any smooth rational curve $R$ on $X$\nand\n$$d(\\theta|_{E_2})_{P_{32}} = 1\\,\\, ,\\,\\, d(\\theta|_{C_{32}})_{P_{32}} = -1\\,\\, .$$\nIn particular,\n$$T_{X, P_{32}} = T_{E_2, P_{32}} \\oplus T_{C_{32}, P_{32}}\\,\\, .$$\nNote that $\\tilde{f}(E_2) = E_2$ as observed above. Let $C_{32}' := \\tilde{f}(C_{32})$. Then $P_{32} \\in C_{32}' \\simeq \\P^1$ and the induced action $\\theta|_{C_{32}'}$ satisfies\n$$d(\\theta|_{C_{32}'})_{P_{32}} = -1$$\nby Lemma \\ref{lem21} (1). Thus, $d\\tilde{f}|_{T_{X, P_{32}}}$ for all $\\tilde{f}$ preserve both $T_{E_2, P_{32}}$ and $T_{C_{32}, P_{32}}$. This implies (2).\n\nThe assertion (3) is now obvious.\n\\end{proof}\nRecall that for $Q \\in \\P^1$,\n$${\\rm Ine}\\, (\\P^1, Q, T_{\\P^1, Q}) := \\{f \\in {\\rm Ine}(\\P^1, Q)\\, |\\, df|_{T_{\\P^1, Q}} = \\id_{T_{\\P^1, Q}}\\} \\simeq (\\C, +)\\,\\, .$$\nHere $(\\C, +)$ is the additive group, in particular, an abelian group. The last isomorphism is given by\n$$\\C \\ni c \\mapsto (z \\mapsto z + c) \\in {\\rm Ine}\\, (\\P^1, Q, T_{\\P^1, Q})\\,\\, ,$$\nif we choose an affine coordinate $z$ of $\\P^1$ such that $z(Q) = \\infty$.\nBy Lemma \\ref{lem31} (3), we have then a representation\n$$\\rho : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}}) \\to {\\rm Ine}\\, (H_2, Q_{32}, T_{H_{2}, Q_{32}}) \\simeq (\\C, +)\\,\\, .$$\nHere, for the last isomorphism, we can use the affine coordinate $x$ of $H_2$ fixed at the beginning of this section.\n\n\\begin{proposition}\\label{prop32}\n\\begin{enumerate}\n\\item There is $a \\in \\C \\setminus \\{0\\}$ such that $t^{-2n}a \\in {\\rm Im}\\,\\rho$ for all positive integers $n$.\n\\item ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$\nis not finitely generated.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe assertion (2) follows from the assertion (1).\nIndeed, the additive subgroup $M$ generated by $\\{t^{-2n}a\\, |\\, n \\in \\Z_{\\ge 0}\\}$ is not finitely generated as $a \\not= 0$ and $t$ is transcendental over $\\Q$ by our assumption (Assumption \\ref{ass21}). The assertion (1) says that $M \\subset {\\rm Im}\\,\\rho$. Since ${\\rm Im}\\, \\rho \\subset (\\C, +)$, the group ${\\rm Im}\\, \\rho$ is also an abelian group. It follows that the abelian group ${\\rm Im}\\,\\rho$ is not finitely generated, either,\nregardless of $[{\\rm Im}\\, \\rho : M]$.\nHence ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated as claimed.\n\nIn the rest, we will show the assertion (1) by constructing two genus one fibrations on $Z$ and by considering their Jacobian fibrations.\n\nConsider the following two divisors $M_1$ and $M_2$ of Kodaira's type $I_8$ and $IV^*$ on $Z$:\n$$M_1 := H_2 + D_{32} + H_3 + D_{31} + H_1 + D_{41} + H_4 + D_{42}\\,\\, ,$$\n$$M_2 := H_2 + 2D_{32} + H_1 + 2D_{31} + H_4 + 2D_{34} + 3H_3\\,\\, .$$\nThen $|M_1|$ and $|M_2|$ define genus one fibrations\n$$\\varphi_{M_1} : Z \\to \\P^1\\,\\, ,\\,\\, \\varphi_{M_2} : Z \\to \\P^1\\,\\, .$$\n$\\varphi_{M_1}$ is the genus one fibration induced from an elliptic fibration\n$\\Phi_1 : X \\to \\P^1$ on $X$ given by the divisor of Kodaira's type $I_8$\n$$N_1 :=  E_2 + C_{32} + F_3 + C_{31} + E_1 + C_{41} + F_4 + C_{42}\\,\\, ,$$\nand $\\varphi_{M_2}$ is the genus one fibration induced from an elliptic fibration $\\Phi_2 : X \\to \\P^1$ on $X$ given by the divisor of Kodaira's type $IV^*$\n$$N_2 :=  E_2 + 2C_{32} + E_1 + 2C_{31} + E_4 + 2C_{34} + 3F_3\\,\\, .$$\nBy the classification of \\cite[Theorem 2.1]{Og89}, $\\Phi_1$ then belongs to Type ${\\mathcal J}_1$ and $\\Phi_2$ belongs to Type ${\\mathcal J}_3$ in \\cite[Theorem 2.1]{Og89}. By the definition of the action of our Enriques involution $\\epsilon$ on $X$ and the classification of \\cite[Theorem 2.1]{Og89}, it follows that the reducible fibers of $\\Phi_1$ are exactly $N_1$ and $\\epsilon(N_1)$, and the reducible fibers of $\\Phi_2$ are exactly $N_2$ and $\\epsilon(N_2)$. Thus, $\\varphi_{M_1}$ has no reducible fibers other than $M_1$ and $\\varphi_{M_2}$ has also no reducible fibers other than $M_2$.\n\nLet us consider the (proper non-singular, relatively minimal) Jacobian fibration $\\varphi_i : R_i \\to \\P^1$ of $\\varphi_{M_i}$ for $i=1$ and $2$. Then the fiber $R_{i, p}$ of $\\varphi_i$ over general $p \\in \\P^1$ is ${\\rm Pic}^0\\,(Z_{i, p})$, i.e., the identity component of the Picard group of the corresponding fiber $Z_{i, t}$ of $\\varphi_{M_i}$. Therefore, the Mordell-Weil group ${\\rm MW}\\,(\\varphi_i)$ of $\\varphi_i$ acts on $\\varphi_{M_i}$, which is the unique biregular extension of the translation action of ${\\rm Pic}^0\\,(Z_{i, p})$ on $Z_{i, p}$ where $p \\in \\P^1$ runs through general points. Note also that the types of singular fibers are the same for $\\varphi_{M_i}$ and $\\varphi_i$ up to multiplicities \\cite[Theorem 5.3.1]{CD89}. Therefore $c_2(R_i) = c_2(Z) = 12$. In particular, $R_i \\to \\P^1$ are rational elliptic surfaces.\n\nHere and hereafter, we will use basic notions and properties of Mordell-Weil lattices due to Shioda \\cite{Sh90}.\n\nLet us consider first the action of ${\\rm MW}\\,(\\varphi_2)$ on $\\varphi_{M_2} : Z \\to \\P^1$. From the fact that $\\varphi_{M_2}$ has also no reducible fibers other than $M_2$, we see that $\\varphi_2 : R_2 \\to \\P^1$ belongs to\nNo. 27 in the classfication of \\cite[Main Theorem]{OS91}. Then, the narrow Mordell-Weil lattice ${\\rm MW}^0\\,(\\varphi_2)$ of $\\varphi_2$ is isomorphic to the positive definite root lattice $A_2$. In particular, there is $P \\in {\\rm MW}^0\\,(\\varphi_2)$ such that $\\langle P, P\\rangle =2$ for the height pairing of ${\\rm MW}^0(\\varphi_2)$ \\cite[Section 8]{Sh90}. For this $P$, we have $(P) \\cap (O) = \\emptyset$ by \\cite[Formula 8.19]{Sh90}.  Here $(P)$ is the divisor on $R_2$ corresponding to $P$. The action $t_P$ of $P$ on $\\varphi_{M_2} : Z \\to \\P^1$ then preserves each irreducible component of $M_2$ as $P \\in {\\rm MW}^0\\,(\\varphi_2)$, particularly the curve $H_2$ and the point $Q_{32} \\in H_2$, and the action $t_P|_{H_2}$ is of the form\n$$x \\mapsto x + a$$\nfor some $a \\not= 0$ under the affine coordinate $x$ of $H_2$. Recall that the action of $d(t_P)$ on $T_{Z, Q_{32}}$ is diagonalizable (Lemma \\ref{lem31}). Then, by the finiteness of bicanonical representation \\cite[Theorem 14.10]{Ue75}, by replacing $t_P$ by some power $t_P^k$ ($k \\not= 0$) and $a$ by $ka$ if necessary, we obtain an element\n$$f_2 \\in {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$$\nsuch that  $\\rho(f_2) = a \\not= 0$.\n\nNext we consider the Jacobian fibration $\\varphi_1 : R_1 \\to \\P^1$.\nWe need an explicit geometric construction of $\\varphi_1$ from $\\varphi_{M_1}$ explained by \\cite[Lemma 2.6]{Ko86} and \\cite[Section 3]{HS11}. Note that $D_{21}$ is a $2$-section of $\\varphi_{M_1}$ and $\\pi^{-1}(D_{21}) = C_{12} \\cup C_{21}$. The curves $C_{12}$ and $C_{21}$ are sections of $\\Phi_1$.\nWe may and do choose $C_{21}$ as the zero section of $\\Phi_1$ and set\n$$0 := [C_{21}] \\in {\\rm MW}(\\Phi_1)\\,\\, .$$\nHere and hereafter, we use the following notation:\n\n\\medskip\\noindent\n{\\bf Notation.} \n\\begin{enumerate}\\item For a section $D$ of $\\Phi_1$, we denote by $[D]$ the corresponding element of ${\\rm MW}(\\Phi_1)$ with respect to the zero section $C_{21}$. \n\\item We denote by $T(R) \\in {\\rm Aut}\\, (X)$ the automorphism corresponding to $R \\in {\\rm MW}\\, (\\Phi_1)$.\n\\end{enumerate}\n\n\\medskip\\noindent\nThen the element $[C_{12}] \\in {\\rm MW}(\\Phi_1)$ is a $2$-torsion, because\n$$\\langle [C_{12}], [C_{12}] \\rangle = 2\\cdot2 + 2\\cdot2 - \\frac{4(8-4)}{8} - \\frac{4(8-4)}{8} =0$$\nfor the height pairing \\cite[Theorem 8.6, Formula 8.10]{Sh90} and ${\\rm MW}(\\Phi_1) \\simeq \\Z^{\\oplus 2} \\oplus \\Z\/2$ by \\cite[Theorem 2.1, Case\n${\\mathcal J}_1$]{Og89}. \nSet\n$$\\iota := T([C_{12}]) \\circ \\epsilon \\in {\\rm Aut}\\, (X)\\,\\, .$$\nThen $\\iota$ is an involution on $X$ (\\cite[Lemma 2.6]{Ko86}) such that $X^{\\iota}$ consists of two elliptic curves corresponding to the multiple fibers of $\\varphi_{M_{2}}$ by Assumption \\ref{ass21}. Then, by \\cite[Lemma 2.6]{Ko86} (see also \\cite[Section 3]{HS11}), the Jacobian fibration $\\varphi_1$ of $\\varphi_{M_1}$ is given by\n$$\\varphi_1 : R_1 = X\/\\langle \\iota \\rangle \\to \\P^1\/\\langle \\epsilon \\rangle\n\\,\\, . $$\nHere $\\P^1\/\\langle \\epsilon \\rangle$ is the quotient of the base space $\\P^1$ of $\\Phi_1$ on which $\\epsilon$ acts equivariantly as an involution. Let us denote by $\\pi_{R_1} : X \\to R_1$ the quotient morphism and the fibers $\\pi_{R_1}(N_1)$ by $N_{1, R_1}$ and $\\pi_{R_1}(X_p)$ by $R_{1, \\overline{p}}$.\n\nWe may and do identify both $X_{p}$ and $X_{\\epsilon(p)}$ with $R_{1, \\overline{p}}$ for general $p \\in \\P^1$ via $\\pi_{R_1}$.\n\nSince $\\iota$, $T([C_{12}])$ and $\\epsilon$ are involutions, we have\n$$\\iota := T([C_{12}]) \\circ \\epsilon = \\epsilon \\circ T([C_{12}])\\,\\, .$$\nAlso by the construction, we find that\n$$\\iota(C_{21}) = \\epsilon \\circ T([C_{12}])(C_{21}) = \\epsilon (C_{12}) = C_{21}\\,\\, ,$$\ni.e., preservation of the zero section $C_{21}$ under $\\iota$. Therefore, $Q \\in {\\rm MW}\\,(\\Phi_1)$ is induced from some element $Q'\\in {\\rm MW}(\\varphi_1)$ exactly when\n$$\\iota \\circ T([Q]) = T([Q]) \\circ \\iota\\,\\, {\\rm ,i.e.,}\\,\\, \\iota \\circ T([Q]) \\circ  \\iota = T([Q])\\,\\, .$$\n\\begin{lemma}\\label{lem32}\n\\begin{enumerate}\n\\item $\\iota(C_{11}) = C_2$ and $\\iota(C_2) = C_{11}$.\n\\item $[C_{11}]+[C_2]\\in {\\rm MW}\\,(\\Phi_1)$ is induced from some element $Q'\\in {\\rm MW}(\\varphi_1)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} By preservation of the zero section $C_{21}$ under $\\iota$, we obtain that\n$$\\iota \\circ T([C_{11}]+[C_2]) \\circ \\iota (x) = \\iota (\\iota(x) + [C_{11}] + [C_2]) = x + [\\iota(C_{11})] + [\\iota(C_2)] \\,\\, ,$$\nfor any $x \\in X_p$ on each smooth fiber $X_p$. Hence\n$$\\iota \\circ T(([C_{11}]+[C_2])) \\circ \\iota = T([\\iota(C_{11})] + [\\iota(C_2)])\\,\\, .$$\nSo, the assertion (2) follows from the assertion (1). We show the assertion (1).\nNote that the torsion group of ${\\rm MW}\\, (\\Phi_1)$ is isomorphic to $\\Z\/2$ by \\cite[Theorem 2.1, Case ${\\mathcal J}_1$]{Og89}. In particular, the non-zero torsion element is only $[C_{12}]$.\n\nIf we choose $C_{11}$ (instead of $C_{21}$) as the zero section of $\\Phi_1$, then, the height pairing of the section $C_{22}$ with respect to the zero section $C_{11}$ is computed as\n$$\\langle C_{22}, C_{22}\\rangle = 2\\cdot2 + 2\\cdot2 - \\frac{4(8-4)}{8} - \\frac{4(8-4)}{8} =0\\,\\, .$$\nThus $[C_{22}] - [C_{11}]$ is a non-zero torsion element in ${\\rm MW}\\, (\\Phi_1)$ and therefore coincides with $[C_{12}]$, i.e.,\n$$[C_{22}] = [C_{11}] + [C_{12}]$$\nin ${\\rm MW}\\, (\\Phi_1)$.\nSince $\\iota = \\epsilon \\circ T([C_{12}])$, it follows that\n$$\\iota (C_{11}) = \\epsilon \\circ T([C_{12}])(C_{11}) = \\epsilon (C_{22}) = C_{2}\\,\\, $$\nas claimed. Then\n$$\\iota(C_2) = \\iota (\\iota(C_{11})) = C_{11}\\,\\, ,$$\nas $\\iota$ is an involution. This completes the proof of Lemma \\ref{lem32}.\n\\end{proof}\nLet $Q' \\in {\\rm MW}\\,({\\rm MW}(\\varphi_1)$ be as in Lemma \\ref{lem32}. Then $Q'$ induces an automorphism $f_2 \\in {\\rm Aut}\\, (Z)$ preserving each fiber of $\\varphi_{M_1}$. The action of $f_2$ on $M_1 \\setminus {\\rm Sing}\\, M_1 = \\C^{\\times} \\times \\Z\/8$ \\cite[Page 604]{Ko63} is then the same action of $Q'$ on $N_{1, R} \\setminus {\\rm Sing}\\, (N_{1, R})$ and therefore also the same action of $[C_{11}] + [C_2]$ on $N_1 \\setminus {\\rm Sing}\\, N_1$ under the identifications of these\nthree fibers by $\\pi$ and $\\pi_{R_1}$.\nThus, representing points on $M_1 \\setminus {\\rm Sing}\\, M_1 = \\C^{\\times} \\times \\Z\/8$\nby $(x, m\\,{\\rm mod}\\, 8)$, we have by \\cite[Theorem 9.1, Page 604]{Ko63}\n$$f_2 : (x,m\\,{\\rm mod}\\, 8) \\mapsto (tx, m\\,{\\rm mod}\\, 8) \\mapsto (tx, m + 4 \\,{\\rm mod}\\, 8)\\,\\, .$$\nHere we recall that $C_{22} \\cap E_2 = t$ (resp. $C_2 \\cap F_2 = t$) with respect to the affine coordinate $x$ on $E_2$ (resp. $u$ on $F_2$). Hence\n$f_2^2(H_2) = H_2$, $f_2^2(Q_{32}) = Q_{32}$ and\n$$f_2^2(x) = t^2x$$\non $H_2$. Then\n$$(f_2^2)^{-n} \\circ f_1 \\circ (f_2^2)^{n} \\in {\\rm Ine}\\,(Z, Q_{32}, T_{Z, Q_{32}})$$\nand\n$$(f_2^2)^{-n} \\circ f_1 \\circ (f_2^2)^{n}|_{H_2} : x \\mapsto t^{2n}x \\mapsto t^{2n}x +a \\mapsto x + t^{-2n}a\\,\\,$$\non $H_2$. Thus\n$$t^{-2n}a = \\rho((f_2^2)^{-n} \\circ f_1 \\circ (f_2^2)^{n}) \\in {\\rm Im}\\, \\rho\\,\\, ,$$\nas claimed.\n This completes the proof.\n\\end{proof}\n\nTheorem \\ref{thm31}, hence Theorem \\ref{thm1}, now follows from Propositions \\ref{prop31} (2) and \\ref{prop32} (2).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Electron nuclei coupled system in a coupled double QD \\label{sec_dqd}}\n We consider an electrically gated double quantum dot(QD)\noccupied by two electrons. Under a high magnetic field, s.t. the\nelectron Zeeman splitting is much greater than the hyperfine fields\nand the exchange energy, dynamics takes place in the spin singlet\nground state $|S\\rangle$ and triplet state of zero magnetic quantum\nnumber $|T\\rangle$,\n\\begin{eqnarray} H_e=JS_z+ r\\delta h_z S_x,\n\\label{eq_hf} \\end{eqnarray} where ${\\bf S}$ is the pseudospin\noperator with  $|T\\rangle$ and $|S\\rangle$ forming the $S_z$ basis.\n $\\delta h_z=h_{1z}-h_{2z}$, where\n$h_{1z}$ and $h_{2z}$ are the components of nuclear HF field along\nthe external magnetic field in the first and second dot,\nrespectively\\cite{Coish05,Merkulov02}.\n$r=t\/\\sqrt{t^2+(\\delta\/2+\\sqrt{\\delta^2\/4+t^2})^2}$ is the amplitude\nof the hyperfine coupling, which is determined by the gate voltages.\n$\\delta$ is the detuning which is a linear function of gate voltage\ndifferences, and $t$ is the tunneling coefficient. When $\\delta\\gg\nt$, the ground state singlet state corresponds to the case where\nboth electrons are localized in the same dot and HF coupling is\nswitched off, $r\\rightarrow 0$. The opposite limit $\\delta\\ll -t$\ncorresponds to the singlet state where electrons are located in\ndifferent dots, and HF coupling is maximized $r\\rightarrow 1$.\n\n\\subsection{Bunching in electron spin measurements}\nNow we show that by electron spin measurements the coherent behavior\nof nuclear spins can be demonstrated.\n Electron spins are initialized in the singlet state and the nuclear spin states are initially in a mixture of  $\\delta h_z$ eigenstates,\n$\\rho(t=0)=\\sum_n p_n\\rho_n|S\\rangle\\langle S|$, where $\\rho_n$ is a\nnuclear state with an eigenvalue of $\\delta h_z=h_n$ and satisfies\n$Tr(\\rho_n)=1$. $p_n$ is the probability of the hyperfine field\n$\\delta h_z$ having the value $h_n$.\n\nIn the unbiased regime $\\delta\\ll -t$, the nuclear spins and the\nelectron spins interact for a time span of $\\tau$. Then the gate\nvoltage is swept adiabatically to a high value(s.t. $\\delta\\gg t$),\nin a time scale much shorter than HF interaction time, leading to\nthe state, $ \\rho=\\sum_n p_n\\rho_n|\\Psi_n\\rangle\\langle \\Psi_n|$,\n where $|\\Psi_n\\rangle=\\alpha_n|S\\rangle+\\beta_n|T\\rangle$, with\n $\n \\alpha_n=\\cos(\\Omega_n \\tau\/2)+iJ\/\\Omega_n \\sin(\\Omega_n \\tau\/ 2)$,\n $\\beta_n=\n -ih_n\/\\Omega_n\\sin(\\Omega_n \\tau\/2)\\label{ampl}\n $\n and  $\\Omega_n=\\sqrt{J^2+h_n^2}$ is the Rabi frequency.\n\nNext a charge state measurement is performed which detects a singlet\nor triplet state\\cite{Petta05}. Probability to detect the singlet\nstate is $\\sum_n p_n|\\alpha_n|^2$, and the triplet state is $\\sum_n\np_n|\\beta_n|^2$. Subsequently one can again initialize the system in\nthe singlet state of electron spins, and  turn on the hyperfine\ninteraction for a time span of $\\tau$, and perform a second\nmeasurement. In general over $N$ measurements, the probability of\n$k$ times singlet outcomes is\n \\begin{eqnarray}\n P_{N,k}=\\bigl(^N_{\\,k}\\bigr)\\langle\n|\\alpha|^{2k}|\\beta|^{2(N-k)}\\rangle.\\label{Pqm}\n \\end{eqnarray}\nwhere $\\langle\\ldots\\rangle$ is the ensemble averaging over the\nhyperfine field $h_n$\\cite{Merkulov02}.\n Here the key assumption is\nthat nuclear states preserve their coherence over $N$ measurements,\nthus the measurements are not independent due to nuclear memory. One\ncan easily contrast this result with the semiclassical(SC) result\nfor which nuclear spins are assumed to be purely classical, whereas\nelectron spins are taken to obey quantum mechanics\\cite{Merkulov02}.\nIn SC case results of successive measurements\n are independent and the probability for obtaining $k$ times singlet results over $N$ measurements is given by,\n\\begin{eqnarray}\n P'_{N,k}=\\bigl(^N_{\\,k}\\bigr)\\langle|\\alpha|^2\\rangle^{k}\\langle|\\beta|^2\\rangle^{(N-k)}.\\label{Psc}\n\\end{eqnarray}\n\nIn the SC case  the probability distribution (\\ref{Psc}) obeys\nsimply a Gaussian distribution with mean\n$k=N\\langle|\\alpha|^2\\rangle$, and variance\n$N\\langle|\\alpha|^2\\rangle\\langle|\\beta|^2\\rangle$, as\n$N\\rightarrow\\infty$. However, in quantum mechanical(QM) treatment\nof nuclear spins, the probability distribution (\\ref{Pqm}) may\nexhibit different statistics depending on the initial nuclear state.\nIf the SC distribution of $h_n$ is characterized by the same\ndistribution as in QM case, the two probability distributions\n(\\ref{Pqm}) and (\\ref{Psc}) yield the same mean value, $\n\\overline{k}=N\\langle|\\alpha|^2\\rangle$, however with distinct\nhigher order moments. If the distribution of initial nuclear state\n$p_n$ has a width $\\Delta$, then for HF interaction time $\\tau\\geq\n1\/\\Delta$, the SC and QM distributions start to deviate from each\nother. They yield the same distribution only when the initial\nnuclear state is in a well defined eigenstate of $\\delta h_z$, i.e.\nwhen $\\Delta=0$.\n\nIn particular we are going to consider the case when the nuclear\nspins are initially randomly oriented; probability distribution for\nhyperfine fields obeying a Gaussian distribution $p_n\\rightarrow\np[h]=1\/\\sqrt{2\\pi\\sigma^2}\\exp[-h^2\/2\\sigma^2]$. In Fig.\n\\ref{Fig_20meas}, for $N=20$ measurements, $P_{N,k}$ is shown for HF\ninteraction times $\\sigma\\tau=0.5, 1.5, \\infty$. For $\\tau=0$, the\nprobability for both SC and QM cases is peaked at $k=20$. However,\nimmediately after the HF interaction is introduced, the probability\ndistributions show distinct behavior. The SC distribution converges\nto a Gaussian distribution. In QM case the probabilities bunch at\n$k=0,20$ for $J=0$, and when $J\/\\sigma=0.5$ those bunch at $k=20$\nonly. As $J$ is increased above some critical value, no bunching\ntakes place at $k=0$ singlet measurement.\n\\begin{figure}[h!]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{fig_2}\n\\caption{Double QD: Probability distribution  at $N=20$ measurements\nfor $k={0,1,\\ldots,20}$ times singlet detections, for QM(solid\nlines), SC(dashed lines). Two cases of the exchange energy are\nconsidered a) $J=0$ b) $J\/\\sigma=0.5$ for HF interaction times\n$\\sigma\\tau=$ i)$0.5$, ii)$1.5$, iii)$\\infty$. \\label{Fig_20meas}}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\psfrag{a}{$k$} \\psfrag{b}{$P$}\n\\includegraphics[width=18pc]{figSQD}\n\\caption{Single QD: Probability distribution  at $N=40$ measurements\nfor $k={0,1,\\ldots,20}$ times $|+\\rangle$ detections at HF\ninteraction times  $\\sigma\\tau=$ i)$0.3$, ii)$0.6$, iii)$0.9$\niv)$\\infty$. \\label{figSQD}}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Electron spin revivals\\label{sec_esr}}\n\nThe modified nuclear spectrum leads to correlations between the\nsuccessive electron spin measurements. Depending on the results of\nprevious measurement, one may decrease the singlet-triplet mixing.\n As a particular example consider the\ncase: Starting from a random spin configuration, $N$ successive\nelectron spin measurements are performed, each following\ninitialization of electron spins in the spin singlet state and a HF\ninteraction of duration $\\{\\tau_i,i=1\\ldots N\\}$ and all outcomes\nturn out to be singlet. Then again HF interaction is switched on for\na time $t$, and the $(N+1)$th measurement is carried out. The\nconditional probability to detect the singlet state is given by\n \\begin{eqnarray}\n P=\\frac{\\sum(^{~2}_{s_1})(^{~2}_{s_2})\\ldots(^{~2}_{s_{N+1}})e^{-\\frac{1}{2}\\bigl[(s_1-1)\\tilde{\\tau}_1+(s_2-1)\\tilde{\\tau}_2+\\ldots\n+(s_N-1)\\tilde{\\tau}_N+(s_{N+1}-1)\\tilde{t}\\bigr]^2}}{\n4\\sum(^{~2}_{s_1})(^{~2}_{s_2})\\ldots(^{~2}_{s_N})e^{-\\frac{1}{2}\\bigl[(s_1-1)\\tilde{\\tau}_1+(s_2-1)\\tilde{\\tau}_2+\\ldots\n+(s_N-1)\\tilde{\\tau}_N\\bigr]^2} },\n \\label{condprob}\n \\end{eqnarray}\n where the sums run over $s_i=0\\ldots 2$ and $\\tilde{\\tau}_i=\\sigma\\tau_i$.\nFor the particular case $\\tau_1=\\tau_2=\\ldots=\\tau_N=\\tau\\gg\n1\/\\sigma$,\n the initial state is revived at $t=n\\tau, \\,(n=1,2,\\ldots,N)$ with\n a decreasing amplitude, $P\\simeq1\/2+\\sum_{s=0}^{N}(\n^{2N}_{~s})e^{\\frac{-\\sigma^2}{2}(t-(N-s)\\tau)^2}\/4(^{2N}_{~N})$. In\n Fig. \\ref{Fig-cond} the conditional probabilities(\\ref{condprob})\n are shown for $\\sigma\\tau=1.0, 3.0, 6.0$ subject to $N=0,1,2,5,10$\n prior singlet measurements in each. Revivals are observable only\n for $\\sigma\\tau>1$, because the modulation period of the nuclear\n state spectrum characterized by $1\/\\tau$ should be smaller than the\n variance $\\sigma$.\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.9\\textwidth]{fig_3.eps}\n\\caption{Conditional probability for singlet state detection as a\nfunction of HF interaction time $\\sigma t$, subject to\n$N=0,1,2,5,10$ times prior singlet state measurements and for HF\ninteraction times a)$\\sigma\\tau=1.0$, b)$\\sigma\\tau=3.0$,\nc)$\\sigma\\tau=6.0$. \\label{Fig-cond}}\n\\end{figure}\n\nFrom (\\ref{condprob}), number of revivals can be increased with\nvarious choices for the ratios of HF interaction times $\\tau_i$.\n The underlying mechanism of revivals is purification of nuclear\nspins by the electron spin measurements. The purity of a system\ncharacterized by the density matrix $\\hat{\\rho}$ is given by ${\\cal\nP}=Tr\\rho^2$. As an example we are again going to consider the\nnuclear state prepared by $N$ successive electron spin measurements\nwith singlet outcomes, each followed by HF interaction times\n$\\tau_{1}\\ldots \\tau_N$,\n\\begin{eqnarray}\n{\\cal P}=\\frac{1}{\\cal D}\\frac{\\sum_{s_i=0}^\n4(^{~4}_{s_1})(^{~4}_{s_2})\\ldots(^{~4}_{s_N})e^{-\\frac{1}{2}\\bigl[(s_1-2)\\tilde{\\tau}_1+(s_2-2)\\tilde{\\tau}_2+\\ldots\n+(s_N-2)\\tilde{\\tau}_N\\bigr]^2}}{\\bigl[ \\sum_{s_i=0}^\n2(^{~2}_{s_1})(^{~2}_{s_2})\\ldots(^{~2}_{s_N})e^{-\\frac{1}{2}\\bigl[(s_1-1)\\tilde{\\tau}_1+(s_2-1)\\tilde{\\tau}_2+\\ldots\n+(s_N-1)\\tilde{\\tau}_N\\bigr]^2}\\bigr]^2 }.\\label{purity}\n\\end{eqnarray}\n${\\cal D}$ is the dimension of the Hilbert space for the nuclear spins. For a fixed ratio of $\\tau_1:\\tau_2:\\ldots:\\tau_N$, purity\n(\\ref{purity}) is a monotonically increasing function of time. For\n$\\sigma\\tau_i\\gg 1$, one can attain various asymptotic limits for\nthe purity.  For instance for $N=2$, there are three asymptotic\nlimits;when a)$\\tau_1=2\\tau_2$ then ${\\cal P}=11\/4{\\cal D}$,\nb)$\\tau_1=\\tau_2$ then ${\\cal P}=35\/18{\\cal D}$, c)otherwise ${\\cal P}=9\/4{\\cal D}$.\nFor $N=2$ with $\\tau_1=2\\tau_2=2\\tau\\gg 1\/\\sigma$,  the conditional\nprobability (\\ref{condprob}) is given as follows,\n\\begin{eqnarray}\nP\\simeq\\frac{1}{2}+\\frac{1}{8}\\bigl\\{ e^{-\\frac{(\\tilde{t}-3\\tilde{\\tau})^2}{2} }+\n2e^{-\\frac{(\\tilde{t}-2\\tilde{\\tau})^2}{2} }+ 3e^{-\\frac{(\\tilde{t}-\\tilde{\\tau})^2}{2} }+\n4e^{-\\frac{\\tilde{t}^2}{2} } \\bigr\\},\\label{cond2}\n\\end{eqnarray}\nwhereas for $\\tau_2=\\tau_1=\\tau\\gg 1\/\\sigma$,\n\\begin{eqnarray}\nP\\simeq\\frac{1}{2}+\\frac{1}{12}\\bigl\\{ e^{-\\frac{(\\tilde{t}-2\\tilde{\\tau})^2}{2} }+\n4e^{-\\frac{(\\tilde{t}-\\tilde{\\tau})^2}{2} }+6e^{-\\frac{\\tilde{t}^2}{2} }\n\\bigr\\}.\\label{cond3}\n\\end{eqnarray}\nAs the purity of nuclear spins increase, more revivals are present\nwith an increased amplitude.\n\n\\section{Electron spin bunching and revivals in a single QD\\label{sec_sqd}}\nNow we are going\nto consider a single electron on a single QD. Under external field\n$B$, the system is governed by the Hamiltonian,\n\\begin{eqnarray}\nH=g_e\\mu_BB S_z +g_n\\mu_n B\\sum I^{(j)}_z  + {\\bf h}\\cdot{\\bf\nS}.\\label{hfsqd}\n\\end{eqnarray}\nIn (\\ref{hfsqd}), the first two terms are electron and nuclear\nZeeman energies respectively, and the last term is the HF\ninteraction where ${\\bf h}$ is the HF field. When electron Zeeman\nenergy is much greater than rms value of HF fields, viz.\n$g_e\\mu_BB\\gg\\sqrt{\\langle h^2\\rangle} $, flip-flop terms are\nsuppressed and the Hamiltonian (\\ref{hfsqd}) becomes, $H\\simeq\ng_e\\mu_BB S_z + h_z S_z$. Up to a unitary rotation, with $B=0$, this\nis equivalent to the Hamiltonian (\\ref{eq_hf}) with $J=0$.\n$|\\pm\\rangle=(|\\uparrow\\rangle)\\pm|\\downarrow\\rangle\/\\sqrt{2}$ states\nare coupled by HF interaction  with $|\\uparrow(\\downarrow)\\rangle$\nbeing the eigenstates of $S_z$. Each time the electron is prepared\nin $|+\\rangle$. Next it is loaded onto the QD, then removed from the\nQD after some dwelling time $\\tau$. Next spin measurement is\nperformed in $|\\pm\\rangle$ basis. Essentially the same predictions\nas that of double QD can be made for this system, namely electron\nspin bunching and revival.\n\nIn Fig. \\ref{figSQD}, for $N=40$ measurements, the QM probability\ndistribution of $P_{N,k}$ is shown at electron Zeeman energy\n$\\epsilon=g_e\\mu_B\\hbar\/2=3\\sigma$, for $\\sigma\\tau=0.3,~,0.6,~0.9,~\\infty$. It is\nseen that contrary to the double QD, the population bunches at\n$|-\\rangle$ states at times $\\tau\\sim \\pi\/\\epsilon$, but then relaxes to the\nequilibrium distribution cf. Fig. \\ref{Fig_20meas}.\n\nNext we are going to consider electron spin revivals.  For instance\nafter $N$ times HF interaction of duration $\\tau\\gg 1\/\\sigma$, each\nfollowed by $|+\\rangle$ measurement, the conditional probability for\nobtaining $|+\\rangle$ in the $(N+1)$th step followed by a HF\ninteraction of duration $t$ is given as, $ P\\simeq\n1\/2+\\sum_{s=0}^{N}(\n^{2N}_{~s})e^{-\\sigma^2(t-(N-s)\\tau)^2\/2}\\cos\\epsilon[t-(N-s)\\tau]\/4(^{2N}_{~N})$.\n This is\nessentially the same result for that of a double QD discussed in\nsection-\\ref{sec_esr}.\n\n\n\\section{Discussion and conclusion}\n\nThe randomization of nuclear spins will lead to loss of memory\neffects described above. The nuclear state conditioned on the\nelectron spin measurements will decohere during time interval\nbetween the successive measurements, i.e., when the HF interaction\nis switched off. Thus, the main decoherence mechanism of nuclear\nspins is due to intrinsic nuclear dipole-dipole interactions. In\ndouble quantum dots the duration of the cycle involving electron\nspin initialization and measurement is about $10\n\\mu$s\\cite{Petta05}. Since the nuclear spin coherence time\ndetermined mostly by the nuclear spin diffusion is longer than about\nseveral tens of ms\\cite{Paget82,Giedke05,Paget77}, the bunching for\n$N$ successive measurements up to $N>1000$ can be observed. The same\nholds for the number of revivals that can be observed.\n\nWe have studied the quantum dynamics of the electron-nuclei coupled\nsystem in QD's. The bunching of results of the electron spin\nmeasurements and the revival in the conditional probabilities are\nemerging features of coherence of nuclear spins. The underlying\nmechanism is the correlations between successive measurements\ninduced via nuclear spins and the increase in the purity of the\nnuclear spin state through the electron spin measurements. This\nmechanism is expected to lead to the extension of the electron spin\ncoherence time.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA priori, the boundary of a graph is not a meaningful concept, as graphs do not have an interior or complement. However, given that the usual boundary for subsets in $\\mathbb{R}^n$ is induced by a metric, one interesting question is if it is possible in a finite metric space to find an axiomatic approach to defining a subset that behaves in a boundary-like manner. Chartrand, Erwins, Johns, and Zhang proposed one definition for graphs in \\cite{chartrand}, and Steinerberger proposed another in \\cite{steinerberger}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\scalebox{0.3}{\n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (4,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (30) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (31) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (32) at (6,4) {};\n    \\node[shape=circle,draw=black, fill=red] (n11) at (-1,-1) {};\n    \\node[shape=circle,draw=black, fill=red] (44) at (7,7) {};\n    \\node[shape=circle,draw=black, fill=red] (n14) at (-1,7) {};\n    \\node[shape=circle,draw=black, fill=red] (n41) at (7,-1) {};\n    \\node[shape=circle,draw=black, fill=blue] (0515) at (1,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (2505) at (5,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (0505) at (1,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (2515) at (5,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (1505) at (3,1) {};\n   \n    \\node[shape=circle,draw=black, fill=blue] (2525) at (5,5) {};\n    \\node[shape=circle,draw=black, fill=blue] (1525) at (3,5) {};\n    \\node[shape=circle,draw=black, fill=blue] (0525) at (1,5) {};\n    \\node[shape=circle,draw=black, fill=blue] (03) at (0,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (2,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (23) at (4,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (6,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (1515) at (3,3) {};\n    \\path (1515) edge node {} (11);\n    \\path (1515) edge node {} (21);\n    \\path (1515) edge node {} (12);\n    \\path (1515) edge node {} (22);\n    \\path (1515) edge node {} (1505);\n    \\path (1515) edge node {} (2515);\n    \\path (1515) edge node {} (0515);\n    \\path (1515) edge node {} (1525);\n    \\path (0505) edge node {} (1505);\n    \\path (1505) edge node {} (2505);\n    \\path (0505) edge node {} (0515);\n    \\path (2505) edge node {} (2515);\n    \\path (2515) edge node {} (2525);\n    \\path (0515) edge node {} (0525);\n    \\path (0525) edge node {} (1525);\n    \\path (1525) edge node {} (2525);\n    \\path (00) edge node {} (10);\n    \\path (10) edge node {} (20);\n    \\path (00) edge node {} (01);\n    \\path (01) edge node {} (11);\n    \\path (11) edge node {} (21);\n    \\path (02) edge node {} (12);\n    \\path (12) edge node {} (22);\n    \\path (01) edge node {} (02);\n    \\path (21) edge node {} (22);\n    \\path (20) edge node {} (21);\n    \\path (10) edge node {} (11);\n    \\path (11) edge node {} (12);\n    \\path (22) edge node {} (32);\n    \\path (21) edge node {} (31);\n    \\path (20) edge node {} (30);\n    \\path (30) edge node {} (31);\n    \\path (31) edge node {} (32);\n    \\path (20) edge node {} (2505);\n    \\path (30) edge node {} (2505);\n    \\path (21) edge node {} (2505);\n    \\path (31) edge node {} (2505);\n    \\path (10) edge node {} (1505);\n    \\path (20) edge node {} (1505);\n    \\path (11) edge node {} (1505);\n    \\path (21) edge node {} (1505);\n    \\path (01) edge node {} (0515);\n    \\path (11) edge node {} (0515);\n    \\path (02) edge node {} (0515);\n    \\path (12) edge node {} (0515);\n    \\path (00) edge node {} (0505);\n    \\path (10) edge node {} (0505);\n    \\path (01) edge node {} (0505);\n    \\path (11) edge node {} (0505);\n    \\path (21) edge node {} (2515);\n    \\path (31) edge node {} (2515);\n    \\path (22) edge node {} (2515);\n    \\path (32) edge node {} (2515);\n   \n   \n   \n   \n    \\path (02) edge node {} (0525);\n    \\path (03) edge node {} (0525);\n    \\path (12) edge node {} (0525);\n    \\path (13) edge node {} (0525);\n    \\path (12) edge node {} (1525);\n    \\path (13) edge node {} (1525);\n    \\path (22) edge node {} (1525);\n    \\path (23) edge node {} (1525);\n    \\path (22) edge node {} (2525);\n    \\path (23) edge node {} (2525);\n    \\path (32) edge node {} (2525);\n    \\path (33) edge node {} (2525);\n    \\path (00) edge node {} (n11);\n    \\path (30) edge node {} (n41);\n    \\path (03) edge node {} (n14);\n    \\path (33) edge node {} (44);\n    \\path (03) edge node {} (02);\n    \\path (13) edge node {} (12);\n    \\path (23) edge node {} (22);\n    \\path (33) edge node {} (32);\n    \\path (03) edge node {} (13);\n    \\path (13) edge node {} (23);\n    \\path (23) edge node {} (33);\n    \\node at (0,-2) {};\n\\end{tikzpicture} \n\\qquad \\qquad \\qquad \n\\begin{tikzpicture}\n   \\node[shape=circle,draw=black, fill=blue] (00) at ({3-2*sqrt(2)},{3-2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at ({3+4*sin(45)\/sin(112.5)*cos(11*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(11*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (40) at ({3+4*sin(45)\/sin(112.5)*cos(13*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(13*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at ({3+4*sin(45)\/sin(112.5)*cos(9*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(9*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (04) at ({3+4*sin(45)\/sin(112.5)*cos(7*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(7*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (60) at ({3+2*sqrt(2)},{3-2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (62) at ({3+4*sin(45)\/sin(112.5)*cos(15*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(15*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (64) at ({3+4*sin(45)\/sin(112.5)*cos(180\/8)},{3+4*sin(45)\/sin(112.5)*sin(180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (06) at ({3-2*sqrt(2)},{3+2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (26) at ({3+4*sin(45)\/sin(112.5)*cos(5*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(5*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (46) at ({3+4*sin(45)\/sin(112.5)*cos(3*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(3*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (66) at ({3+2*sqrt(2)},{3+2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (37) at (3,7) {};\n    \\node[shape=circle,draw=black, fill=blue] (3n1) at (3,-1) {};\n    \\node[shape=circle,draw=black, fill=blue] (73) at (7,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (n13) at (-1,3) {};\n    \\node[shape=circle,draw=black, fill=red] (38) at (3,8) {};\n    \\node[shape=circle,draw=black, fill=red] (3n2) at (3,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (83) at (8,3) {};\n    \\node[shape=circle,draw=black, fill=red] (n23) at (-2,3) {};\n    \\node[shape=circle,draw=black, fill=red] (77) at ({3+2.5*sqrt(2)},{3+2.5*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (7n1) at ({3+2.5*sqrt(2)},{3-2.5*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (n1n1) at ({3-2.5*sqrt(2)},{3-2.5*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (n17) at ({3-2.5*sqrt(2)},{3+2.5*sqrt(2)}) {};\n    \\path (00) edge node {} (20);\n    \\path (20) edge node {} (40);\n    \\path (40) edge node {} (60);\n    \\path (06) edge node {} (26);\n    \\path (26) edge node {} (46);\n    \\path (46) edge node {} (66);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (04);\n    \\path (04) edge node {} (06);\n    \\path (60) edge node {} (62);\n    \\path (62) edge node {} (64);\n    \\path (64) edge node {} (66);\n    \\path (20) edge node {} (02);\n    \\path (04) edge node {} (26);\n    \\path (46) edge node {} (64);\n    \\path (62) edge node {} (40);\n    \\path (26) edge node {} (37);\n    \\path (37) edge node {} (38);\n    \\path (46) edge node {} (37);\n    \\path (20) edge node {} (3n1);\n    \\path (3n1) edge node {} (3n2);\n    \\path (40) edge node {} (3n1);\n    \\path (02) edge node {} (n13);\n    \\path (04) edge node {} (n13);\n    \\path (n13) edge node {} (n23);\n    \\path (62) edge node {} (73);\n    \\path (64) edge node {} (73);\n    \\path (73) edge node {} (83);\n    \\path (66) edge node {} (77);\n    \\path (7n1) edge node {} (60);\n    \\path (n1n1) edge node {} (00);\n    \\path (n17) edge node {} (06);\n    \\path (20) edge node {} (46);\n    \\path (40) edge node {} (26);\n    \\path (02) edge node {} (64);\n    \\path (04) edge node {} (62);\n\\end{tikzpicture}\n\\qquad \\qquad \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (30) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=red] (40) at (8,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (31) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (41) at (8,2) {};\n    \\node[shape=circle,draw=black, fill=red] (02) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (32) at (6,4) {};\n    \\node[shape=circle,draw=black, fill=red] (42) at (8,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (03) at (0,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (2,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (23) at (4,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (6,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (43) at (8,6) {};\n    \\node[shape=circle,draw=black, fill=red] (04) at (0,8) {};\n    \\node[shape=circle,draw=black, fill=blue] (14) at (2,8) {};\n    \\node[shape=circle,draw=black, fill=red] (24) at (4,8) {};\n    \\node[shape=circle,draw=black, fill=blue] (34) at (6,8) {};\n    \\node[shape=circle,draw=black, fill=red] (44) at (8,8) {};\n    \\node at (0,-1) {};\n    \\path (00) edge node {} (10);\n    \\path (10) edge node {} (20);\n    \\path (20) edge node {} (30);\n    \\path (30) edge node {} (40);\n    \n    \\path (01) edge node {} (11);\n    \\path (11) edge node {} (21);\n    \\path (21) edge node {} (31);\n    \\path (31) edge node {} (41);\n    \n    \\path (02) edge node {} (12);\n    \\path (32) edge node {} (42);\n    \n    \\path (03) edge node {} (13);\n    \\path (13) edge node {} (23);\n    \\path (23) edge node {} (33);\n    \\path (33) edge node {} (43);\n    \n    \\path (04) edge node {} (14);\n    \\path (14) edge node {} (24);\n    \\path (24) edge node {} (34);\n    \\path (34) edge node {} (44);\n    \n    \\path (01) edge node {} (02);\n    \\path (02) edge node {} (03);\n    \\path (03) edge node {} (04);\n    \\path (00) edge node {} (01);\n    \n    \\path (11) edge node {} (12);\n    \\path (12) edge node {} (13);\n    \\path (13) edge node {} (14);\n    \\path (10) edge node {} (11);\n    \n    \\path (21) edge node {} (20);\n    \\path (23) edge node {} (24);\n    \n    \\path (31) edge node {} (32);\n    \\path (32) edge node {} (33);\n    \\path (33) edge node {} (34);\n    \\path (30) edge node {} (31);\n    \n    \\path (41) edge node {} (42);\n    \\path (42) edge node {} (43);\n    \\path (43) edge node {} (44);\n    \\path (40) edge node {} (41);\n\\end{tikzpicture} \n}\\\\\n\\phantom{-} \\\\\n\\scalebox{0.3}{\n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=red] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (02) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (4,4) {};\n    \\node[shape=circle,draw=black, fill=red] (30) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=red] (31) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (32) at (6,4) {};\n    \\node[shape=circle,draw=black, fill=red] (n11) at (-1,-1) {};\n    \\node[shape=circle,draw=black, fill=red] (44) at (7,7) {};\n    \\node[shape=circle,draw=black, fill=red] (n14) at (-1,7) {};\n    \\node[shape=circle,draw=black, fill=red] (n41) at (7,-1) {};\n    \\node[shape=circle,draw=black, fill=blue] (0515) at (1,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (2505) at (5,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (0505) at (1,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (2515) at (5,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (1505) at (3,1) {};\n   \n    \\node[shape=circle,draw=black, fill=blue] (2525) at (5,5) {};\n    \\node[shape=circle,draw=black, fill=blue] (1525) at (3,5) {};\n    \\node[shape=circle,draw=black, fill=blue] (0525) at (1,5) {};\n    \\node[shape=circle,draw=black, fill=red] (03) at (0,6) {};\n    \\node[shape=circle,draw=black, fill=red] (13) at (2,6) {};\n    \\node[shape=circle,draw=black, fill=red] (23) at (4,6) {};\n    \\node[shape=circle,draw=black, fill=red] (33) at (6,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (1515) at (3,3) {};\n    \\path (1515) edge node {} (11);\n    \\path (1515) edge node {} (21);\n    \\path (1515) edge node {} (12);\n    \\path (1515) edge node {} (22);\n    \\path (1515) edge node {} (1505);\n    \\path (1515) edge node {} (2515);\n    \\path (1515) edge node {} (0515);\n    \\path (1515) edge node {} (1525);\n    \\path (0505) edge node {} (1505);\n    \\path (1505) edge node {} (2505);\n    \\path (0505) edge node {} (0515);\n    \\path (2505) edge node {} (2515);\n    \\path (2515) edge node {} (2525);\n    \\path (0515) edge node {} (0525);\n    \\path (0525) edge node {} (1525);\n    \\path (1525) edge node {} (2525);\n    \\path (00) edge node {} (10);\n    \\path (10) edge node {} (20);\n    \\path (00) edge node {} (01);\n    \\path (01) edge node {} (11);\n    \\path (11) edge node {} (21);\n    \\path (02) edge node {} (12);\n    \\path (12) edge node {} (22);\n    \\path (01) edge node {} (02);\n    \\path (21) edge node {} (22);\n    \\path (20) edge node {} (21);\n    \\path (10) edge node {} (11);\n    \\path (11) edge node {} (12);\n    \\path (22) edge node {} (32);\n    \\path (21) edge node {} (31);\n    \\path (20) edge node {} (30);\n    \\path (30) edge node {} (31);\n    \\path (31) edge node {} (32);\n    \\path (20) edge node {} (2505);\n    \\path (30) edge node {} (2505);\n    \\path (21) edge node {} (2505);\n    \\path (31) edge node {} (2505);\n    \\path (10) edge node {} (1505);\n    \\path (20) edge node {} (1505);\n    \\path (11) edge node {} (1505);\n    \\path (21) edge node {} (1505);\n    \\path (01) edge node {} (0515);\n    \\path (11) edge node {} (0515);\n    \\path (02) edge node {} (0515);\n    \\path (12) edge node {} (0515);\n    \\path (00) edge node {} (0505);\n    \\path (10) edge node {} (0505);\n    \\path (01) edge node {} (0505);\n    \\path (11) edge node {} (0505);\n    \\path (21) edge node {} (2515);\n    \\path (31) edge node {} (2515);\n    \\path (22) edge node {} (2515);\n    \\path (32) edge node {} (2515);\n   \n   \n   \n   \n    \\path (02) edge node {} (0525);\n    \\path (03) edge node {} (0525);\n    \\path (12) edge node {} (0525);\n    \\path (13) edge node {} (0525);\n    \\path (12) edge node {} (1525);\n    \\path (13) edge node {} (1525);\n    \\path (22) edge node {} (1525);\n    \\path (23) edge node {} (1525);\n    \\path (22) edge node {} (2525);\n    \\path (23) edge node {} (2525);\n    \\path (32) edge node {} (2525);\n    \\path (33) edge node {} (2525);\n    \\path (00) edge node {} (n11);\n    \\path (30) edge node {} (n41);\n    \\path (03) edge node {} (n14);\n    \\path (33) edge node {} (44);\n    \\path (03) edge node {} (02);\n    \\path (13) edge node {} (12);\n    \\path (23) edge node {} (22);\n    \\path (33) edge node {} (32);\n    \\path (03) edge node {} (13);\n    \\path (13) edge node {} (23);\n    \\path (23) edge node {} (33);\n    \\node at (0,-2) {};\n\\end{tikzpicture}  \\qquad  \\qquad \\qquad \n\\begin{tikzpicture}\n   \\node[shape=circle,draw=black, fill=red] (00) at ({3-2*sqrt(2)},{3-2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at ({3+4*sin(45)\/sin(112.5)*cos(11*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(11*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (40) at ({3+4*sin(45)\/sin(112.5)*cos(13*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(13*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at ({3+4*sin(45)\/sin(112.5)*cos(9*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(9*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (04) at ({3+4*sin(45)\/sin(112.5)*cos(7*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(7*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=red] (60) at ({3+2*sqrt(2)},{3-2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (62) at ({3+4*sin(45)\/sin(112.5)*cos(15*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(15*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (64) at ({3+4*sin(45)\/sin(112.5)*cos(180\/8)},{3+4*sin(45)\/sin(112.5)*sin(180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=red] (06) at ({3-2*sqrt(2)},{3+2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (26) at ({3+4*sin(45)\/sin(112.5)*cos(5*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(5*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=blue] (46) at ({3+4*sin(45)\/sin(112.5)*cos(3*180\/8)},{3+4*sin(45)\/sin(112.5)*sin(3*180\/8)}) {};\n    \\node[shape=circle,draw=black, fill=red] (66) at ({3+2*sqrt(2)},{3+2*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (37) at (3,7) {};\n    \\node[shape=circle,draw=black, fill=red] (3n1) at (3,-1) {};\n    \\node[shape=circle,draw=black, fill=red] (73) at (7,3) {};\n    \\node[shape=circle,draw=black, fill=red] (n13) at (-1,3) {};\n    \\node[shape=circle,draw=black, fill=red] (38) at (3,8) {};\n    \\node[shape=circle,draw=black, fill=red] (3n2) at (3,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (83) at (8,3) {};\n    \\node[shape=circle,draw=black, fill=red] (n23) at (-2,3) {};\n    \\node[shape=circle,draw=black, fill=red] (77) at ({3+2.5*sqrt(2)},{3+2.5*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (7n1) at ({3+2.5*sqrt(2)},{3-2.5*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (n1n1) at ({3-2.5*sqrt(2)},{3-2.5*sqrt(2)}) {};\n    \\node[shape=circle,draw=black, fill=red] (n17) at ({3-2.5*sqrt(2)},{3+2.5*sqrt(2)}) {};\n    \\path (00) edge node {} (20);\n    \\path (20) edge node {} (40);\n    \\path (40) edge node {} (60);\n    \\path (06) edge node {} (26);\n    \\path (26) edge node {} (46);\n    \\path (46) edge node {} (66);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (04);\n    \\path (04) edge node {} (06);\n    \\path (60) edge node {} (62);\n    \\path (62) edge node {} (64);\n    \\path (64) edge node {} (66);\n    \\path (20) edge node {} (02);\n    \\path (04) edge node {} (26);\n    \\path (46) edge node {} (64);\n    \\path (62) edge node {} (40);\n    \\path (26) edge node {} (37);\n    \\path (37) edge node {} (38);\n    \\path (46) edge node {} (37);\n    \\path (20) edge node {} (3n1);\n    \\path (3n1) edge node {} (3n2);\n    \\path (40) edge node {} (3n1);\n    \\path (02) edge node {} (n13);\n    \\path (04) edge node {} (n13);\n    \\path (n13) edge node {} (n23);\n    \\path (62) edge node {} (73);\n    \\path (64) edge node {} (73);\n    \\path (73) edge node {} (83);\n    \\path (66) edge node {} (77);\n    \\path (7n1) edge node {} (60);\n    \\path (n1n1) edge node {} (00);\n    \\path (n17) edge node {} (06);\n    \\path (20) edge node {} (46);\n    \\path (40) edge node {} (26);\n    \\path (02) edge node {} (64);\n    \\path (04) edge node {} (62);\n\\end{tikzpicture}\\qquad \\qquad \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=red] (30) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=red] (40) at (8,0) {};\n    \\node[shape=circle,draw=black, fill=red] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=red] (21) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (31) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (41) at (8,2) {};\n    \\node[shape=circle,draw=black, fill=red] (02) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=red] (12) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=red] (32) at (6,4) {};\n    \\node[shape=circle,draw=black, fill=red] (42) at (8,4) {};\n    \\node[shape=circle,draw=black, fill=red] (03) at (0,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (2,6) {};\n    \\node[shape=circle,draw=black, fill=red] (23) at (4,6) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (6,6) {};\n    \\node[shape=circle,draw=black, fill=red] (43) at (8,6) {};\n    \\node[shape=circle,draw=black, fill=red] (04) at (0,8) {};\n    \\node[shape=circle,draw=black, fill=red] (14) at (2,8) {};\n    \\node[shape=circle,draw=black, fill=red] (24) at (4,8) {};\n    \\node[shape=circle,draw=black, fill=red] (34) at (6,8) {};\n    \\node[shape=circle,draw=black, fill=red] (44) at (8,8) {};\n    \\node at (0,-1) {};\n    \\path (00) edge node {} (10);\n    \\path (10) edge node {} (20);\n    \\path (20) edge node {} (30);\n    \\path (30) edge node {} (40);\n    \n    \\path (01) edge node {} (11);\n    \\path (11) edge node {} (21);\n    \\path (21) edge node {} (31);\n    \\path (31) edge node {} (41);\n    \n    \\path (02) edge node {} (12);\n    \\path (32) edge node {} (42);\n    \n    \\path (03) edge node {} (13);\n    \\path (13) edge node {} (23);\n    \\path (23) edge node {} (33);\n    \\path (33) edge node {} (43);\n    \n    \\path (04) edge node {} (14);\n    \\path (14) edge node {} (24);\n    \\path (24) edge node {} (34);\n    \\path (34) edge node {} (44);\n    \n    \\path (01) edge node {} (02);\n    \\path (02) edge node {} (03);\n    \\path (03) edge node {} (04);\n    \\path (00) edge node {} (01);\n    \n    \\path (11) edge node {} (12);\n    \\path (12) edge node {} (13);\n    \\path (13) edge node {} (14);\n    \\path (10) edge node {} (11);\n    \n    \\path (21) edge node {} (20);\n    \\path (23) edge node {} (24);\n    \n    \\path (31) edge node {} (32);\n    \\path (32) edge node {} (33);\n    \\path (33) edge node {} (34);\n    \\path (30) edge node {} (31);\n    \n    \\path (41) edge node {} (42);\n    \\path (42) edge node {} (43);\n    \\path (43) edge node {} (44);\n    \\path (40) edge node {} (41);\n\\end{tikzpicture} \n}\n    \\caption{The CEJZ (top) and the Steinerberger (bottom) boundaries of three graphs, with boundary vertices shown in red.}\n    \\label{boundaryexamples}\n\\end{figure}\n\n\nIn the Chartrand, Erwin, Johns, and Zhang definition, a vertex $v$ is in the boundary of $G$ if there is a vertex $u$ such that no neighbor of $v$ is farther away from $u$ than $v$. \nHowever, when considering subsets in $\\mathbb{R}^n$ with the Euclidean metric, many boundary points do not satisfy a condition analogous to this. Motivated by this, Steinerberger proposed an alternative notion, where $v$ is a boundary vertex if there is a vertex $u$ such that on average, the neighbors of $v$ are closer to $u$ than $v$ is. \nExamples of both boundaries are shown in \\cref{boundaryexamples}.\n\n\nFormally, the \\emph{Chartrand-Erwin-Johns-Zhang (CEJZ) boundary} of a connected graph $G=(V,E)$ is\n\\begin{equation}\n    (\\partial G)'=\\left\\{v \\in V\\; \\bigg\\vert \\;\\exists\\;u\\in V \\text{ such that } d(w,u) \\leq d(v,u) \\text{ for all } (v,w)\\in E \\right\\}.\n\\end{equation}\nThe \\emph{Steinerberger boundary} of a connected graph $G = (V, E)$ is defined as\n\\begin{equation}\\label{partialG}\n    \\partial G = \\bigg\\{v \\in V\\; \\bigg\\vert \\;\\exists\\;u\\in V \\text{ such that }\\frac{1}{\\text{deg}(v)}\\sum_{(v, w) \\in E} d(w,u) < d(v,u)\\bigg\\},\n\\end{equation}\nwhere we use the convention $\\partial G=V$ when $|V|=1$. We call any $u\\in V$ establishing that $v\\in \\partial G$ or $v\\in (\\partial G)'$ a \\emph{witness} for $v$. \nIt is an immediate consequence of the definitions that the condition for $\\partial G$ is a relaxation of the condition for $(\\partial G)'$.\n\n\\begin{proposition}[Steinerberger \\cite{steinerberger}, Proposition 1]\\label{containment}\nFor any connected graph $G$, $(\\partial G)'\\subseteq \\partial G$. \n\\end{proposition}\n\nThe CEJZ boundary has been studied in a variety of contexts, and we refer the reader to \\cite{allgeier,  caceres,  caceres3, caceres2, chartrand2, hernando, kang, rodriguez, zejnilovic} for details. When discussing notions of boundary, one natural question is what sets minimize or maximize the size of the boundary. Our work will parallel prior work on characterizing graphs with small CEJZ boundary. The cases of $|(\\partial G)'|=2$ or $|(\\partial G)'|=3$ were classified by Hasegawa and Saito \\cite{hasegawa}, and graphs with $|(\\partial G)'|=4$ were classified by M\\\"{u}ller, P\\'{o}r, and Sereni \\cite{mullerfour}. The main result of our paper is the following theorem classifying graphs with Steinerberger boundary size at most 4. \n\n\\begin{theorem}\\label{maintheorem}\nLet $G=(V,E)$ be a connected graph. \n\\begin{enumerate}[label=(\\alph*)]\n    \\item $|\\partial G|=2$ if and only if $G$ is a path graph with at least two vertices.\n    \\item $|\\partial G|=3$ if and only if $G$ is a tree with three leaves or a tripod.\n    \\item $|\\partial G|=4$ if and only if $G$ is a tree with four leaves or one of the graphs in \\cref{partialG4} with paths of arbitrary lengths attached to boundary vertices $v$ with boundary stability number $\\max_{u\\in V} \\sum_{w\\in N(v)}[d(v,u)-d(w,u)]=1.$\n\\end{enumerate}\n\\end{theorem}\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\scalebox{0.3}{\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n    \\node at (0,2.5) {\\Huge{$3$}};\n    \\node at (2.5,0) {\\Huge{$3$}};\n    \\node at (0,-2.5) {\\Huge{$3$}};\n    \\node at (-2.5,0) {\\Huge{$3$}};\n    \\node at (0.5,0.5) {\\Huge{$0$}};\n    \\path (A) edge node {} (M);\n    \\path (M) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (A) edge node {} (B);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n    \\path (B) edge node {} (M);\n    \\path (C) edge node {} (M);\n   \n\\end{tikzpicture}\\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$2$}};\n    \\node at (2.5,0) {\\Huge{$2$}};\n    \\node at (0,-2.5) {\\Huge{$2$}};\n    \\node at (-2.5,0) {\\Huge{$2$}};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n   \n\\end{tikzpicture}\\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$1$}};\n    \\node at (2.5,0) {\\Huge{$1$}};\n    \\node at (0,-2.5) {\\Huge{$1$}};\n    \\node at (-2.5,0) {\\Huge{$1$}};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (D);\n    \\path (C) edge node {} (D);\n   \n\\end{tikzpicture} \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$2$}};\n    \\node at (2.5,0) {\\Huge{$2$}};\n    \\node at (0,-2.5) {\\Huge{$2$}};\n    \\node at (-2.5,0) {\\Huge{$2$}};\n    \\node at (0.5,0.5) {\\Huge{$0$}};\n    \\path (A) edge node {} (M);\n    \\path (M) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (A) edge node {} (B);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (M);\n    \\path (C) edge node {} (M);\n   \n\\end{tikzpicture}\\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$1$}};\n    \\node at (2.5,0) {\\Huge{$2$}};\n    \\node at (0,-2.5) {\\Huge{$1$}};\n    \\node at (-2.5,0) {\\Huge{$2$}};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n   \n\\end{tikzpicture}}\\\\\n\\phantom{-}\\\\\n\\scalebox{0.3}{\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (-4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (-4,-2) {};\n    \\node[shape=circle,draw=black, fill=blue] (M1) at (-2,0) {};\n    \\node (P) at (0,0) {$\\cdots$};\n    \\node[shape=circle,draw=black, fill=blue] (M2) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (4,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (4,2) {};\n    \\node at (-4.5,2) {\\Huge{$1$}};\n    \\node at (4.5,2) {\\Huge{$1$}};\n    \\node at (-4.5,-2) {\\Huge{$1$}};\n    \\node at (4.5,-2) {\\Huge{$1$}};\n    \\node at (1.75,0.5) {\\Huge{$0$}};\n    \\node at (-1.75,0.5) {\\Huge{$0$}};\n    \\path (A) edge node {} (M1);\n    \\path (M1) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (M2);\n    \\path (C) edge node {} (M2);\n    \\path (M1) edge node {} (P);\n    \\path (P) edge node {} (M2);\n   \n\\end{tikzpicture}\\qquad\n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (-4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (-4,-2) {};\n    \\node[shape=circle,draw=black, fill=blue] (M1) at (-2,0) {};\n    \\node (P) at (0,0) {$\\cdots$};\n    \\node[shape=circle,draw=black, fill=blue] (M2) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (4,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (4,2) {};\n    \\node at (-4.5,2) {\\Huge{$1$}};\n    \\node at (4.5,2) {\\Huge{$1$}};\n    \\node at (-4.5,-2) {\\Huge{$1$}};\n    \\node at (4.5,-2) {\\Huge{$1$}};\n    \\node at (1.75,0.5) {\\Huge{$0$}};\n    \\node at (-1.75,0.5) {\\Huge{$0$}};\n    \\path (A) edge node {} (M1);\n    \\path (M1) edge node {} (D);\n   \n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (M2);\n    \\path (C) edge node {} (M2);\n    \\path (M1) edge node {} (P);\n    \\path (P) edge node {} (M2);\n   \n\\end{tikzpicture}}\n\\caption{Non-tree graphs with $|\\partial G|=4$, where $\\partial G$ is shown in red. The boundary stability number is given beside each vertex.}\n\\label{partialG4}\n\\end{figure}\n\nThe boundary stability number is a new parameter that we introduce to establish this theorem. For any $v\\in \\partial G$, this parameter is an integer that measures how stable the condition in \\cref{partialG} is under certain operations. It allows us to more accurately describe what happens when we build larger graphs from smaller ones, which is one of the techniques used in the classification of graphs with small CEJZ boundary.\n\nObserve that in each of the graphs in Theorem \\ref{maintheorem}, the maximum degree is small, and few vertices are not on long paths. In Euclidean space, larger regions have larger boundary. One formal statement of this is the isoperimetric \ninequality, which gives a lower bound on the surface area or perimeter of a region in terms of its volume. Steinerberger established a corresponding isoperimetric inequality for $\\partial G$ in \\cite[Theorem 1]{steinerberger} given by\n\\[|\\partial G|\\geq \\frac{1}{2\\Delta}\\cdot \\frac{|V|}{\\diam(G)},\\]\nwhere $\\Delta$ is the maximum degree of $G$. Hence, graphs with a large number of vertices have more boundary vertices unless these graphs contain large paths. We see that the results of Theorem \\ref{maintheorem} are consistent with the above isoperimetric inequality for $\\partial G$. \n\nNote that the CEJZ boundary does not exhibit this general behavior. From generalizations of the first graph in \\cref{boundaryexamples} and other grid-like graphs, $(\\partial G)'$ can remain very small for very large graphs that are not formed from paths. A lower bound on $|(\\partial G)'|$ in terms of the maximum degree was previously studied in \\cite{mullerbound}, where the authors established a bound that is logarithmic with respect to $\\Delta$ and showed that this bound is sharp up to constants. However, this bound is entirely independent of $|V|$.\n\n\nWe start in \\cref{preliminaries} by summarizing previous results for the CEJZ and Steinerberger boundaries. In particular, we will summarize the characterizations of graphs with small CEJZ boundary. In \\cref{stability}, we will introduce the boundary stability number of a vertex and establish several lemmas. We then apply these results in \\cref{boundarysize4} to prove \\cref{maintheorem}. In \\cref{largeboundary}, we apply some of our results to describe some graphs with large Steinerberger boundary. We then conclude with open questions in \\cref{futurework}.\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\n\nIn this section, we outline necessary definitions and notation, and we then summarize previous results for both the CEJZ and Steinerberger boundaries. Throughout this paper, all graphs are assumed to be simple and undirected. We assume basic familiarity with these graphs and refer the reader to \\cite{bona} or \\cite{bondy} for this information. We summarize the definitions and notation relevant for this paper. \n\nGraphs will be denoted $G=(V,E)$. An edge in $E$ will be denoted as a pair, such as $(v,w)$. Note that graphs in this paper are undirected, so this is equivalent to $(w,v)$. We use $N_G(v)$ to denote the set of neighbors of $v$ in the graph $G$ and $\\deg_G(v)$ to denote the degree of $v$ in $G$. \n\nFor any graph $G=(V,E)$, a \\emph{walk} is a sequence of vertices $W=v_0v_1\\ldots v_\\ell$ such that $(v_i,v_{i+1})\\in E$ for $i=0,1,\\ldots,\\ell-1$. We call $v_0$ the \\emph{initial vertex} and $v_\\ell$ the \\emph{terminal vertex} of $W$. We also call $\\ell$ the \\emph{length} of the walk $W$. \nA walk is a \\emph{path} if all vertices are distinct, and a \\emph{shortest} path from $v$ to $w$ is a path of minimum length with $v$ as its initial vertex and $w$ as its terminal vertex. A walk is \\emph{closed} if $v_0=v_{\\ell}$, and a closed walk is called a \\emph{cycle} if no vertices are repeated except for $v_0=v_{\\ell}$. \n\nA graph $G=(V,E)$ is \\emph{connected} if for every $v,w\\in V$, there exists a path with $v$ as its initial vertex and $w$ as its terminal vertex. The \\emph{connected components} of $G$ are the maximal connected subgraphs of $G$. For a connected graph $G=(V,E)$, a vertex $v\\in V$ is a \\emph{cut vertex} if $G-v$, the graph obtained by deleting $v$ and all edges incident to it, is not connected.\n\nFor a connected graph $G=(V,E)$ and any $v,w\\in V$, the \\emph{distance} from $v$ to $w$, denoted $d_G(v,w)$, is the length of a shortest path with $v$ as its initial vertex and $w$ as its terminal vertex. It is straightforward to verify that $d_G$ satisfies the properties of a metric. For any $v\\in V$, the \\emph{eccentricity} of $v$ is\n$\\ecc_G(v)=\\max_{w\\in V} \\{d_G(v,w)\\}$. The \\emph{diameter} of a graph is $\\diam(G)=\\max_{v\\in V} \\{\\ecc_G(v)\\}$.\nIf $v\\in V$ satisfies $\\ecc_G(v)=\\diam(G)$, then we call $v$ a \\emph{peripheral} vertex. When the context is clear, we omit the subscript $G$ in $N_G,\\deg_G,d_G$, and $\\ecc_G$. An example of these definitions is given in \\cref{graphexample}.\n\n\n\\begin{figure}[h!]\n\\centering\n    \\scalebox{0.4}{\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=blue] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=blue] (D) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (2) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (3) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (4) at (8,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (5) at (10,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (6) at (8,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (7) at (8,-2) {};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (D);\n    \\path (C) edge node {} (D);\n    \\path (D) edge node {} (2);\n    \\path (2) edge node {} (3);\n    \\path (3) edge node {} (4);\n    \\path (4) edge node {} (5);\n    \\path (4) edge node {} (6);\n    \\path (4) edge node {} (7);\n    \\node at (-2.75,0) {\\Huge{$v_1$}};\n    \\node at (0,2.75) {\\Huge{$v_2$}};\n    \\node at (0,-2.75) {\\Huge{$v_3$}};\n    \\node at (2,-0.75) {\\Huge{$v_4$}};\n    \\node at (4,-0.75) {\\Huge{$v_5$}};\n    \\node at (6,-0.75) {\\Huge{$v_6$}};\n    \\node at (8,2.75) {\\Huge{$v_7$}};\n    \\node at (8.75,-0.75) {\\Huge{$v_8$}};\n    \\node at (8,-2.75) {\\Huge{$v_9$}};\n    \\node at (11,0) {\\Huge{$v_{10}$}};\n\\end{tikzpicture}}\n\\caption{A graph $G$ with $\\diam(G)=5$, where $v_1,v_2,v_3,v_7,v_9$, and $v_{10}$ are all peripheral vertices. The vertices $v_4,v_5,v_6$, and $v_8$ are cut vertices, as removing any of them disconnects $G$.}\n\\label{graphexample}\n\\end{figure}\n\nWe now summarize previous results. We start with the following  observation, which is used throughout the study of both the CEJZ and Steinerberger boundaries.\n\n\\begin{observation}\\label{peripheral}\nFor any connected graph $G=(V,E)$, peripheral vertices are always in $(\\partial G)'$. Combined with \\cref{containment}, this implies that peripheral vertices are always in $\\partial G$. \n\\end{observation}\n\nFor $\\partial G$, the following results were established by Steinerberger in \\cite{steinerberger}. The first result immediately implies part (a) of \\cref{maintheorem}. Hence, it remains to show parts (b) and (c).\n\n\\begin{proposition}[Steinerberger \\cite{steinerberger}, Proposition 3]\\label{boundary2}\nFor any connected graph with at least\ntwo vertices, we have $|\\partial G|\\geq  2$. If $|\\partial G|=2$, then $G$ is a path.\n\\end{proposition}\n\n\\begin{proposition}[Steinerberger \\cite{steinerberger}, Proposition 2]\\label{treeandleaves}\nIf $G$ is a tree, then $\\partial G$ are the vertices of\ndegree $1$. For any connected graph, vertices of degree $1$ are in $\\partial G$. \n\\end{proposition}\n\nTo establish the remaining parts of our main theorem, we will use the characterization of graphs with small CEJZ boundary, which we summarize here. We start with the results of Hasegawa and Saito characterizing graphs with CEJZ boundary size $2$ or $3$. Note that a \\emph{tripod} is a graph formed by starting with the complete graph on three vertices $K_3$ and attaching a path of arbitrary length (possibly 0) to each vertex.\n\n\n\n\\begin{theorem}[Hasegawa and Saito \\cite{hasegawa}, Theorem 7]\\label{CEJZ2}\nLet $G$  be a connected graph. If $|(\\partial G)'|=2$, then $G$ is a path.\n\\end{theorem}\n\n\\begin{theorem}[Hasegawa and Saito \\cite{hasegawa}, Theorem 9]\\label{CEJZ3}\nA connected graph $G$ has $|(\\partial G)'|=3$ if and only if $G$ is either a tree with three leaves or a\ntripod.\n\\end{theorem}\n\nThe characterization of graphs with CEJZ boundary size $4$ is significantly more complex. We start by defining several families of graphs and axis slice convex sets in $\\mathbb{R}^2$.\n\n\\begin{definition}[M\\\"{u}ller, P\\'{o}r, and Sereni, \\cite{mullerfour}, Definition 2]\nLet $a$ and $c$ be two positive integers. Define the vertex sets  \n\\[V_{a\\times c}^0=\\{(x,y)\\in \\mathbb{N}^2\\, | \\, 0\\leq x\\leq a \\text{ and } 0\\leq y\\leq c\\}\\]\n\\[ V_{a\\times c}^1=\\left\\{\\left(x+\\frac{1}{2},y+\\frac{1}{2}\\right)\\, \\bigg| \\, (x,y)\\in \\mathbb{N}^2,0\\leq x< a, \\text{ and } 0\\leq y< c\\right\\}.\\]\n\\begin{enumerate}[label=(\\alph*)]\n    \\item The grid graph $G_{a\\times c}$ has vertex set $V_{a\\times c}^0$ and edges between any vertices of Euclidean distance 1. Note that $|V_{a\\times c}^0|=(a+1)(c+1)$.\n    \\item The graph $N_{a\\times c}$ has vertex set $V_{a\\times c}=V_{a\\times c}^0\\cup V_{a\\times c}^1$ and edges between vertices of Euclidean distance at most $1$.\n    \\item For $a>2$, the graph $X_{a\\times c}$ is the subgraph of $N_{(a-1)\\times c}$ induced by \\[V_{(a-1)\\times c}\\setminus \\{(x,y)\\in \\mathbb{N}^2\\, | \\, 0<x<a-1 \\text{ and } y\\in \\{0,c\\}\\}.\\]\n    If $a=2$, then $X_{2\\times c}$ is the subgraph of $N_{1\\times c}$ obtained by removing the edge between the vertices $(0,0)$ and $(1,0)$, and the edge between the vertices $(0,c)$ and $(1,c)$. If $a\\geq 1$ and $c>1$, define $X_{a\\times c}=X_{c\\times a}$. Finally, define $X_{1\\times 1}=K_4$.\n    \\item The graph $T_{a\\times c}$ is the subgraph of $N_{a\\times (c+1)}$ induced by \n    \\[V_{a\\times (c+1)}\\setminus (\\{(0,y)\\, |\\, y\\in \\mathbb{N}\\}\\cup \\{(x,y)\\, |\\, x<a \\text{ and } y\\in \\{0,c+1\\}\\}).\\]\n    \\item Let $G_{a\\times c}^1$ and $G_{a\\times c}^2$ be two grid graphs with vertex sets respectively labeled by $v_{x,y}$ and $w_{x,y}$ for $0\\leq x\\leq a,0\\leq y\\leq c$. The graph $D_{a\\times c}$ is obtained by identifying $v_{x,y}$ with $w_{x,y}$ whenever $x\\in \\{0,a\\}$ or $y\\in \\{0,c\\}$, adding an edge between $v_{x,y}$ and $w_{x,y}$ for $0\\leq x\\leq a,0\\leq y\\leq c$, and adding an edge between $v_{x+1,y}$ and $w_{x,y+1}$ whenever $0\\leq x<a$ and $0\\leq y<c$. \n    \\item The graph $L_{a\\times c}$ is obtained from $D_{a\\times c}$ by removing the vertices $w_{x,y}$ for $0< x<a$ and $0< y<c$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{definition}[M\\\"{u}ller, P\\'{o}r, and Sereni \\cite{mullerfour}, Definition 3]\nA set $W\\subseteq \\mathbb{R}^2$ is {axis slice convex} if \n\\begin{itemize}\n    \\item whenever $(x_1,y),(x_2,y)\\in W$ and $x_1<x_2$, then $(x,y)\\in W$ for all $x$ in $\\{x_1,x_1+1,\\ldots,x_2\\}$, and\n    \\item whenever $(x,y_1),(x,y_2)\\in W$ and $y_1<y_2$, then $(x,y)\\in W$ for all $y$ in $\\{y_1,y_1+1,\\ldots,y_2\\}$. \n\\end{itemize}\n\\end{definition}\n\nAn example of these definitions is shown in \\cref{graphexamples}. We now state the main theorem of M\\\"{u}ller, P\\'{o}r, and Sereni. \n\n\\begin{figure}[h!]\n    \\centering\n    \\scalebox{0.4}{\n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (40) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (42) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (60) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (62) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (04) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (24) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (44) at (4,4) {};\n    \\node[shape=circle,draw=black, fill=red] (64) at (6,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (1,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (31) at (3,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (51) at (5,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (1,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (3,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (53) at (5,3) {};\n    \\path (00) edge node {} (20);\n    \\path (20) edge node {} (40);\n    \\path (40) edge node {} (60);\n    \\path (02) edge node {} (22);\n    \\path (22) edge node {} (42);\n    \\path (42) edge node {} (62);\n    \\path (04) edge node {} (24);\n    \\path (24) edge node {} (44);\n    \\path (44) edge node {} (64);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (04);\n    \\path (20) edge node {} (22);\n    \\path (22) edge node {} (24);\n    \\path (40) edge node {} (42);\n    \\path (42) edge node {} (44);\n    \\path (60) edge node {} (62);\n    \\path (62) edge node {} (64);\n    \\path (00) edge node {} (11);\n    \\path (20) edge node {} (11);\n    \\path (02) edge node {} (11);\n    \\path (22) edge node {} (11);\n    \\path (20) edge node {} (31);\n    \\path (40) edge node {} (31);\n    \\path (22) edge node {} (31);\n    \\path (42) edge node {} (31);\n    \\path (40) edge node {} (51);\n    \\path (60) edge node {} (51);\n    \\path (42) edge node {} (51);\n    \\path (62) edge node {} (51);\n    \\path (02) edge node {} (13);\n    \\path (22) edge node {} (13);\n    \\path (04) edge node {} (13);\n    \\path (24) edge node {} (13);\n    \\path (22) edge node {} (33);\n    \\path (42) edge node {} (33);\n    \\path (24) edge node {} (33);\n    \\path (44) edge node {} (33);\n    \\path (42) edge node {} (53);\n    \\path (62) edge node {} (53);\n    \\path (44) edge node {} (53);\n    \\path (64) edge node {} (53);\n    \\path (11) edge node {} (31);\n    \\path (31) edge node {} (51);\n    \\path (13) edge node {} (33);\n    \\path (33) edge node {} (53);\n    \\path (11) edge node {} (13);\n    \\path (31) edge node {} (33);\n    \\path (51) edge node {} (53);\n    \\node at (0,-1) {};\n    \\node at (0,5) {};\n    \\end{tikzpicture}\n    \\qquad \\qquad \n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n   \n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=red] (40) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (42) at (4,2) {};\n   \n   \n    \\node[shape=circle,draw=black, fill=red] (04) at (0,4) {};\n   \n    \\node[shape=circle,draw=black, fill=red] (44) at (4,4) {};\n   \n    \\node[shape=circle,draw=black, fill=blue] (11) at (1,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (31) at (3,1) {};\n   \n    \\node[shape=circle,draw=black, fill=blue] (13) at (1,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (3,3) {};\n   \n   \n   \n   \n    \\path (02) edge node {} (22);\n    \\path (22) edge node {} (42);\n   \n   \n   \n   \n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (04);\n   \n   \n    \\path (40) edge node {} (42);\n    \\path (42) edge node {} (44);\n   \n   \n    \\path (00) edge node {} (11);\n   \n    \\path (02) edge node {} (11);\n    \\path (22) edge node {} (11);\n   \n    \\path (40) edge node {} (31);\n    \\path (22) edge node {} (31);\n    \\path (42) edge node {} (31);\n   \n   \n   \n   \n    \\path (02) edge node {} (13);\n    \\path (22) edge node {} (13);\n    \\path (04) edge node {} (13);\n   \n    \\path (22) edge node {} (33);\n    \\path (42) edge node {} (33);\n   \n    \\path (44) edge node {} (33);\n   \n   \n   \n   \n    \\path (11) edge node {} (31);\n   \n    \\path (13) edge node {} (33);\n   \n    \\path (11) edge node {} (13);\n    \\path (31) edge node {} (33);\n   \n    \\node at (0,-1) {};\n    \\node at (0,5) {};\n    \\end{tikzpicture}\n    \\qquad \\qquad \n    \\begin{tikzpicture}\n    \\node at (-0.5,0) {};\n    \\node at (5.5,0) {};\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (40) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (42) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (60) at (5,-1) {};\n   \n    \\node[shape=circle,draw=black, fill=red] (04) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (24) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (44) at (4,4) {};\n    \\node[shape=circle,draw=black, fill=red] (64) at (5,5) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (1,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (31) at (3,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (51) at (5,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (1,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (3,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (53) at (5,3) {};\n    \\path (00) edge node {} (20);\n    \\path (20) edge node {} (40);\n    \\path (40) edge node {} (60);\n    \\path (02) edge node {} (22);\n    \\path (22) edge node {} (42);\n   \n    \\path (04) edge node {} (24);\n    \\path (24) edge node {} (44);\n    \\path (44) edge node {} (64);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (04);\n    \\path (20) edge node {} (22);\n    \\path (22) edge node {} (24);\n    \\path (40) edge node {} (42);\n    \\path (42) edge node {} (44);\n   \n   \n    \\path (00) edge node {} (11);\n    \\path (20) edge node {} (11);\n    \\path (02) edge node {} (11);\n    \\path (22) edge node {} (11);\n    \\path (20) edge node {} (31);\n    \\path (40) edge node {} (31);\n    \\path (22) edge node {} (31);\n    \\path (42) edge node {} (31);\n    \\path (40) edge node {} (51);\n    \\path (60) edge node {} (51);\n    \\path (42) edge node {} (51);\n   \n    \\path (02) edge node {} (13);\n    \\path (22) edge node {} (13);\n    \\path (04) edge node {} (13);\n    \\path (24) edge node {} (13);\n    \\path (22) edge node {} (33);\n    \\path (42) edge node {} (33);\n    \\path (24) edge node {} (33);\n    \\path (44) edge node {} (33);\n    \\path (42) edge node {} (53);\n   \n    \\path (44) edge node {} (53);\n    \\path (64) edge node {} (53);\n    \\path (11) edge node {} (31);\n    \\path (31) edge node {} (51);\n    \\path (13) edge node {} (33);\n    \\path (33) edge node {} (53);\n    \\path (11) edge node {} (13);\n    \\path (31) edge node {} (33);\n    \\path (51) edge node {} (53);\n    \\end{tikzpicture}}\\\\\n    \\phantom{-}\\\\\n    \\scalebox{0.4}{\n    \\begin{tikzpicture}\n        \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n        \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n        \\node[shape=circle,draw=black, fill=blue] (11') at (2.5,2.5) {};\n        \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n        \n        \\node[shape=circle,draw=black, fill=blue] (21') at (4.5,2.5) {};\n        \\node[shape=circle,draw=black, fill=red] (02) at (0,4) {};\n        \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n        \\node[shape=circle,draw=black, fill=blue] (22) at (4,4) {};\n        \n        \\node[shape=circle,draw=black, fill=red] (30) at (6,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (31) at (6,2) {};\n        \\node[shape=circle,draw=black, fill=red] (32) at (6,4) {};\n        \n        \\path (00) edge node {} (10);\n        \\path (20) edge node {} (10);\n        \\path (01) edge node {} (11);\n        \\path (21) edge node {} (11);\n        \\path (00) edge node {} (01);\n        \\path (10) edge node {} (11);\n        \\path (20) edge node {} (21);\n        \\path (20) edge node {} (30);\n        \\path (21) edge node {} (31);\n        \\path (30) edge node {} (31);\n        \\path (22) edge node {} (32);\n        \\path (32) edge node {} (31);\n        \n        \\path (21') edge node {} (11');\n        \\path (22) edge node {} (21');\n        \\path (21') edge node {} (32);\n        \\path (21') edge node {} (31);\n       \n        \n        \\path (11) edge node {} (11');\n        \\path (21) edge node {} (21');\n        \\path (01) edge node {} (11');\n        \\path (10) edge node {} (11');\n        \\path (12) edge node {} (11');\n        \\path (21') edge node {} (20);\n        \n        \\path (02) edge node {} (12);\n        \\path (12) edge node {} (11);\n        \\path (02) edge node {} (01);\n        \\path (21) edge node {} (22);\n        \\path (22) edge node {} (12);\n        \n       \n        \\path (22) edge node {} (11');\n       \n       \n        \n        \\path (01) edge node {} (12);\n        \\path (00) edge node {} (11);\n        \\path (10) edge node {} (21);\n        \\path (20) edge node {} (31);\n    \\end{tikzpicture}\n    \\qquad \\qquad \n        \\begin{tikzpicture}\n        \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n        \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {}; {};\n        \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n        \n        \\node[shape=circle,draw=black, fill=red] (02) at (0,4) {};\n        \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n        \\node[shape=circle,draw=black, fill=blue] (22) at (4,4) {};\n        \n        \\node[shape=circle,draw=black, fill=red] (30) at (6,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (31) at (6,2) {};\n        \\node[shape=circle,draw=black, fill=red] (32) at (6,4) {};\n        \n        \\path (00) edge node {} (10);\n        \\path (20) edge node {} (10);\n        \\path (01) edge node {} (11);\n        \\path (21) edge node {} (11);\n        \\path (00) edge node {} (01);\n        \\path (10) edge node {} (11);\n        \\path (20) edge node {} (21);\n        \\path (20) edge node {} (30);\n        \\path (21) edge node {} (31);\n        \\path (30) edge node {} (31);\n        \\path (22) edge node {} (32);\n        \\path (32) edge node {} (31);\n\n        \n        \n        \\path (02) edge node {} (12);\n        \\path (12) edge node {} (11);\n        \\path (02) edge node {} (01);\n        \\path (21) edge node {} (22);\n        \\path (22) edge node {} (12);\n        \n        \\path (01) edge node {} (12);\n        \\path (00) edge node {} (11);\n        \\path (10) edge node {} (21);\n        \\path (20) edge node {} (31);\n    \\end{tikzpicture}\n    \\qquad \\qquad \n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (40) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (42) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (60) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (62) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (04) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (24) at (2,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (44) at (4,4) {};\n    \\node[shape=circle,draw=black, fill=red] (64) at (6,4) {};\n   \n   \n    \\node[shape=circle,draw=black, fill=blue] (51) at (5,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (1,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (33) at (3,3) {};\n   \n    \\path (00) edge node {} (20);\n    \\path (20) edge node {} (40);\n    \\path (40) edge node {} (60);\n    \\path (02) edge node {} (22);\n    \\path (22) edge node {} (42);\n    \\path (42) edge node {} (62);\n    \\path (04) edge node {} (24);\n    \\path (24) edge node {} (44);\n    \\path (44) edge node {} (64);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (04);\n    \\path (20) edge node {} (22);\n    \\path (22) edge node {} (24);\n    \\path (40) edge node {} (42);\n    \\path (42) edge node {} (44);\n    \\path (60) edge node {} (62);\n    \\path (62) edge node {} (64);\n   \n   \n   \n   \n   \n   \n   \n   \n    \\path (40) edge node {} (51);\n    \\path (60) edge node {} (51);\n    \\path (42) edge node {} (51);\n    \\path (62) edge node {} (51);\n    \\path (02) edge node {} (13);\n    \\path (22) edge node {} (13);\n    \\path (04) edge node {} (13);\n    \\path (24) edge node {} (13);\n    \\path (22) edge node {} (33);\n    \\path (42) edge node {} (33);\n    \\path (24) edge node {} (33);\n    \\path (44) edge node {} (33);\n   \n   \n   \n   \n   \n   \n    \\path (13) edge node {} (33);\n   \n   \n   \n   \n    \\end{tikzpicture}\n    }\n    \\caption{The graphs $N_{3\\times 2}$, $X_{3\\times 2}$, and $T_{3\\times 2}$ are shown in the top row. The graphs $D_{3\\times 2}$, $L_{3\\times 2}$, and a subgraph of $N_{3\\times 2}$ induced by $V_{3\\times 2}^0\\cup (W\\cap V_{3\\times 2}^1)$ for some axis slice convex $W\\subseteq \\mathbb{R}^2$ are shown in the second row. Vertices in $(\\partial G)'$ are shown in red. Observe that in each case, $\\partial G\\setminus (\\partial G)'\\neq \\emptyset$}\n    \\label{graphexamples}\n\\end{figure}\n\n\\begin{theorem}[M\\\"{u}ller, P\\'{o}r, and Sereni \\cite{mullerfour}, Theorem 4]\\label{mullerthm}\nA connected graph $G$ has $|(\\partial G)'|=4$ if and only if $G$ is one of the following graphs:\n\\begin{enumerate}[label=(\\alph*)]\n    \\item a subdivision of the star graph $K_{1,4}$,\n    \\item a subdivision of the tree with exactly four leaves and two vertices of degree~3,\n    \\item a graph obtained from one of the trees in (b) by removing a vertex of degree 3 and adding all edges between its neighbors,\n   \n    \\item a subgraph of $N_{a\\times c}$ induced by $V_{a\\times c}^0\\cup (W\\cap V_{a\\times c}^1)$ for some axis slice convex set $W\\subseteq \\mathbb{R}^2$, with exactly one path of arbitrary length attached to each of its CEJZ boundary vertices,\n    \\item the graph $X_{a\\times c}$ with exactly one path of arbitrary length attached to each of its CEJZ boundary vertices,\n    \\item a subgraph of $T_{a\\times c}$ induced by $V_{a\\times (c+1)}^1\\cup (W\\cap V_{a\\times (c+1)}^0)$ for some axis-slice convex set $W\\subseteq \\mathbb{R}^2$ that contains $(a,0)$ and $(a,c+1)$, with exactly one path of arbitrary length attached to each of its CEJZ boundary vertices, \n    \\item the graph $D_{a\\times c}$ with exactly one path of arbitrary length attached to each of its CEJZ boundary vertices, or\n    \\item the graph $L_{a\\times c}$ with exactly one path of arbitrary length attached to each of its CEJZ boundary vertices.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\\label{mullerremark}\nM\\\"{u}ller, P\\'{o}r, and Sereni show that the graphs in (d)-(h) without any attached paths have CEJZ boundary vertices given by the points with extreme $x$ and $y$ coordinates. For example, $N_{a\\times c}$ has CEJZ boundary $\\{(0,0),(0,c),(a,0),(a,c)\\}$, and $T_{a\\times c}$ has boundary $\\{(1\/2,1\/2),(1\/2,c+1\/2),(a,0),(a,c+1)\\}$. Attaching paths to the vertices $v\\in (\\partial G)'$ replaces $v$ with the new leaf produced from the attachment, and hence preserves the number of CEJZ boundary vertices.\n\\end{remark}\n\n\n\n\\section{The boundary stability number of a vertex}\\label{stability}\n\n\nIn this section, we introduce a new parameter called the boundary stability number of a vertex. We will establish several results involving this parameter, some of which we will use for the proof of \\cref{maintheorem}. We start with the following observation, which can be verified directly from definitions.\n\n\n\\begin{observation}\\label{neighbordistance}\nLet $G=(V,E)$ be a connected graph, and let $v\\in V$. If $w~\\in~N(v)$, then for any $u\\in V$, $|d(v,u)-d(w,u)|\\leq 1$. Additionally, if $v\\neq u$, then there exists some $w\\in N(v)$ such that $d(w,u)=d(v,u)-1$, as some neighbor of $v$ is on a shortest path from $v$ to $u$.\n\\end{observation}\n\nWith this observation in mind, notice that for graphs on at least two vertices, $u$ being a witness for $v\\in \\partial G$ corresponds to when more neighbors of $v$ are distance $d(v,u)-1$ than $d(v,u)+1$. We define a new parameter that records precisely this difference.\n\n\\begin{definition}\nFor a  connected graph $G=(V,E)$ and $v,u\\in V$, define\n\\[\\beta_G(v,u)=\\sum_{w\\in N_G(v)}[d_G(v,u)-d_G(w,u)],\\]\nand define the {boundary stability number} of $v$ in $G$ to be $\\beta_G(v)=\\max_{u\\in V}\\beta_G(v,u)$. When the context is clear, we omit the subscript in $\\beta_G$.\n\\end{definition}\n\nObserve that $\\beta(v)$ is always an integer, and it is straightforward to show that when a connected graph $G$ has at least two vertices, $\\beta(v)\\geq 1$ is equivalent to $v\\in \\partial G$. When this occurs, any $u\\in V$ with $\\beta(v,u)\\geq 1$ is a witness for $v\\in \\partial G$. However, the exact value of $\\beta(v)$ provides some additional information, which we will use later to consider the effect of certain graph operations. Note that if $G=(V,E)$ is the single vertex graph, then $\\beta(v)=0$ for the unique $v\\in V$. Since we use the convention that $\\partial G=V$ when $|V|=1$, this is the only case in which a boundary vertex does not satisfy $\\beta(v)\\geq 1$. Examples of $\\beta(v)$ are shown in \\cref{stabilityexample}.\n\n\\begin{figure}[h!]\n\\centering\n    \\scalebox{0.4}{\n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.75) {\\Huge{$1$}};\n    \\node at (2.75,0) {\\Huge{$1$}};\n    \\node at (0,-2.75) {\\Huge{$1$}};\n    \\node at (-2.75,0) {\\Huge{$1$}};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (D);\n    \\path (C) edge node {} (D);\n   \n\\end{tikzpicture}\n\\qquad \n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n    \\node at (0,2.75) {\\Huge{$3$}};\n    \\node at (2.75,0) {\\Huge{$3$}};\n    \\node at (0,-2.75) {\\Huge{$3$}};\n    \\node at (-2.75,0) {\\Huge{$3$}};\n    \\node at (0.5,0.5) {\\Huge{$0$}};\n    \\path (A) edge node {} (M);\n    \\path (M) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (A) edge node {} (B);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n    \\path (B) edge node {} (M);\n    \\path (C) edge node {} (M);\n   \n\\end{tikzpicture} \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (3) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (4) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=red] (5) at (8,0) {};\n    \\node[shape=circle,draw=black, fill=red] (6) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (7) at (6,-2) {};\n    \\path (3) edge node {} (4);\n    \\path (4) edge node {} (5);\n    \\path (4) edge node {} (6);\n    \\path (4) edge node {} (7);\n    \\node at (3.25,0) {\\Huge{$1$}};\n    \\node at (6,2.75) {\\Huge{$1$}};\n    \\node at (6.5,.5) {\\Huge{$-2$}};\n    \\node at (6,-2.75) {\\Huge{$1$}};\n    \\node at (8.75,0) {\\Huge{$1$}};\n    \\end{tikzpicture}\n    \n   }\n    \\caption{Several graphs with the boundary stability number of each vertex shown next to it. In general, leaves in any graph and all vertices in the complete graph $K_n$ for $n\\geq 2$ have boundary stability number 1.}\n    \\label{stabilityexample}\n\\end{figure}\n\nWe now show some lemmas, starting with cut vertices and followed by a characterization of vertices of degree 2. The method for establishing \\cref{degree2lemma} is based on part of the proof of \\cite[Proposition 1.2]{chartrand}.\n\n\\begin{lemma}\\label{cutdistance}\nIf $G=(V,E)$ is connected, $v$ is a cut vertex of $G$, and $u$ and $w$ are in different components of $G-v$, then $d(u,w)=d(u,v)+d(v,w)$.\n\\end{lemma}\n\n\\begin{proof}\n   \n    The bound $d(u,w)\\leq d(u,v)+d(v,w)$ follows by triangle inequality. For equality, note that $u$ and $w$ are in different components of $G-v$, so every path in $G$ from $u$ to $w$ must contain $v$. Hence, $d(u,w)\\geq d(u,v)+d(v,w)$, and equality must hold.\n\\end{proof}\n\n\n\\begin{lemma}\\label{degree2lemma}\nLet $G=(V,E)$ be a connected graph.  A vertex $v\\in V$ with $\\deg(v)=2$ is a boundary vertex if and only if there exists a cycle in $G$ that contains $v$.\n\\end{lemma}\n\n\\begin{proof}\nLet $N(v)=\\{v_1,v_2\\}$.  Suppose there exists a cycle in $G$ that contains $v$. Then choose such a cycle $C$ so that the number of vertices in it is minimal and let $k$ denote the number of vertices in $C$. Observe that both neighbors of $v$ are in this cycle, and $d(v,w)\\leq \\lceil k\/2\\rceil$ for any $w\\in C$. Since $C$ is a minimal cycle containing $v$, there exists $u\\in C$ where equality holds. Additionally, $d(v_i,u)\\leq \\lceil k\/2\\rceil$ for $i\\in \\{1,2\\}$, and combined with \\cref{neighbordistance}, we see that \\[\\beta(v,u)=[d(v,u)-d(v_1,u)]+[d(v,u)-d(v_2,u)] \\geq 1.\\]\nHence, $\\beta(v)\\geq 1$ and $v\\in \\partial G$.\n\nConversely, suppose $v\\in \\partial G$. Fix a witness $u\\in V$, and assume without loss of generality that some shortest path from $u$ to $v$ has the form $P_1=vv_1w_2\\ldots w_{\\ell-1}u$, so $d(v_1,u)=d(v,u)-1$. Combining \\cref{neighbordistance} with $v\\in \\partial G$, it must be that $d(v_2,u)\\leq d(v,u)$. This implies that a shortest path $P_2=v_2w_1'w_2'w_3'\\ldots w_{k-1}'u$ from $v_2$ to $u$ cannot contain $v$. Then $W=vv_1w_2\\ldots w_{\\ell-1} uw_{k-1}'\\ldots w_1'v_2v$ is a closed walk that contains $v,v_1$, and $v_2$. Finding the first vertex $w$ in $w_2w_3\\ldots w_{\\ell-1}u$ that also appears in $v_2w_1'w_2'w_3'\\ldots w_{k-1}'u$ and removing all vertices in between them in $W$ produces a cycle containing $v$.\n\\end{proof}\n\n\\begin{corollary}\\label{deg2cut}\nLet $G=(V,E)$ be a connected graph. A vertex $v\\in V$ with $\\deg(v)=2$ is either a boundary vertex of $G$ or a cut vertex of $G$. \n\\end{corollary}\n\n\n\\begin{proof} \nSuppose $v\\in \\partial G$ has $\\deg(v)=2$. Then by \\cref{degree2lemma} there is a cycle $C$ in $G$ which contains $v$. For any vertices $u,w\\in V\\setminus \\{v\\}$, there is a path $P$ in $G$ from $u$ to $w$. If $P$ contains $v_1vv_2$ or $v_2vv_1$, then we can replace these with paths between $v_1$ and $v_2$ in $C\\setminus \\{v\\}$ to produce a path $P'$ from $u$ to $w$ in $G-v$. Hence, $G-v$ is connected, and $v$ is not a cut vertex. \n\n\nIf $v$ is a degree 2 cut vertex, then $v_1$ and $v_2$ must be in different components of $G-v$. Furthermore, any connected subgraph of $G$ containing $v$ and its neighbors also has $v$ as a cut vertex. Because any cycle containing $v$ must also contain $v_1,v_2$ and the subgraph correspond to a cycle does not have cut vertices, there cannot be a cycle in $G$ that contains $v$. By \\cref{degree2lemma}, we conclude that $v\\notin \\partial G$.\n\\end{proof}\n\n\nWe now consider the effects of certain graph operations. Observe that in \\cref{mullerthm}, attaching paths to vertices in $(\\partial G)'$ results in a graph $H$ with $|(\\partial H)'|=|(\\partial G)'|$. This is not always true for vertices in $\\partial G$, and we will see that the boundary stability number allows us to determine precisely when the number of boundary vertices is preserved.\n\n\n{\n\\begin{lemma}\\label{graphedgeattachment}\nLet $G_1=(V_1,E_1)$ be a connected graph on at least two vertices with $v_1\\in V_1$. Let $G_2=(V_2,E_2)$ a connected graph, let $v_2\\in V_2$, and define $G=(V_1\\cup V_2,E_1\\cup E_2\\cup(v_1,v_2))$. Then $\\beta_G(v_1)=\\beta_{G_1}(v_1)-1$, and for $v\\in V_1\\setminus \\{v_1\\}$, we have that $\\beta_G(v)=\\beta_{G_1}(v)$.\n\\end{lemma}\n\n\\begin{proof}\nObserve that $v_1$ is a cut vertex of $G$, and $N_G(v_1)=N_{G_1}(v_1)\\cup \\{v_2\\}$. Additionally, for $u,w\\in V_1$, we have that $d_G(w,u)=d_{G_1}(w,u)$. Combining this with \\cref{cutdistance}, we see that for any $u\\in V_1$,\n\\begin{equation*}\n    \\begin{split}\n        \\beta_G(v_1, u)&=\\beta_{G_1}(v_1,u)+ [d_G(v_1,u)-d_G(v_2,u)] \\\\\n    & = \\beta_{G_1}(v_1,u)+[d_G(v_1,u)-(1+d_G(v_1,u))] \\\\\n    &=\\beta_{G_1}(v_1,u)-1.\n    \\end{split}\n\\end{equation*}\nAdditionally, for any $u\\in V_2$, we again use \\cref{cutdistance} to find\n\\begin{equation*}\n    \\begin{split}\n        \\beta_G(v_1, u) & = [d_G(v_1,u)-d_G(v_2,u)]+\\sum_{w\\in N_{G_1}(v)} [d_G(v_1,u)-d_G(w,u)]\\\\\n    & = [d_G(v_1,u)-d_G(v_2,u)]+\\sum_{w\\in N_{G_1}(v_1)} [d_G(v_1,u)-(1+d_G(v_1,u))]\\\\\n    & = 1-|N_{G_1}(v_1)|.\n    \\end{split}\n\\end{equation*}\nSince $|V_1|\\geq 2$, choosing some $u\\in V_1$ distinct from $v_1$ and applying \\cref{neighbordistance} implies that \\[\\beta_{G_1}(v_1)\\geq \\beta_G(v_1,u)\\geq  1-(|N_{G_1}(v_1)|-1)=2-|N_{G_1}(v_1)|.\\]\nHence, \\[\\beta_G(v_1)=\\max_{u\\in V_1\\cup V_2} \\{\\beta_G(v_1,u)\\}=\\max_{u\\in V_1}\\{\\beta_{G_1}(v_1,u)-1\\}=\\beta_{G_1}(v_1)-1.\\]\n\nNow consider $v\\in V_1\\setminus \\{v_1\\}$, where\n$N_{G}(v)=N_{G_1}(v)$. As before, for $u,w\\in V_1$, we have that $d_G(w,u)=d_{G_1}(w,u)$, so $\\beta_G(v,u)=\\beta_{G_1}(v,u)$. For $u\\in V_2$, \\cref{cutdistance} and $d_{G}(v,v_1)=d_{G_1}(v,v_1)$ imply\n\\begin{align*}\n    \\beta_G(v, u) & = \\sum_{w\\in N_{G_1}(v)} [d_G(v,u)-d_G(w,u)]\\\\\n    & = \\sum_{w\\in N_{G_1}(v)} [(d_G(v,v_1)+d_G(v_1,u))-(d_G(w,v_1)+d_G(v_1,u))] \\\\\n    & = \\sum_{w\\in N_{G_1}(v)} [d_{G}(v,v_1)-d_{G}(w,v_1)] \\\\\n    & = \\sum_{w\\in N_{G_1}(v)} [d_{G_1}(v,v_1)-d_{G_1}(w,v_1)] \\\\\n    & = \\beta_{G_1}(v,v_1).\n\\end{align*}\nHence, we conclude that \\[\\beta_G(v)=\\max_{u\\in V_1\\cup V_2} \\{\\beta_G(v,u)\\}=\\max_{u\\in V_1}\\{\\beta_{G_1}(v,u)\\}=\\beta_{G_1}(v).\\qedhere\\]\n\\end{proof}\n\n\nAn example of \\cref{graphedgeattachment} is shown in \\cref{attachmentexample}. One consequence of this result is that we can determine precisely how $\\partial G$ relates to $\\partial G_1$ and $\\partial G_2$. \n\n\\begin{figure}[h!]\n\\begin{subfigure}{0.4\\textwidth}\n    \\centering\n    \\scalebox{0.4}{\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n   \n    \\node[shape=circle,draw=black, fill=red] (3) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (4) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=red] (5) at (8,0) {};\n    \\node[shape=circle,draw=black, fill=red] (6) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (7) at (6,-2) {};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n   \n    \\path (C) edge node {} (D);\n   \n   \n    \\path (3) edge node {} (4);\n    \\path (4) edge node {} (5);\n    \\path (4) edge node {} (6);\n    \\path (4) edge node {} (7);\n    \\node at (-2.75,0) {\\huge{$2$}};\n    \\node at (0,2.75) {\\huge{$1$}};\n    \\node at (0,-2.75) {\\huge{$1$}};\n    \\node at (2,-0.75) {\\huge{$2$}};\n    \\node at (2,0.75) {\\huge{$v_1$}};\n   \n    \\node at (4,-0.75) {\\huge{$1$}};\n    \\node at (4,0.75) {\\huge{$v_2$}};\n    \\node at (6,2.75) {\\huge{$1$}};\n    \\node at (6.75,-0.75) {\\huge{$-2$}};\n    \\node at (6,-2.75) {\\huge{$1$}};\n    \\node at (8.75,0) {\\huge{$1$}};\n    \\end{tikzpicture}}\n\\end{subfigure}$~~~$\n\\begin{subfigure}{0.4\\textwidth}\n    \\centering\n    \\scalebox{0.4}{\n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n   \n    \\node[shape=circle,draw=black, fill=blue] (3) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (4) at (6,0) {};\n    \\node[shape=circle,draw=black, fill=red] (5) at (8,0) {};\n    \\node[shape=circle,draw=black, fill=red] (6) at (6,2) {};\n    \\node[shape=circle,draw=black, fill=red] (7) at (6,-2) {};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n   \n    \\path (C) edge node {} (D);\n    \\path (D) edge node {} (3);\n   \n    \\path (3) edge node {} (4);\n    \\path (4) edge node {} (5);\n    \\path (4) edge node {} (6);\n    \\path (4) edge node {} (7);\n    \\node at (-2.75,0) {\\huge{$2$}};\n    \\node at (0,2.75) {\\huge{$1$}};\n    \\node at (0,-2.75) {\\huge{$1$}};\n    \\node at (2,-0.75) {\\huge{$1$}};\n    \\node at (2,0.75) {\\huge{$v_1$}};\n   \n    \\node at (4,-0.75) {\\huge{$0$}};\n    \\node at (4,0.75) {\\huge{$v_2$}};\n    \\node at (6,2.75) {\\huge{$1$}};\n    \\node at (6.75,-0.75) {\\huge{$-2$}};\n    \\node at (6,-2.75) {\\huge{$1$}};\n    \\node at (8.75,0) {\\huge{$1$}};\n    \\end{tikzpicture}\n    }\n\\end{subfigure}\n\\caption{Example of \\cref{graphedgeattachment}. The graphs $G_1, G_2,$ and $G$ are given with boundary vertices shown in red. For each $v\\in V_1,V_2,V$, $\\beta(v)$ is shown next to $v$.}\n\\label{attachmentexample}\n\\end{figure}\n\n\\begin{theorem}\\label{edgeattachment}\nLet $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be connected graphs, and let $v_1\\in V_1$ and $v_2\\in V_2$. Define $G=(V_1\\cup V_2,E_1\\cup E_2\\cup(v_1,v_2))$.\n\\begin{enumerate}[label=(\\alph*)]\n            \\item If $\\beta_{G_1}(v_1)\\neq 1$ and $\\beta_{G_2}(v_2)\\neq 1$, then $\\partial G=\\partial G_1\\cup \\partial G_2$.\n        \\item If $\\beta_{G_1}(v_1)=1$ and $\\beta_{G_2}(v_2)\\neq 1$, then $\\partial G=(\\partial G_1\\cup \\partial G_2) \\setminus \\{v_1\\}$.\n        \\item If $\\beta_{G_1}(v_1)\\neq 1$ and $\\beta_{G_2}(v_2)= 1$, then $\\partial G=(\\partial G_1\\cup \\partial G_2) \\setminus \\{v_2\\}$.\n        \\item If $\\beta_{G_1}(v_1)=1$ and $\\beta_{G_2}(v_2)=1$, , then $\\partial G=(\\partial G_1\\cup \\partial G_2) \\setminus \\{v_1,v_2\\}$.\n\\end{enumerate}\nIn particular, we have that $|\\partial G|\\geq \\max\\{|\\partial G_1|,|\\partial G_2|\\}$.\n\\end{theorem}\n\\begin{proof}\nFirst, suppose that $|V_1|>1$ and $|V_2|>1$. Then \\cref{graphedgeattachment} implies that for any vertices $v\\in V_i\\setminus \\{v_i\\}$, we have $v\\in \\partial G_i$ if and only if $v\\in \\partial G$. For $v_i$, observe that the same is true except when $\\beta_{G_i}(v_i)=1$, and in this case $v_i\\in \\partial G_i$ and $v_i\\notin \\partial G$. The theorem then follows from the various cases.\n\nNow suppose $|V_1|>1$ and $|V_2|=1$. In this case, $\\partial G_2=\\{v_2\\}$ and $\\beta_{G_2}(v_2)=0$, so it suffices to show (a) and (b) with $\\partial G_2=\\{v_2\\}$. Since $v_2$ is a leaf in $G$, \\cref{treeandleaves} implies $v_2\\in \\partial G$. Applying the above argument with \\cref{graphedgeattachment} on $V_1$, we conclude (a) and (b). The case $|V_1|=1$ and $|V_2|>1$ is similar. Finally, when $|V_1|=|V_2|=1$, we see that $\\partial G_i=\\{v_i\\}$ and $\\beta_{G_i}(v_i)=0$ for all $i$. In this case, $G$ is a path graph on two vertices, and (a) is clear.\n\\end{proof}\n}\n\nWe conclude this section with results on subgraphs. Using the boundary stability number, we can sometimes determine when boundary vertices in a subgraph of $G$ are also boundary vertices in $G$ itself.\n\n\\begin{lemma}\\label{boundarysubgraph}\nLet $G$ be a connected graph with a connected subgraph $H$ on at least two vertices. Suppose $v\\in \\partial H$, and let $u$ be a witness for $v$. If $N_{H}(v)=N_G(v)$ and $d_{H}(v,u)=d_G(v,u)$, then $v \\in \\partial G$.\n\\end{lemma}\n\n\\begin{proof}\nNotice that for any $w\\in N_G(v)=N_{H}(v)$, we have that $d_G(w,u)\\leq d_{H}(w,u)$. Then a direct calculation shows \n    \\begin{align*}\n        \\beta_{G}(v)&\\geq \\beta_G(v,u) \\\\\n        & =\\sum_{w\\in N_G(v)}[d_G(v,u)-d_G(w,u)] \\\\\n       \n        & \\geq \\sum_{w\\in N_{H}(v)}[d_{H}(v,u)-d_{H}(w,u)]\\\\\n        & = \\beta_{H}(v,u) \\\\\n        & \\geq 1. \\qedhere\n    \\end{align*}\n\\end{proof}\n\n\n\\begin{corollary}\\label{boundarysubgraphcor}\nLet $G$ be a connected graph with a connected subgraph $H$ on at least two vertices. Suppose $v\\in \\partial H$, and let $u$ be a witness for $v$. If $N_{H}(v)=N_G(v)$ and $d_{H}(v,u)=2$, then $v \\in \\partial G$.\n\\end{corollary}\n\\begin{proof}\nSince $H$ is a subgraph of $G$, we have that $d_G(v,u)\\leq d_H(v,u)=2$. Observe that $d_H(v,u)=2$ implies $u\\notin N_H(v)$, and the assumption $N_{H}(v)=N_G(v)$ implies $u\\notin N_G(v)$. Then $d_G(v,u)>1$, and combined, we conclude that $d_G(v,u)=2$. The result then follows from \\cref{boundarysubgraph}.\n\\end{proof}\n\n\n\n\n\\section{Graphs with at most four boundary vertices}\\label{boundarysize4}\n\nIn this section, we establish the proof of \\cref{maintheorem}. We start by applying the results of \\cref{stability} to establish lemmas for graphs with $|\\partial G|=3$ or $|\\partial G|=4$.\n\n\\begin{lemma}\\label{boundary3lemma}\nLet $G=(V,E)$ be a connected graph. If $|(\\partial G)'|=3$, then $|\\partial G|=3$. \n\\end{lemma}\n\n\\begin{proof}\nWe use the characterization of $|(\\partial G)'|=3$ given in \\cref{CEJZ3}. If $G$ is a tree on three leaves, then \\cref{treeandleaves} implies that $|\\partial G|=3$. For tripods, we start by considering the complete graph $K_3$. A direct calculation shows that $|\\partial K_3|=3$, and each vertex $v\\in V$ has $\\beta(v)=1$. Tripods are formed by attaching arbitrary length paths to the vertices of $K_3$. By applying \\cref{graphedgeattachment} for each nontrivial path attached, we conclude that tripods have three boundary vertices. \n\\end{proof}\n\n\n\n\\begin{lemma}\\label{boundary4lemma}\nLet $G=(V,E)$ be a connected graph with $|(\\partial G)'|=4$. Then $|\\partial G|=4$ if and only if $G$ is one of the following graphs:\n\\begin{enumerate}[label=(\\alph*)]\n    \\item a subdivision of the star graph $K_{1,4}$,\n    \\item a subdivision of the tree with exactly four leaves and two vertices of degree~3,\n    \\item a graph obtained from one of the trees in (b) by removing a vertex of degree 3 and adding edges between all of its neighbors, or\n    \\item one of the graphs in \\cref{partialG4base} with a path of arbitrary length attached to each $v\\in \\partial G$ with $\\beta(v)=1$. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{figure}[h!]\n    \\centering\n    \\scalebox{0.4}{\n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n    \\path (A) edge node {} (M);\n    \\path (M) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (A) edge node {} (B);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n    \\path (B) edge node {} (M);\n    \\path (C) edge node {} (M);\n    \\node at (0,-3) {\\Huge{$N_{1,1}$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n    \\node at (0,-3) {\\Huge{$C_4$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (D);\n    \\path (C) edge node {} (D);\n    \\node at (0,-3) {\\Huge{$K_4=X_{1,1}$}};\n\\end{tikzpicture}} \n\\\\\n\\phantom{-}\\\\\n\\scalebox{0.4}{\n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (-4,2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (-4,-2) {};\n    \\node[shape=circle,draw=black, fill=blue] (M1) at (-2,0) {};\n    \\node (P) at (0,0) {$\\cdots$};\n    \\node[shape=circle,draw=black, fill=blue] (M2) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (4,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (4,2) {};\n\n    \\path (A) edge node {} (M1);\n    \\path (M1) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (M2);\n    \\path (C) edge node {} (M2);\n    \\path (M1) edge node {} (P);\n    \\path (P) edge node {} (M2);\n    \\node at (0,-3) {\\Huge{$X_{1\\times c}$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n    \\path (A) edge node {} (M);\n    \\path (M) edge node {} (D);\n    \\path (A) edge node {} (D);\n    \\path (A) edge node {} (B);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (M);\n    \\path (C) edge node {} (M);\n    \\node at (0,-3) {\\Huge{$T_{1,1}$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n    \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (C) edge node {} (D);\n    \\node at (0,-3) {\\Huge{$D_{1,1}=L_{1,1}$}};\n\\end{tikzpicture}}\n\n\\caption{Graphs from \\cref{mullerthm} (d)-(h) with $|\\partial G|=4$.}\n\\label{partialG4base}\n\\end{figure}\n\n\n\\begin{proof}\nWe show this by considering each case of \\cref{mullerthm}. Note that graphs (a), (b), and (c) match \\cref{mullerthm}, and the results for (a) and (b) follow directly from \\cref{treeandleaves}. For (c), notice that these graphs $G$ can alternatively be constructed by starting with a tripod $H$ and attaching two nontrivial paths to some $v\\in \\partial H$. Observe that $H$ is constructed by attaching paths of arbitrary lengths to $K_3$. Using \\cref{graphedgeattachment} on $K_3$ with paths attached, the boundary stability number of any vertex in $\\partial H$ is 1. Using \\cref{graphedgeattachment} once for each path attached to $v\\in \\partial H$, we conclude that $|\\partial G|=4$.\n\nNow consider the remaining cases in \\cref{mullerthm}. Notice that if $G$ is constructed as a subgraph of $N_{a\\times c}$, $X_{a\\times c}$, $T_{a\\times c}$, $D_{a\\times c}$, or $L_{a\\times c}$ and $|\\partial G|\\geq |(\\partial G)'|\\geq 5$, then by \\cref{edgeattachment}, a graph $H$ formed by attaching paths to $G$ has $|\\partial H|\\geq |\\partial G|\\geq 5$. Hence, we must consider constructions where $|(\\partial G)'|=4$ to obtain a graph $H$ with $|\\partial H|=4$. We do this using the graphs in \\cref{basecases}, which will allow us to show the existence of additional Steinerberger boundary vertices in addition to the four CEJZ boundary vertices described in \\cref{mullerremark}. Notice that for each $G_i$ in the figure, $v\\in \\partial G_i$ with witness $u$, and $d_{G_i}(v,u)=2$.\n\n\\begin{figure}[h!]\n    \\centering\n    \\scalebox{0.45}{\n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\path (00) edge node {} (10);\n    \\path (20) edge node {} (10);\n    \\path (01) edge node {} (11);\n    \\path (21) edge node {} (11);\n    \\path (00) edge node {} (01);\n    \\path (10) edge node {} (11);\n    \\path (20) edge node {} (21);\n    \\node[shape=circle,draw=black, fill=blue] (05) at (1,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (15) at (3,1) {};\n    \\path (00) edge node {} (05);\n    \\path (10) edge node {} (05);\n    \\path (01) edge node {} (05);\n    \\path (11) edge node {} (05);\n    \\path (10) edge node {} (15);\n    \\path (11) edge node {} (15);\n    \\path (20) edge node {} (15);\n    \\path (21) edge node {} (15);\n    \\path (05) edge node {} (15);\n    \\node at (2,-0.5) {\\huge{$v$}};\n    \\node at (4,2.5) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_1$}};\n    \\end{tikzpicture}\n    \\quad \n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\path (00) edge node {} (10);\n    \\path (20) edge node {} (10);\n    \\path (01) edge node {} (11);\n    \\path (21) edge node {} (11);\n    \\path (00) edge node {} (01);\n    \\path (10) edge node {} (11);\n    \\path (20) edge node {} (21);\n    \\node[shape=circle,draw=black, fill=blue] (05) at (1,1) {};\n    \\path (00) edge node {} (05);\n    \\path (10) edge node {} (05);\n    \\path (01) edge node {} (05);\n    \\path (11) edge node {} (05);\n     \\node at (2,-0.5) {\\huge{$v$}};\n    \\node at (4,2.5) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_2$}};\n    \\node at (-0.5,2) {};\n    \\end{tikzpicture}\n    \\quad \n    \\begin{tikzpicture}\n    \n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\path (00) edge node {} (10);\n    \\path (20) edge node {} (10);\n    \\path (01) edge node {} (11);\n    \\path (21) edge node {} (11);\n    \\path (00) edge node {} (01);\n    \\path (10) edge node {} (11);\n    \\path (20) edge node {} (21);\n     \\node at (2,-0.5) {\\huge{$v$}};\n    \\node at (4,2.5) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_3$}};\n    \\end{tikzpicture} \n    }\n    \\\\\n    \\phantom{-}\\\\\n    \\scalebox{0.45}{\n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (4,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (0,4) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,4) {};\n    \\path (00) edge node {} (10);\n    \\path (20) edge node {} (10);\n    \\path (01) edge node {} (11);\n    \\path (21) edge node {} (11);\n   \n    \\path (10) edge node {} (11);\n   \n    \\node[shape=circle,draw=black, fill=blue] (05) at (2,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (15) at (2,3) {};\n    \\path (00) edge node {} (05);\n    \\path (10) edge node {} (05);\n    \\path (01) edge node {} (05);\n    \\path (11) edge node {} (05);\n    \\path (10) edge node {} (15);\n    \\path (11) edge node {} (15);\n    \\path (20) edge node {} (15);\n    \\path (21) edge node {} (15);\n    \\path (05) edge node {} (15);\n    \\node at (-0.5,2) {\\huge{$v$}};\n    \\node at (4.5,0) {\\huge{$u$}};\n    \\node at (2,-0.75) {\\Huge{$G_4$}};\n    \\end{tikzpicture}\n    \\quad \n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (4n) at (4,-1) {};\n    \\node[shape=circle,draw=black, fill=blue] (41) at (4,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (43) at (4,3) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (1,1) {};\n    \\path (00) edge node {} (20);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (22);\n    \\path (22) edge node {} (20);\n    \\path (22) edge node {} (43);\n    \\path (22) edge node {} (41);\n    \\path (20) edge node {} (41);\n    \\path (43) edge node {} (41);\n    \\path (20) edge node {} (4n);\n    \\path (41) edge node {} (4n);\n    \\path (11) edge node {} (00);\n    \\path (11) edge node {} (20);\n    \\path (11) edge node {} (02);\n    \\path (11) edge node {} (22);\n    \\node at (2,2.5) {\\huge{$v$}};\n    \\node at (4.5,-1) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_5$}};\n    \\end{tikzpicture}\n    \\quad \n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (4n) at (4,-1) {};\n    \\node[shape=circle,draw=black, fill=blue] (41) at (4,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (43) at (4,3) {};\n    \\path (00) edge node {} (20);\n    \\path (00) edge node {} (02);\n    \\path (02) edge node {} (22);\n    \\path (22) edge node {} (20);\n    \\path (22) edge node {} (43);\n    \\path (22) edge node {} (41);\n    \\path (20) edge node {} (41);\n    \\path (43) edge node {} (41);\n    \\path (20) edge node {} (4n);\n    \\path (41) edge node {} (4n);\n    \\node at (2,2.5) {\\huge{$v$}};\n    \\node at (4.5,-1) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_6$}};\n    \\end{tikzpicture}}\n    \\\\\n     \\phantom{-}\\\\\n     \\scalebox{0.45}{\n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (02) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (04) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (1n) at (-1,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (1,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (13) at (3,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (15) at (5,2) {};\n    \\path (00) edge node {} (02);\n    \\path (00) edge node {} (1n);\n    \\path (00) edge node {} (11);\n    \\path (02) edge node {} (11);\n    \\path (02) edge node {} (13);\n    \\path (02) edge node {} (04);\n    \\path (04) edge node {} (13);\n    \\path (04) edge node {} (15);\n    \\path (13) edge node {} (15);\n    \\path (1n) edge node {} (11);\n    \\path (11) edge node {} (13);\n    \\node at (3,2.5) {\\huge{$v$}};\n    \\node at (-0.5,0) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_7$}};\n    \\end{tikzpicture}\n    \\quad \n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n        \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n        \\node[shape=circle,draw=black, fill=blue] (11') at (2.5,2.5) {};\n        \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n        \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n        \\node[shape=circle,draw=black, fill=blue] (02) at (0,4) {};\n        \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n        \\node[shape=circle,draw=black, fill=blue] (22) at (4,4) {};\n        \\path (00) edge node {} (10);\n        \\path (20) edge node {} (10);\n        \\path (01) edge node {} (11);\n        \\path (21) edge node {} (11);\n        \\path (00) edge node {} (01);\n        \\path (10) edge node {} (11);\n        \\path (20) edge node {} (21);\n        \\path (11) edge node {} (11');\n        \\path (01) edge node {} (11');\n        \\path (10) edge node {} (11');\n        \\path (12) edge node {} (11');\n        \\path (02) edge node {} (12);\n        \\path (12) edge node {} (11);\n        \\path (02) edge node {} (01);\n        \\path (21) edge node {} (22);\n        \\path (22) edge node {} (12);\n       \n        \\path (22) edge node {} (11');\n       \n       \n        \\path (01) edge node {} (12);\n        \\path (00) edge node {} (11);\n        \\path (10) edge node {} (21);\n        \\path (11') edge node {} (21);\n        \\node at (-0.5,2) {\\huge{$v$}};\n        \\node at (1.625,0.325) {\\huge{$u$}};\n        \\node at (2,-0.75) {\\Huge{$G_8$}};\n    \\end{tikzpicture}\n    \\quad \n    \\begin{tikzpicture}\n    \\node at (-1,0) {};\n    \\node at (5,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n    \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n    \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n    \\path (00) edge node {} (10);\n    \\path (20) edge node {} (10);\n    \\path (01) edge node {} (11);\n    \\path (21) edge node {} (11);\n    \\path (00) edge node {} (01);\n    \\path (10) edge node {} (11);\n    \\path (20) edge node {} (21);\n    \\path (00) edge node {} (11);\n    \\path (10) edge node {} (21);\n    \\node at (2,-0.5) {\\huge{$v$}};\n    \\node at (0,2.5) {\\huge{$u$}};\n    \\node at (2,-1.75) {\\Huge{$G_9$}};\n    \\end{tikzpicture}\n    }\n    \\caption{In each $G_i$, observe that $v\\in \\partial G_i$ with witness $u$, and $d_{G_i}(v,u)=2$.}\n    \\label{basecases}\n\\end{figure}\n\n\nFirst, consider $N_{a\\times c}$, and suppose $G$ is the subgraph induced by $V_{a\\times c}^0\\cup (W\\cap V_{a\\times c}^1)$ for axis slice convex $W$. \nIf $a>1$ and $W$ contains both, one, or none of the vertices in $\\{(1\/2,1\/2),(3\/2,1\/2)\\}$, then $G$ will respectively contain $G_1$, $G_2$, or $G_3$ from \\cref{basecases} as a subgraph with $N_G(v)=N_{G_i}(v)$, where in each case, $v$ corresponds to the vertex $(1,0)$ in $N_{a\\times c}$. A similar argument applies if $c>1$, where $v$ corresponds to the vertex $(0,1)$.\nIn these cases, \\cref{boundarysubgraphcor} implies that $v\\in \\partial G$ and $|\\partial G|\\geq 5$. From this, we see that the only graphs $G$ that have four boundary vertices must be constructed using $N_{1\\times 1}$, which are $N_{1\\times 1}$ itself or the cycle graph $C_4$. A direct verification shows $|\\partial G|=4$ for both of these graphs, and they are shown in \\cref{partialG4base}.\n\nNow consider $G=X_{a\\times c}$. Recall that $X_{a\\times c}$ is isomorphic to $X_{c\\times a}$, so assume without loss of generality that $a\\leq c$. \nIf $c\\geq 2$ and $a=2$, then $X_{a\\times c}$ contains $G_4$ as a subgraph with $N_G(v)=N_{G_4}(v)$, where $v$ corresponds to the vertex $(0,1)$ in $X_{a\\times c}$. If $c\\geq a\\geq 2$, then $X_{a\\times c}$ contains $G_5$ as a subgraph with $N_G(v)=N_{G_5}(v)$, where $v$ corresponds to the vertex at $(1\/2,1\/2)$. In these cases, \\cref{boundarysubgraphcor} implies $|\\partial G|\\geq 5$. The graphs $X_{1\\times 1}=K_4$ and $X_{1\\times c}$ are depicted in \\cref{partialG4base}, and a direct calculation shows that when $G$ is one of these graphs, $|\\partial G|=4$.\n\n\nNext, consider $T_{a\\times c}$, and suppose that $G$ is the subgraph of $T_{a\\times c}$ induced by $V_{a\\times (c+1)}^1\\cup (W\\cap V_{a\\times (c+1)}^0)$ for some axis slice convex $W$ that contains $(a,0)$ and $(a,c+1)$. If $a>1$ and $(a-1,c)\\in W$, then $G$ will contain $G_5$ from \\cref{basecases} as a subgraph with $N_G(v)=N_{G_5}(v)$, where $v$ corresponds to $(a-1\/2,c+1\/2)$. If $a>1$ and $(a-1,c)\\not\\in W$, then a similar statement is true for $G_6$, where $v$ again corresponds to $(a-1\/2,c+1\/2)$. If $a=1$ and $c>1$, then $G$ contains $G_7$ from \\cref{basecases} as a subgraph with $N_G(v)=N_{G_7}(v)$, where $v$ corresponds to $(1,1)$. In all of these cases, \\cref{boundarysubgraphcor} implies that $|\\partial G|\\geq 5$. Finally, when $a=c=1$,\nnote that any axis slice convex set containing $(1,0)$ and $(1,2)$ must also contain $(1,1)$. Hence, the only graph in this case is $T_{1\\times 1}$. Direct verification shows $|\\partial T_{1\\times 1}|=4$, and this graph is shown in \\cref{partialG4base}.\n\n\nNow consider $G=D_{a\\times c}$. If $a\\geq 2$ and $c\\geq 2$, then the graph $G_8$ from \\cref{basecases} is a subgraph with $N_G(v)=N_{G_8}(v)$, where $v$ corresponds to the $(0,1)$ in $D_{a\\times c}$.\nIf either $a=1$ and $c\\geq 2$, or $a\\geq 2$ and $c=1$, then $G_9$ from \\cref{basecases} is a subgraph of $G$ with $N_G(v)=N_{G_9}(v)$. Thus in each of these cases, \\cref{boundarysubgraphcor} implies $v\\in \\partial G$ and $|\\partial G|\\geq 5$. Finally, if $a=c=1$ then $|\\partial G|=4$, and $D_{1\\times 1}$ is depicted in \\cref{partialG4base}. \n\n\nFinally, consider $G=L_{a\\times c}$. If $a>1$, then $G$ contains $G_{9}$ as a subgraph with $N_G(v)=N_{G_{9}}(v)$, where $v$ corresponds to $(1,0)$. When $a=1$, observe that by definition, $D_{1\\times c}=L_{1\\times c}$, and we have already considered these cases above.\n\nCombined, we see that the only graphs described in \\cref{mullerthm} parts (d)-(h) with $|\\partial G|=4$ are those formed by attaching paths to the graphs in \\cref{partialG4base} at the vertices in $\\partial G=(\\partial G)'$. By applying \\cref{graphedgeattachment} repeatedly, we conclude that attaching nontrivial paths at $v\\in \\partial G$ preserves the number of boundary vertices if and only if $\\beta(v)=~1$.\n\\end{proof}\n\n\nWe are now able to prove our main theorem characterizing graphs with small Steinerberger boundary.\n\n\\begin{proof}[Proof of \\cref{maintheorem}]  \nBy \\cref{CEJZ2} and \\cref{boundary2}, $|\\partial G|=2$ and $|(\\partial G')|=2$ are equivalent, corresponding precisely to paths. This establishes \\cref{maintheorem}(a). Combined with $(\\partial G)'\\subseteq \\partial G$, we also see that a necessary condition for $|\\partial G|=3$ is that $|(\\partial G)'|=3$. Applying \\cref{boundary3lemma}, we conclude that $|\\partial G|=3$ if and only if $|(\\partial G)'|=3$, implying \\cref{maintheorem}(b). \n\nUsing similar reasoning, we see that a necessary condition for $|\\partial G|=4$ is that $|(\\partial G)'|=4$. We consider each case in \\cref{boundary4lemma}. By \\cref{treeandleaves}, any tree on four leaves has $|\\partial G|=4$, and this accounts for \\cref{boundary4lemma} (a) and (b). For \\cref{boundary4lemma} (c), observe that this is the last graph in \\cref{partialG4}. The remaining graphs in \\cref{boundary4lemma} (d) are also given in \\cref{partialG4}. Hence, we conclude \\cref{maintheorem}(c). \n\\end{proof}\n\n\n\n\\section{Some graphs with large boundary}\\label{largeboundary}\n\nIn this section, we consider some graphs with large Steinerberger boundary. Observe that the cycle and complete graphs consist entirely of boundary vertices, as each vertex is peripheral. In the case of $\\diam(G)=2$, Chartrand, Erwin, Johns, and Zhang showed the following result on the CEJZ boundary.\n\n\\begin{lemma}[Chartrand, Erwin, Johns, and Zhang \\cite{chartrand}, Lemma 2.1]\\label{diameter2}\nLet $G=(V,E)$ be connected graph of diameter 2. Then every vertex $v$ is in $(\\partial G)'$ unless $v$ is the unique vertex of $G$ having eccentricity $1$.\n\\end{lemma}\n\nThe same result holds for the Steinerberger boundary. We now show this, with some additional results.\n\n\\begin{theorem}\\label{diameter}\nLet $G=(V,E)$ be a connected graph on at least two vertices with $\\diam(G)\\leq 2$. If $G$ has a single vertex $v$ with eccentricity 1, then $\\partial G=V\\setminus \\{v\\}$. Otherwise, $\\partial G=V$. Furthermore, the bound $\\diam(G)\\leq 2$ is sharp.\n\\end{theorem}\n\n\n\\begin{proof}\nIf $G$ has diameter 1, then $G$ is the complete graph $K_n$, which satisfies $\\partial G=V$. Otherwise, by \\cref{containment} and \\cref{diameter2}, every vertex in G with eccentricity greater than 1 is a Steinerberger boundary vertex. If $G$ has no vertices of eccentricity 1, then $\\partial G=V$. If $G$ has two or more vertices of eccentricity 1, then by \\cref{diameter2}, these vertices are in $\\partial G$. Thus, $\\partial G=V$. \n\nNow suppose that $G$ has a single vertex $v$ of eccentricity $1$. Then $N_G(v)=V\\setminus \\{v\\}$, implying $d(v,w)=1$ for all $w\\in V\\setminus \\{v\\}$.\nLet $u\\in N_G(v)$ satisfy $\\beta(v)=\\beta(v,u)$. Because no vertex other than $v$ has eccentricity $1$, we know that $\\ecc(u)\\geq 2$. In particular, there exists at least one vertex $x\\in N(v)$ such that $d(u,x)= 2$. Therefore,\n\\begin{align*}\n        \\beta_{G}(v) & =\\beta_{G}(v,u)\\\\\n        &=\\sum_{w\\in N_G(v)}[d(v,u)-d(w,u) ]\\\\\n       \n        & = [d(v,u)-d(u,u)]+[d(v,u)-d(x,u)]  +\\sum_{w\\in N_G(v)\\setminus \\{u,x\\}}[d(v,u)-d(w,u)] \\\\\n        &= 1 + (-1) +\\sum_{w\\in N_G(v)\\setminus \\{u,x\\}}[1-d(w,u)]\\\\\n       \n        &\\leq 0. \\qedhere\n\\end{align*}\n\nTo show that our bound is sharp, let $G=(V,E)$ be the $n$-barbell graph for $n\\geq 2$, which is formed by adding an edge between two disjoint copies of $K_n$, as shown in \\cref{barbell}. Observe that $\\diam(G)=3$. Denote the two $K_n$ graphs $G_1=(V_1,E_1),G_2=(V_2,E_2)$, and let $v_1\\in G_1$, $v_2\\in G_2$ be the vertices where an edge is added. Since $\\beta_{G_1}(v_1)=\\beta_{G_2}(v_2)=1$, \\cref{edgeattachment}, implies that $\\partial G = (\\partial G_1\\cup\\partial G_2)\\setminus \\{v_1,v_2\\}$. Hence, $|\\partial G|=|V|-2$.\n\n\\begin{figure}[h!]\n    \\centering\n    \\scalebox{0.5}{\n    \\begin{tikzpicture}\n    \\node[shape=circle,draw=black, fill=red] (A) at (0,1) {};\n    \\node[shape=circle,draw=black, fill=red] (B) at (-1,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C) at (0,-1) {};\n    \\node[shape=circle,draw=black, fill=blue] (D) at (1,0) {};\n    \\node[shape=circle,draw=black, fill=red] (A2) at (4,1) {};\n    \\node[shape=circle,draw=black, fill=blue] (B2) at (3,0) {};\n    \\node[shape=circle,draw=black, fill=red] (C2) at (4,-1) {};\n    \\node[shape=circle,draw=black, fill=red] (D2) at (5,0) {};\n    \\path (A) edge node {} (B);\n    \\path (A) edge node {} (C);\n    \\path (A) edge node {} (D);\n    \\path (B) edge node {} (C);\n    \\path (B) edge node {} (D);\n    \\path (C) edge node {} (D);\n    \\path (D) edge node {} (B2);\n    \\path (A2) edge node {} (B2);\n    \\path (A2) edge node {} (C2);\n    \\path (A2) edge node {} (D2);\n    \\path (B2) edge node {} (C2);\n    \\path (B2) edge node {} (D2);\n    \\path (C2) edge node {} (D2);\n\\end{tikzpicture}}\n    \\caption{The 4-barbell graph with $\\partial G$ shown in red.}\n    \\label{barbell}\n\\end{figure}\n\\end{proof}\n\n\n\\begin{corollary}\nLet $G=(V,E)$ be a graph on at least two vertices with minimum degree $\\delta(G)\\geq \\frac{1}{2}(|V|-1)$. If $G$ has a single vertex $u$ of eccentricity $1$, then $\\partial G=V\\setminus \\{u\\}$. Otherwise, $\\partial G=V$. Furthermore, the bound $\\frac{1}{2}(|V|-1)$ is sharp.\n\\end{corollary}\n\n\\begin{proof}\nSuppose $v\\in G$ has $\\ecc(v)>1$. We claim that $\\ecc(v)=2$ by showing that $d(v,u)\\leq 2$ for any $u\\in V$. Let $u\\in V$ be any vertex. If $u\\in N_G(v)$, then $d(v,u)=1$. Otherwise, $u\\notin N_G(v)$ and $v\\notin N_G(u)$. The assumption $\\delta(G)\\geq \\frac{1}{2}(|V|-1)$ implies $|N_G(v)|\\geq \\frac{1}{2}(|V|-1)$ and $|N_G(u)|\\geq \\frac{1}{2}(|V|-1)$. If $N_G(v)\\cap N_G(u)=\\emptyset$, then\n\\[|V|\\geq |N_G(u)\\cup N_G(v)\\cup \\{u,v\\}|=|N_G(u)|+|N_G(v)|+2 \\geq |V|+1,\\]\nwhich is a contradiction. Hence, $d(v,u)\\leq 2$ for all $u$ and $\\ecc(v)=2$. We conclude that when $\\diam(G)\\neq 1$, it must be that $\\diam(G)=2$. The result now follows from \\cref{diameter}, where the $n$-barbell graph also establishes that the bound $\\frac{1}{2}(|V|-1|)$ is sharp, as the minimum degree in the $n$-barbell graph is $\\frac{1}{2}|V|-1<\\frac{1}{2}(|V|-1)$.\n\\end{proof}\n\n\n\n\\section{Open questions}\\label{futurework}\n\nWe have characterized the graphs with $|\\partial G|$ at most 4. Hence, we propose the natural next step.\n\n\\begin{problem}\nClassify connected graphs $G=(V,E)$ for which $|\\partial G|=5$.\n\\end{problem}\n\nOur characterization of $|\\partial G|=4$ relied on a characterization of $|(\\partial G)'|=4$. However, a characterization of graphs with $|(\\partial G)'|=5$ is not currently known. Hence, this problem requires new methods for studying Steinerberger boundary vertices. Since the Steinberberger boundary has more vertices, we expect the characterization problem to be easier than for the CEJZ boundary. \n\nIn \\cref{largeboundary}, we described some graphs where all or almost all vertices are in $\\partial G$. We propose identifying some additional cases when this occurs. Note that a complete characterization is likely difficult. Data on random graphs suggests that they often consist entirely of boundary vertices. \n\\begin{problem}\nDescribe some additional cases when $|\\partial G|=|V|-1$ or $|\\partial G|=|V|$.\n\\end{problem}\n\nIn \\cref{stability}, we characterized when vertices of degree 2 are in $\\partial G$, and one case of our results in \\cref{largeboundary} was on graphs with large minimum degree. We propose a problem related to vertices with low degree.\n\n\\begin{problem}\nCharacterize when vertices of degree $3$ or $4$ are in $\\partial G$. Apply this characterization to find properties of $\\partial G$ for graphs with maximum degree $\\Delta =3$ or $\\Delta= 4$.\n\\end{problem}\n\n\n\nFinally, we propose a question of the boundary stability number, a central tool in establishing our results. To prove our characterization of graphs with small boundary, we explicitly characterized the effect on boundary stability number when adding an edge between two graphs, and we described a case when boundary vertices of a subgraph are also boundary vertices in the graph itself. One natural question is the effect on boundary stability number for other operations.\n\n\\begin{problem}\nDescribe the effect on the boundary stability number of a vertex under other graph operations, such as edge contraction, Cartesian products, etc.\n\\end{problem}\n\n\n\\section*{Acknowledgements}\nWe would like to thank Stefan Steinerberger for suggesting this problem, helpful discussions, and valuable feedback. We would also like to thank Catherine Babecki for valuable feedback and Sam Millard for helpful discussions. Finally, we would like to thank the Washington eXperimental Mathematics Lab for organizing and supporting this project.\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Prolog language was invented 50 years ago and today in 2022 it is still widely used.\nMany possible extensions were and are being defined for Prolog to allow\nit to evolve as a living language.\nThe Oz language was developed starting in 1991\nand it led to a major system release in 1999 called Mozart \\cite{hopl}.\nThe main goal of Oz was to cleanly support many programming paradigms.\nThe Oz language and Mozart system saw wide use for about one decade,\nbut then its use declined.\nAfter 2009 and up to the present day, it is mainly used only for programming education.\nThis is unlike Prolog, which still enjoys wide use up to the present day.\n\nDespite its decline in usage, Oz pioneered many important concepts that\nhave moved into the mainstream since its release.\nWe mention deterministic dataflow computation, actors that return futures,\ndeep support for constraint programming through computation spaces,\nand deep embedding of distributed computing.\nThis is relevant for Prolog because of the\nclose semantic relationship between Oz and Prolog.\nThe semantic foundation of Oz is concurrent constraint programming,\nwhich is a generalization of Prolog's execution model \\cite{saraswat90}.\nProlog programs with logical semantics can be directly\ntranslated into Oz, as explained in Section \\ref{syntax}.\nGiven these facts, we believe that\nthe design of Oz can be an inspiration for the future evolution of Prolog,\nso that Prolog can enjoy another 50 years of success.\nThe purpose of this article is to present three important extensions\nto Prolog-style logic programming made by Oz and explain why\nthey can be an inspiration to future Prolog development:\n\\begin{itemize}\n\\item {\\em Deterministic logic programming} (Section \\ref{dlp}).\nWe show how Oz supports writing logic programs that support\nmanaging deterministic execution while maintaining their logical semantics.\nFor a given logical semantics, we show how Oz can express different operational\nsemantics.\n\n\\item {\\em Lazy concurrent functional programming} (Section \\ref{lcfp}).\nWe show how the logic programming core of Oz can be the foundation of a series\nof functional programming paradigms, culminating in lazy deterministic dataflow.\nAll these paradigms keep the strong confluence properties of pure functional\nprogramming as well as their origin in deterministic logic programming.\n\n\\item {\\em Purely functional distributed computing} (Section \\ref{deep}).\nThe Oz community built a distributed implementation of Oz based on a deep embedding\napproach, where each language entity has its own distributed algorithm.\nAs part of this work, we designed and proved correctness of distributed\nrational tree unification.\nBased on the use of logic variables for dataflow synchronization,\nwe showed that asynchronous message passing can be purely functional.\nFinally, we showed that most practical distributed systems used today\ngreatly overuse nondeterminism, and that it would be better to design\nthem as mostly functional, adding nondeterminism only in the few places where\nit is needed.\n\\end{itemize}\nEach of these extensions is presented separately in its own section,\nIn each case, we explain the main ideas and we give references for readers\nwishing to investigate further.\nAs a preliminary step, Section \\ref{syntax} explains how to translate Prolog\nprograms into Oz, to justify our claim that Oz is a syntactic variant of Prolog.\n\n\\section{Oz is a syntactic variant of Prolog}\n\\label{syntax}\n\nWe explain how to translate any pure Prolog program\nextended with green or blue cuts and\nthe \\verb+bagof\/3+ or \\verb+setof\/3+ predicates\ninto an Oz program with\nthe same logical and operational semantics as the Prolog program.\nWe assume the reader has enough knowledge of Prolog to understand our examples.\nFor brevity, we do not explain the semantics of the Oz statements in detail.\nTheir meaning should be clear from the examples.\nFor a more complete and formal explanation of the relationship\nbetween Oz and Prolog, we refer the reader to Chapter 9 of \\cite{ctm}.\n\nBoth Oz and Prolog support symbolic data structures that are bound\nusing unification.\nSimilar to many modern Prolog systems, Oz uses rational tree unification.\nThere are slight differences in syntax and semantics\nbetween Oz and Prolog data structures, mainly because Oz supports additional\ndata structures such as records.\nWe do not explain these differences here;\nwe refer the reader to \\cite{ctm} for a precise definition of Oz data structures.\n\n\\subsection{Deterministic predicates}\nA predicate is deterministic if all its clauses are logically disjoint.\nThis is the case when the clauses have disjoint guards.\nDeterministic predicates are translated using the Oz statements\n\\OzInline{\\OzKeyword{if}} and \\OzInline{\\OzKeyword{case}}, which both have a logical semantics.\nConsider the following deterministic Prolog predicate:\n\\begin{verbatim}\n place_queens(I, _, _, _) :- I==0, !.\n place_queens(I, Cs, Us, [_|Ds]) :-\n    I>0, J is I-1,\n    place_queens(J, Cs, [_|Us], Ds),\n    place_queen(I, Cs, Us, Ds).\n\\end{verbatim}\nThis predicate has a\n\\index{blue cut (in Prolog)|emph}blue cut according\nto \\index{O'Keefe, Richard}O'Keefe~\\cite{craft},\ni.e., the cut is needed \nto inform naive implementations that the\npredicate is deterministic, so they can improve efficiency,\nbut it does not change the program's results.\nThe predicate \\verb+place_queens\/4+ is translated into the following Oz code: \n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueens\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{if}\\OzSpace{1}N==0\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}\\OzKeyword{skip}\\OzEol\n\\OzSpace{4}\\OzKeyword{elseif}\\OzSpace{1}N>0\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}Ds2\\OzSpace{1}Us2=\\OzChar\\_|Us\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}Ds=\\OzChar\\_|Ds2\\OzEol\n\\OzSpace{7}\\OzChar\\{PlaceQueens\\OzSpace{1}N-1\\OzSpace{1}Cs\\OzSpace{1}Us2\\OzSpace{1}Ds2\\OzChar\\}\\OzEol\n\\OzSpace{7}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzSpace{1}\\OzKeyword{fail}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\n\n\\subsection{Nondeterministic predicates}\nA predicate is nondeterministic when its clauses are not logically disjoint.\nSuch predicates can be backtracked into, possibly giving multiple solutions.\nNondeterministic predicates are translated using the Oz statement \\OzInline{\\OzKeyword{choice}}.\nConsider the following nondeterministic Prolog predicate:\n\\begin{verbatim}\n place_queen(N, [N|_],   [N|_],   [N|_]).\n place_queen(N, [_|Cs2], [_|Us2], [_|Ds2]) :-\n    place_queen(N, Cs2, Us2, Ds2).\n\\end{verbatim}\nThis predicate is translated into a nondeterministic Oz procedure:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{choice}\\OzSpace{1}N|\\OzChar\\_\\OzSpace{2}=Cs\\OzSpace{1}N|\\OzChar\\_\\OzSpace{2}=Us\\OzSpace{1}N|\\OzChar\\_\\OzSpace{2}=Ds\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}\\OzChar\\_|Cs2=Cs\\OzSpace{1}\\OzChar\\_|Us2=Us\\OzSpace{1}\\OzChar\\_|Ds2=Ds\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs2\\OzSpace{1}Us2\\OzSpace{1}Ds2\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\n\n\\subsection{First-class Prolog top level}\nThe previous two predicates are part of a program that computes solutions\nto the $n$-queens problem: how to place $n$ queens on an $n \\times n$ chessboard so that\nno queen attacks another.\nFor completeness, we give the full program here:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Queens\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{MakeList\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{7}\\OzKeyword{if}\\OzSpace{1}N==0\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{else}\\OzSpace{1}\\OzChar\\_|\\OzChar\\{MakeList\\OzSpace{1}N-1\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}Qs=\\OzChar\\{MakeList\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueens\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzSpace{1}\\OzComment{\\OzSpace{1}as\\OzSpace{1}before\\OzSpace{1}}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzSpace{1}\\OzComment{\\OzSpace{1}as\\OzSpace{1}before\\OzSpace{1}}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{4}\\OzChar\\{PlaceQueens\\OzSpace{1}N\\OzSpace{1}Qs\\OzSpace{1}\\OzChar\\_\\OzSpace{1}\\OzChar\\_\\OzChar\\}\\OzEol\n\\OzSpace{4}Qs\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThis program is run by calling the system procedure\n\\OzInline{SolveOne} which corresponds to\na Prolog top-level query:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzChar\\{Browse\\OzSpace{1}\\OzChar\\{SolveOne\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}\\OzChar\\{Queens\\OzSpace{1}8\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzChar\\}\\end{oz2texdisplay}\nThe syntax \\OzInline{\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}...\\OzSpace{1}\\OzKeyword{end}} defines a zero-argument\nfunction that calls the one-argument function \\OzInline{Queens}.\nThe system procedure \\OzInline{Browse} displays its argument.\nThe answer displayed by \\OzInline{Browse} is a list giving the first solution:\n\\begin{oz2texdisplay}\\OzSpace{1}[[1\\OzSpace{1}7\\OzSpace{1}5\\OzSpace{1}8\\OzSpace{1}2\\OzSpace{1}4\\OzSpace{1}6\\OzSpace{1}3]]\\end{oz2texdisplay}\nOz also provides \\OzInline{SolveAll} to compute\nall solutions and \\OzInline{Solve} to compute a lazy list of solutions.\nThe \\OzInline{Solve} corresponds closely to a Prolog top level where solutions\nare computed on demand.\n\n\\subsection{Predicates with green or blue cuts}\n\nIf your Prolog program uses cut ``!'', then the translation to Oz is simple if the\ncut is a green or blue cut as defined by O'Keefe \\cite{craft}.  A green cut removes\nirrelevant solutions.  A blue cut indicates to the\ncompiler that the program is deterministic.\nTo show the translation scheme, we translate the following predicate:\n\\begin{verbatim}\n foo(X, Z) :- guard1(X, Y), !, body1(Y, Z).\n foo(X, Z) :- guard2(X, Y), !, body2(Y, Z).\n\\end{verbatim}\nThe two guards must not bind any head variables, i.e., they are quiet guards.\nIt is good Prolog style to postpone\nbinding head variables until after the cut.\nThe translation has two cases, depending on\nwhether the guards are deterministic or not.\nIf a guard is deterministic (it has no \\OzInline{\\OzKeyword{choice}}),\nthen it can be written as a deterministic boolean function.\nThis gives the following simple translation:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Foo\\OzSpace{1}X\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{if}\\OzSpace{5}Y\\OzSpace{1}\\OzKeyword{in}\\OzSpace{1}\\OzChar\\{Guard1\\OzSpace{1}X\\OzSpace{1}Y\\OzChar\\}\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}\\OzChar\\{Body1\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{elseif}\\OzSpace{1}Y\\OzSpace{1}\\OzKeyword{in}\\OzSpace{1}\\OzChar\\{Guard2\\OzSpace{1}X\\OzSpace{1}Y\\OzChar\\}\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}\\OzChar\\{Body2\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzSpace{1}\\OzKeyword{fail}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nIf a guard is nondeterministic (it uses \\OzInline{\\OzKeyword{choice}}),\nthen it can be written with one input\nand one output argument, like this: \\OzInline{\\OzChar\\{Guard1\\OzSpace{1}In\\OzSpace{1}Out\\OzChar\\}}.\nIt should not bind the input argument.\nThis gives the following translation:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Foo\\OzSpace{1}X\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{case}\\OzSpace{1}\\OzChar\\{SolveOne\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}\\OzChar\\{Guard1\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzSpace{1}\\OzKeyword{of}\\OzSpace{1}[Y]\\OzSpace{1}\\OzKeyword{then}\\OzEol\n\\OzSpace{7}\\OzChar\\{Body1\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{elsecase}\\OzSpace{1}\\OzChar\\{SolveOne\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}\\OzChar\\{Guard2\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzSpace{1}\\OzKeyword{of}\\OzSpace{1}[Y]\\OzSpace{1}\\OzKeyword{then}\\OzEol\n\\OzSpace{7}\\OzChar\\{Body2\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzSpace{1}\\OzKeyword{fail}\\OzSpace{1}\\OzKeyword{then}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nIf neither of these two cases apply to your Prolog program, e.g.,\neither your guards bind head variables\nor you use cuts in other ways (i.e., as red cuts),\nthen your program likely does not have a logical semantics.\nA program with red cuts is defined only by its operational semantics\nand this is outside the scope of our translation scheme.\n\n\\subsection{The {\\tt bagof\/3} and {\\tt setof\/3} predicates}\n\nProlog's \\verb+bagof\/3+ predicate\ncorresponds to using \\OzInline{SolveAll} inside an Oz program.\nIts extension \\verb+setof\/3+ sorts the result and removes duplicates.\nThis can be done with the Oz built-in \\OzInline{Sort} operation.\nWe show how to translate \\verb+bagof\/3+ both without and with existential quantification.\nConsider the following small biblical database\n(inspired by~\\cite{artofprolog}):\n\\begin{verbatim}\n father(terach,  abraham).\n father(abraham, isaac).\n father(haran,   milcah).\n father(haran,   yiscah).\n\\end{verbatim}\nThis can be written as follows in Oz:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}F\\OzSpace{1}C\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{choice}\\OzSpace{1}F=terach\\OzSpace{2}C=abraham\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}F=abraham\\OzSpace{1}C=isaac\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}F=haran\\OzSpace{3}C=milcah\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}F=haran\\OzSpace{3}C=yiscah\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}                                                                 \nCalling \\verb+bagof\/3+ without existential quantification, e.g.:\n\\begin{verbatim}\n children1(X, Kids) :- bagof(K, father(X,K), Kids).\n\\end{verbatim}\nis defined as follows with \\OzInline{SolveAll}:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Children1\\OzSpace{1}X\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{SolveAll\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe \\OzInline{Children1} definition is deterministic;\nit assumes \\OzInline{X} is known and it returns \\OzInline{Kids}.\nTo search over different values of \\OzInline{X}\nthe following definition should be used instead:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Children1\\OzSpace{1}X\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}\\OzChar\\_\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{SolveAll\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe call \\OzInline{\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}\\OzChar\\_\\OzChar\\}} creates a choice point on \\OzInline{X}.\nThe ``\\OzInline{\\OzChar\\_}'' is syntactic sugar for \\OzInline{\\OzKeyword{local}\\OzSpace{1}X\\OzSpace{1}\\OzKeyword{in}\\OzSpace{1}X\\OzSpace{1}\\OzKeyword{end}},\nwhich is just a new variable with a very small scope.                           \n\n\\noindent\nCalling \\verb+bagof\/3+ with existential quantification, e.g.:\n\\begin{verbatim}\n children2(Kids) :- bagof(K, X^father(X,K), Kids).\n\\end{verbatim}\nis defined as follows with \\OzInline{SolveAll}:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Children2\\OzSpace{1}?Kids\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{SolveAll\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}\\OzChar\\_\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe Oz solution uses \\OzInline{\\OzChar\\_} to add a new existentially scoped variable.\nThe Prolog solution, on the other hand, introduces\na new syntactic concept, namely the\n``existential quantifier'' \\verb+X^+,\nwhich only has meaning in terms of \\verb+setof\/3+ and \\verb+bagof\/3+.\nThe fact that this notation denotes an existential quantifier\nis defined explicitly in the Prolog semantics.\nThe Oz solution, on the other hand, requires no new semantics.\n\nIn addition to doing all-solutions \\verb+bagof\/3+,\nOz programs can do a lazy \\verb+bagof\/3+,\ni.e., where each new solution is calculated on demand.\nLazy \\verb+bagof\/3+ is done by \\OzInline{Solve}, which\nreturns a lazy list of solutions.\n\n\\section{Deterministic logic programming}\n\\label{dlp}\n\nWriting a logic program in Prolog or another logic language consists in defining\nthe logical semantics and then choosing an operational semantics that gives a\nsatisfactory efficiency.  \nThis follows Kowalski's equation ``Algorithm = Logic + Control''.\nLogic and control need to be balanced.\nThe art of logic programming consists in balancing two \nconflicting tensions: the logical semantics should be simple and the operational\nsemantics should be efficient.  \nWhen done well, this gives an elegant style in Prolog \\cite{craft}.\nOz supports this kind of program design in logic programming by supporting both\ndeterministic and nondeterministic control flow.\nFor example, in Prolog we can define a list append predicate as follows:\n\\begin{verbatim}\n append([], L2, L2).\n append([X|M1], L2, [X|M3]) :- append(M1, L2, M3).\n\\end{verbatim}\nThis definition follows Prolog's operational semantics.\nCompilers can optimize this to make it deterministic in certain cases,\nfor example if the first argument is bound to a list.\nIn Oz we can define the operational semantics more precisely.\nLet us show three ways that the append can be defined in Oz.\n\n\\subsection{Nondeterministic append}\n\nIn our first definition, we define\nthe append nondeterministically:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}L1\\OzSpace{1}L2\\OzSpace{1}L3\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{choice}\\OzEol\n\\OzSpace{7}L1=nil\\OzSpace{2}L3=L2\\OzEol\n\\OzSpace{4}[]\\OzSpace{1}X\\OzSpace{1}M1\\OzSpace{1}M3\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}L1=X|M1\\OzSpace{1}L3=X|M3\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}M1\\OzSpace{1}L2\\OzSpace{1}M3\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThis has the same operational semantics as Prolog.\n\n\\subsection{Deterministic append (first version)}\n\nWe give another definition of append that has the same logical semantics\nas before but a deterministic operational semantics:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}A\\OzSpace{1}B\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{case}\\OzSpace{1}A\\OzEol\n\\OzSpace{4}\\OzKeyword{of}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}B\\OzEol\n\\OzSpace{4}[]\\OzSpace{1}X|As\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}X|\\OzChar\\{Append\\OzSpace{1}As\\OzSpace{1}B\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nIn this case, argument \\OzInline{A} is bound to a list so execution is directional.\nOz allows to use a functional syntax for such definitions.\n\n\\subsection{Deterministic append (second version)}\n\nWe give yet another definition that again has the same logical semantics\nbut a second deterministic operational semantics:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}B\\OzSpace{1}C\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{if}\\OzSpace{1}B==C\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}nil\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzEol\n\\OzSpace{7}\\OzKeyword{case}\\OzSpace{1}C\\OzSpace{1}\\OzKeyword{of}\\OzSpace{1}X|Cs\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}X|\\OzChar\\{Append\\OzSpace{1}B\\OzSpace{1}Cs\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThis version of \\OzInline{Append} takes the last two arguments as inputs\nand returns the first argument as its output.\nFor example, \\OzInline{\\OzChar\\{Append\\OzSpace{1}[3]\\OzSpace{1}[1\\OzSpace{1}2\\OzSpace{1}3]\\OzChar\\}} returns \\OzInline{[1\\OzSpace{1}2]}.\nCorrect execution requires that the second argument is a suffix of the third argument.\n\n\\section{Lazy concurrent functional programming}\n\\label{lcfp}\n\nAn important insight of the Oz project was that the logic programming core\nof Oz can also support functional programming paradigms.\nWe show this in four steps:\n\\begin{itemize}\n\\item {\\em Functional programming with values}.\nWidely used functional programming languages, such as Scheme or Haskell, compute\nwith values.\nThis can be done in Oz simply by not using unbound logic variables.\nTo support higher-order programming,\nwe extend the Oz computation model with function values.\nVariables can be bound to function values, giving traditional eager functional programming.\n\\item {\\em Functional programming with logic variables}.\nWe add logic variables to functional programming with values. \nThe bind operation is unification.\nThis gives exactly the deterministic logic programming paradigm of Section \\ref{dlp}.\nThis paradigm has a surprising benefit: many more function definitions become tail-recursive.\n\\item {\\em Deterministic dataflow}.\nWe add concurrency to functional programming with logic variables.\nWe allow any statement to execute in its own thread where\nthreads synchronize on variable binding.\nThis is exactly the synchronization model of concurrent logic programming.\nCompared to concurrent logic programming, however, we are still purely functional.\n\\item {\\em Lazy deterministic dataflow}.\nThe final extension adds on-demand computation.  We add a new synchronization operation\nthat waits until another thread waits on a variable being bound.\nWith this new primitive operation (called \\OzInline{WaitNeeded}), we have added all the power\nof lazy evaluation to deterministic dataflow.\n\\end{itemize}\nWe examine these four steps in more detail.\n\n\\subsection{Functional programming with values}\n\nWe extend the computation model with function values.\nWe allow variables to be bound to lexically scoped closures.\nThis gives pure functional programming with eager evaluation.\nPure functional programming is the foundation of higher-order\nprogramming, which underlies most of the development of data abstraction, such\nas object-oriented programming, abstract data types,\ncomponents, templates, metaclasses, and so forth.\n\nIn addition to being a foundation for data abstraction,\nthis model has strong formal properties, \nwhich are different from but analogous to the strong properties of logic programming.  \nThe main property is known as the Church-Rosser property, often referred to\nas confluence, which states that the final result of an expression reduction\nis independent of the reduction choices made during the reduction.\nIn what follows, we show how Oz exploits the synergy that is obtained by combining\nconfluence with logic variables.\n\n\\subsection{Functional programming with logic variables}\n\nWe now extend functional programming with logic variables.\nThis corresponds exactly to the \ndeterministic logic programming of Section \\ref{dlp},\nso this form of functional programming is in fact doing logic programming.\nWhat have we gained from this extension?\nFirst of all, it is now possible to write functional programs\nwith partially instantiated data structures, just like in Prolog.\n\nBut we have gained much more than just the use of partial data structures.\nAllowing unbound logic variables makes possible reduction orders that\nare impossible in a functional language based on values only.\nFor  example, the two deterministic append predicates defined in\nSection \\ref{dlp} are both tail-recursive,\nwhich is not possible in a functional language\nbased on values without doing complex program transformation.\n\nLet us examine precisely how adding logic variables makes functions tail-recursive.\nWe start with the first deterministic append from the previous section.\nSection \\ref{dlp} defines it syntactically as a function,\nbut in Oz functions are just syntactic sugar for procedures.\nThe deterministic append is actually defined as a procedure:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}A\\OzSpace{1}B\\OzSpace{1}C\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{case}\\OzSpace{1}A\\OzEol\n\\OzSpace{4}\\OzKeyword{of}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}C=B\\OzEol\n\\OzSpace{4}[]\\OzSpace{1}X|As\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}Cs\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}C=X|Cs\\OzEol\n\\OzSpace{7}\\OzChar\\{Append\\OzSpace{1}As\\OzSpace{1}B\\OzSpace{1}Cs\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nLook at the recursive call, \\OzInline{\\OzChar\\{Append\\OzSpace{1}As\\OzSpace{1}B\\OzSpace{1}Cs\\OzChar\\}}.\nThis call comes after the binding \\OzInline{C=X|Cs} which\nincrementally builds the output of the append.\n\\OzInline{C} is bound to a new cons cell consisting of \\OzInline{X} paired with\na new unbound variable \\OzInline{Cs}.\nThe recursive call can be a tail call because we pass it the unbound variable \\OzInline{Cs}.\nThis is not possible in functional programming with values only.\n\n\\subsection{Deterministic dataflow}\n\nWe extend the previous paradigm by adding threads and using logic variables\nfor synchronization.\nTo understand this paradigm, we first explain the dataflow \nbehavior of logic variables.  Assume we declare a logic variable \\OzInline{X}.\nIn a first thread, we do \\OzInline{X=10}.  In a second thread, we do \\OzInline{Y=X+1}.\nBecause of the scheduler, we do not know which thread will run first.\nThe second thread, if it is scheduled first, will suspend when it attempts\nto do the addition.  It will wait until \\OzInline{X} is bound.\nWhen the first thread binds \\OzInline{X}, then the second thread becomes runnable again\nand the addition will complete, binding \\OzInline{Y} to 11.\nTo summarize, this is a declarative form of dataflow\nwith two basic operations, namely binding a logic variable and waiting\nuntil the variable is bound.\n\nDataflow synchronization is such an important use of logic variables\nthat we call them {\\em dataflow variables}\nwhen we explain the history of Oz \\cite{hopl}.\nThe Oz community did not invent this; it was pioneered in concurrent logic programming,\nwhich was an early development of Prolog for concurrent systems \\cite{lncs259*30,tr003}.\nThe Oz contributions to this concept are its use for purely functional\nconcurrent and distributed computing and its extension to lazy evaluation.\n\nIn deterministic dataflow,\nany instruction can execute in its own thread.\nThis lets us define networks of concurrent agents that communicate through streams,\nwhere we define a {\\em stream} as a list\nthat is built incrementally and that may have an unbound tail.\nFor example, consider the following sequential functional program:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Gen\\OzSpace{1}L\\OzSpace{1}H\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{Delay\\OzSpace{1}1000\\OzChar\\}\\OzSpace{1}\\OzEolComment{\\OzSpace{1}Suspend\\OzSpace{1}thread\\OzSpace{1}execution\\OzSpace{1}for\\OzSpace{1}1000\\OzSpace{1}ms}\\OzSpace{4}\\OzKeyword{if}\\OzSpace{1}L>H\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{else}\\OzSpace{1}L|\\OzChar\\{Gen\\OzSpace{1}L+1\\OzSpace{1}H\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzEol\n\\OzSpace{1}Xs=\\OzChar\\{Gen\\OzSpace{1}1\\OzSpace{1}10\\OzChar\\}\\OzEol\n\\OzSpace{1}Ys=\\OzChar\\{Map\\OzSpace{1}Xs\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}X*X\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzChar\\{Browse\\OzSpace{1}Ys\\OzChar\\}\\end{oz2texdisplay}\nThis computes a list of successive integers, squares each element of this list,\nand displays the result.\nTo follow the execution during an interactive session,\nwe have slowed down the generation so it takes\none second per element.\nWe can make this concurrent by doing\nthe generation and mapping in their own threads:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{thread}\\OzSpace{1}Xs=\\OzChar\\{Gen\\OzSpace{1}1\\OzSpace{1}10\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{thread}\\OzSpace{1}Ys=\\OzChar\\{Map\\OzSpace{1}Xs\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}X*X\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzChar\\{Browse\\OzSpace{1}Ys\\OzChar\\}\\end{oz2texdisplay}\nThis uses the \\OzInline{\\OzKeyword{thread}} statement to create new threads.\nWhat is the difference between the concurrent and\nthe sequential versions?\nThe result of the calculation is the same in both\ncases, namely \\OzInline{[1\\OzSpace{1}4\\OzSpace{1}9\\OzSpace{1}16\\OzSpace{1}...\\OzSpace{1}81\\OzSpace{1}100]}.\nSo what is different?\nIn the sequential version, \\OzInline{Gen} calculates\nthe whole list before \\OzInline{Map} starts.\nThe final result is displayed all at once\nwhen the calculation is complete, which happens after ten seconds.\nIn the concurrent version, \\OzInline{Gen} and \\OzInline{Map} both\nexecute concurrently.\nWhenever \\OzInline{Gen} adds an element to its list,\n\\OzInline{Map} will immediately calculate its square\nbefore the next element is added.\nThe result is displayed incrementally\nas the elements are generated,\none element each second.\nConcurrency has converted a batch computation into an incremental computation\nwithout changing the functional semantics.\n\nThe concurrent executions of \\OzInline{Gen} and \\OzInline{Map} can be considered as\nfunctional agents, where we define an {\\em agent} as a concurrent computation that\nreads zero or more input streams and writes zero or more output streams.\nBecause of logic variables, both \\OzInline{Gen} and \\OzInline{Map} are tail-recursive.\nThis means that the agents execute with constant stack size.\nThis justifies calling these computations ``agents''.\nThe concurrent execution is memory efficient as well as being purely functional.\nThis shows clearly the synergy between logic variables and concurrency.\n\n\\subsection{Lazy deterministic dataflow}\n\nThe final extension adds the ability to do lazy evaluation to deterministic dataflow.\nIn the previous section,\nwe explained how logic variables are used for dataflow synchronization.\nWe now extend this dataflow paradigm to do lazy evaluation.\nWe do this by adding one new operation, \\OzInline{WaitNeeded}, which does by-need synchronization \non logic variables.\nThe operation \\OzInline{\\OzChar\\{WaitNeeded\\OzSpace{1}X\\OzChar\\}} suspends the\ncurrent thread if \\OzInline{X} is unbound and no other thread is waiting for \\OzInline{X} to be bound.\nOtherwise, if another thread is suspended on \\OzInline{X} or if \\OzInline{X} is bound,\nthe operation succeeds.\nWe say that \\OzInline{WaitNeeded} ``waits until \\OzInline{X} is needed''.\nAdding this single operation to the language lets us fully define lazy evaluation.\nOz introduces a syntax sugar to make this easy for the programmer.\nFor example, we define the function \\OzInline{\\OzChar\\{Ints\\OzSpace{1}N\\OzChar\\}}\nthat returns a lazy list of successive integers starting with \\OzInline{N}:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}lazy\\OzSpace{1}\\OzChar\\{Ints\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{4}N|\\OzChar\\{Ints\\OzSpace{1}N+1\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe ``\\OzInline{lazy}'' annotation means that the function does lazy evaluation.\nIt is syntactic sugar for the following procedure:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Ints\\OzSpace{1}N\\OzSpace{1}R\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{thread}\\OzEol\n\\OzSpace{7}\\OzChar\\{WaitNeeded\\OzSpace{1}R\\OzChar\\}\\OzEol\n\\OzSpace{7}R=N|\\OzChar\\{Ints\\OzSpace{1}N+1\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nCalling \\OzInline{S=\\OzChar\\{Ints\\OzSpace{1}1\\OzChar\\}} will suspend in the \\OzInline{WaitNeeded} call\nuntil another thread needs the first element of \\OzInline{S} to run.  When\nthis happens, the \\OzInline{WaitNeeded} call succeeds and \\OzInline{R} is bound to a list\nwith one new element.\nThe recursive call of \\OzInline{Ints} continues this behavior for the next elements.\nWe note that the compiler can optimize the above definition by reusing the thread,\nto avoid the creation of a new thread for each recursive call.\nThe combination of lazy evaluation and concurrency has been known at least since \n1977 \\cite{kahn77}, but as far as we know,\nthe definition of lazy evaluation in a dataflow paradigm and\nits connection to logic programming are both original with Oz.\n\n\\paragraph{Summary} We summarize the paradigms introduced in this section.\nWe start with a first-order paradigm that computes with values.\nWe extend this paradigm in four steps.\nFirst, we add function values, i.e., lexically scoped closures,\nwhich gives pure functional programming with eager evaluation.\nSecond, we add logic variables,\nwhich gives deterministic logic programming as seen in Section \\ref{dlp}.\nA side benefit is that functions that compute recursive data structures such as lists\nbecome tail-recursive.\nThird, we add threads and use\nthe dataflow behavior of logic variables to do synchronization,\nwhich gives deterministic dataflow.\nThis supports concurrent networks of functional agents that communicate\nthrough shared lists used as communication channels, which we call streams.\nNote that because list functions are tail-recursive, these agents use constant stack space.\nFourth, in the final step, we add by-need synchronization using \\OzInline{WaitNeeded}.\nThis gives a paradigm that supports both lazy evaluation and concurrency,\nwhich we call lazy deterministic dataflow.\nAll four paradigms keep the strong confluence properties\nof pure functional programming, as well as being conservative extensions to a\nlanguage supporting deterministic logic programming.\nSince Prolog is identical to a subset of Oz, we propose that some form of these extensions\ncould become part of a future evolution of Prolog.\n\n\\section{Purely functional distributed computing}\n\\label{deep}\n\nThe Oz research community realized early on that \nthe Oz language design would be a good starting point\nfor building a distributed programming system.\nBecause the design cleanly separates immutable, dataflow, and mutable language entities,\nit would be possible to use a {\\em deep embedding approach},\nwhere each language entity would be implemented with its own distributed algorithm.\nThe distribution support was part of the Mozart system at its first release in 1999.\nThis design had many important innovations including the following:\n\\begin{itemize}\n\\item {\\em Deep embedding}.\nEach language concept was implemented with a distributed algorithm.\nThis means that applications written for one distribution structure\ncan easily be ported to another distribution structure without changing the source code.\nThe only differences are failure behavior and timing.  Failure behavior can be handled\nin a modular way, inspired by techniques from Erlang \\cite{collet,erlang}.\n\\item {\\em Distributed rational tree unification}.\nThe concurrent examples of Section \\ref{lcfp} can all be run with unchanged source\ncode if distributed across different compute nodes.\nConcurrent pipelines become asynchronous distributed pipelines, while maintaining\nthe semantics of pure functional programming.\nA key part of this system is the distributed rational tree unification algorithm.\nThe first formal specification and proof of distributed unification was done during\nthis work and published in \\cite{toplas99}.\n\\item {\\em Global references}, {\\em distributed lexical scoping},\n{\\em distributed garbage collection}, and {\\em migratory state}.\nThese are some of the other properties of this design.\n\\end{itemize}\nIn the rest of this section we will focus on what we consider to be\nthe most significant discovery that was made in the work on distributed\ncomputing for Oz.  This discovery is intimately tied to the logic programming\norigins of Oz and it may open a window of opportunity for Prolog.\n\n\\paragraph{Asynchronous message passing can be pure}\nThe deterministic data\\-flow paradigm and its lazy extension\ncan be implemented in a distributed setting.\nA dataflow variable can be read on one compute node and bound on another node.\nThis does a distributed synchronization, which in its general form is implemented\nby distributed unification.\nThe salient property of this distributed synchronization is that it enjoys all\nthe strong properties of pure functional programming.\nSpecifically, it means that {\\em asynchronous message passing between compute nodes\ncan be completely pure}.\nThis property remains mostly unknown in the distributed computing community.\nIn our experience as members of this community,\nwe observe that many often insist that asynchronous message passing\nis intrinsically impure, which is false.\n\n\\paragraph{Overuse of nondeterminism}\nMost of today's large distributed systems do not take advantage of this property.\nIn fact, the contrary is true: {\\em today's systems greatly overuse nondeterminism}.\nThis is one of the main reasons\nwhy it is still difficult to build and debug such systems.\nLarge distributed systems, such as used daily by Google, Facebook, Twitter, and so on,\nmust be continually babysitted by experts.\nThis is not an inevitable property of such systems.\nIn our view, it is mainly due to the massive overuse of nondeterminism in the design\nof these systems.\nEvery single library that is a part of such a system and that has its own API, which\nis true of most libraries, is a point of nondeterminism.\n\n\\paragraph{Purely functional distributed computing}\nThe problem of overuse of nondeterminism and a possible solution\nare explained in a keynote talk given by one of the authors at CodeBEAM 2019,\n{\\em Why Time is Evil in Distributed Systems and What To Do About It} \\cite{evil}.\nThe solution we propose is to use deterministic dataflow programming\nas the default paradigm for distributed systems and to add nondeterminism\nonly where it is needed and nowhere else.\nWith this solution, most distributed systems become mostly functional\nwith a limited use of nondeterminism.\nThis greatly simplifies their development, maintenance, and evolution.\nWe observe that this solution is intimately tied to logic programming,\nsince the basic concept that makes it work is the dataflow variable,\nwhich is simply a logic variable used in a concurrent setting.\n\n\\paragraph{Role of logic programming}\nWe believe that an appropriate extension of Prolog or another logic language\ncan potentially solve this problem\nand play a significant role in distributed computing.\nIf such an extension is not done by the logic programing community,\nthen the distributed computing community will\nreinvent concepts from logic programming as it solves these problems.\nThis reinvention has already started with the development of CRDTs (Conflict-Free\nReplicated Data Types), which have monotonic properties and can replace consensus\nalgorithms, which are nondeterministic,\nby purely functional operations in many cases \\cite{crdt}.\n\n\\section{Conclusions}\n\\label{conclusions}\n\nProlog has enjoyed a relatively large popularity since its conception in 1972\nup to the present day,\nunlike Oz, which despite having many innovations at its release in 1999,\nhas seen a decline since 2009 and is only used today for education.\nOz declined for sociological reasons that have nothing to do with its\ninnovations; in fact we observe that the innovations are all becoming\nadopted in modern programming systems despite the decline of Oz.\nAs members of both the Prolog and Oz communities,\nwe see this as an opportunity to help Prolog evolve\nin order to maintain its popularity.\n\nThis paper presents three innovations of Oz,\nnamely deterministic logic programming, lazy concurrent functional programming,\nand purely functional distributed computing.\nWe propose that it would be straightforward to add these innovations to Prolog\nbecause the core logic languages of Oz and Prolog are quite similar.\nMost of pure Prolog has a direct syntactic translation to Oz.\nThis means that the hard work of formulating and understanding\nthese innovations in a logic programming context has already been done.\nWe hope that the ideas presented in this paper will help the future of Prolog\nas well as help Oz regain some recognition as a source of innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}