diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgexs" "b/data_all_eng_slimpj/shuffled/split2/finalzzgexs" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgexs" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\IEEEPARstart{S}{elf-localization} is an essential capability of autonomous mobile robots, and localization algorithms based on inexpensive commercial vision sensors have become useful and widespread.\nDespite this success, long-term \\emph{metric} localization, where the goal is to continuously estimate the \\mbox{6-dof} pose of the vehicle with respect to a visual map, remains challenging in the presence of appearance change caused by illumination variations over the course of a day, changes in weather conditions, or seasonal shifts.\nThis difficulty is largely due to simplifying assumptions such as brightness constancy and feature descriptor invariance that, when violated, cause visual localization systems to fail.\nIdeally, we would like our systems to function across this `appearance gap', immune to variations in environmental conditions.\n\nLong-term maps based on multiple visual `experiences' of an environment have proven to be effective tools for metric localization through daily and seasonal appearance change~\\cite{Churchill2013-ng,Linegar2015-xs,Paton2016-bz,Paton2018-qa}.\nIn~\\cite{Paton2016-bz}, consecutive visual experiences are recorded in a spatio-temporal pose graph, and localization against a privileged mapping experience proceeds by recalling a relevant experience and tracing through a chain of relative transformations in the graph.\nThis process is often aided by a prior on the vehicle's topological location in the graph, whether from dead reckoning, place recognition, or GNSS, which serves to limit the number of candidate vertices for metric localization.\nHowever, the number of intermediate `bridging' experiences required for reliable long-term localization can be very large, and methods for compressing experience graphs are necessary for this approach to scale.\n\nRecent work in~\\cite{Clement2018-vm,Porav2018-xb} has explored deep image-to-image translation~\\cite{Isola2017-uc,Zhu2017-qc} as a means of directly bridging the `appearance gap' and localizing with fewer experiences.\nHowever these methods rely at least in part on well-aligned training images, which are difficult to obtain at scale in the real world.\nMoreover, the losses used to train these models are not explicitly connected to a target localization pipeline, and provide few assurances that the learned image transformations will ultimately improve localization performance.\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.97\\textwidth]{overview}\n \\caption{We learn an image transformation that improves visual feature matching performance over day-night cycles by maximizing the response of a differentiable proxy network trained to predict the number of inlier feature matches returned by a conventional non-differentiable feature detection\/matching algorithm. The learned transformation can then be used as a pre-processing step in a visual localization pipeline to improve its robustness to appearance change.}\n \\label{fig:pipeline}\n \\vspace{-12pt}\n\\end{figure*}\n\nWe address these limitations by learning an image transformation optimized for a given combination of localization pipeline, sensor, and operating environment.\nRather than translating between arbitrary appearance conditions, we learn to map images to a \\emph{maximally matchable} representation (i.e., one which maximizes the number of inlier feature matches) for a given feature detection\/matching algorithm.\nSpecifically, we learn a drop-in replacement for the standard RGB-to-grayscale colorspace mapping used to pre-process RGB images for use with conventional feature detection\/matching algorithms, which typically operate on single-channel images (\\Cref{fig:pipeline}).\nThis formulation builds upon prior work on color constancy theory~\\cite{Ratnasingam2010-cr}, does not require well-aligned images for training, and naturally admits a self-supervised training approach as training targets can be generated on the fly by the localization pipeline.\nOur main contributions are:\n\\begin{enumerate}\n \\item a technique for improving the robustness of a conventional visual localization pipeline to appearance change using a learned image pre-processing step;\n \\item a method for optimizing the performance of a non-differentiable localization pipeline by approximating the pipeline using a deep neural network;\n \\item experimental results on synthetic and real long-term vision datasets showing that our method enables continuous 6-dof metric visual localization across day-night cycles using a single mapping experience; and\n \\item an open-source implementation of our method using PyTorch~\\cite{paszke2017automatic}.\\footnote{\\url{github.com\/utiasSTARS\/matchable-image-transforms}}\n\\end{enumerate}\n\n\\section{Related Work}\nAppearance robustness in metric visual localization has previously been studied from the perspective of illumination invariance, with methods such as~\\cite{Clement2017-gx, Corke2013-hl, McManus2014-op, Paton2017-fi} making use of hand-engineered image transformations to improve feature matching over time for a given feature detector and descriptor.\nSimilarly, affine models and other simple analytical transformations have been used to improve the robustness of direct visual localization to illumination change~\\cite{Engel2015-il,Park2017-zx}.\nOther approaches such as~\\cite{McManus2015-vj,Linegar2016-cn,Krajnik2017-tz,Zhang2018-ib} have focused on learning feature descriptors that are robust to certain types of appearance change in autonomous route following applications.\nHowever, \\cite{McManus2015-vj,Linegar2016-cn} produce correspondences that are only weakly localized, and \\cite{Krajnik2017-tz,Zhang2018-ib} require sets of true and false point correspondences to train feature descriptors, which are challenging to obtain at scale over long periods.\n\nDeep image-to-image translation~\\cite{Isola2017-uc,Zhu2017-qc} has recently been applied to the problem of metric localization across appearance change.\nIn~\\cite{Gomez-Ojeda2018-ai} the authors train a convolutional encoder-decoder network to enhance the temporal consistency of image streams captured in environments with high dynamic range.\nHere the main source of appearance change is the camera itself as it automatically modulates its imaging parameters in response to the local brightness of a static environment.\nOther work has tackled the problem of localization across \\emph{environmental} appearance change, with \\cite{Clement2018-vm} learning a many-to-one mapping onto a privileged appearance condition and \\cite{Porav2018-xb} learning multiple pairwise mappings between appearance categories such as day and night.\nImage-to-image translation has also been applied to the related task of appearance-invariant place recognition~\\cite{Latif2018-ui,Anoosheh2019-fc}, which typically relies on patch matching or whole-image statistics to identify images corresponding to nearby physical locations rather than estimating the 6-dof pose of the vehicle.\nWhile \\cite{Porav2018-xb,Gomez-Ojeda2018-ai} include loss terms to maximize gradient information, these heuristics are not directly tied to the performance of the localization pipeline.\nMoreover, \\cite{Clement2018-vm,Porav2018-xb,Gomez-Ojeda2018-ai} require well-aligned training images exhibiting appearance variation, which are difficult to obtain at scale in the real world, and it is not clear how categorical appearance mappings such as \\cite{Porav2018-xb,Latif2018-ui,Anoosheh2019-fc} should be applied to continuous appearance change in long-term deployments.\n\nSurrogate-based methods for approximating computationally expensive or non-differentiable objective functions are common in the numerical optimization literature~\\cite{Koziel2011}.\nNeural network surrogates in particular have found applications in a variety of domains including computer graphics~\\cite{Grzeszczuk:1998:FNN:3009055.3009178} and computational oceanography~\\cite{VANDERMERWE2007462}, where high-fidelity physics simulations are available but expensive to compute.\nOur method of learning a differentiable loss function is similar in spirit to Generative Adversarial Networks (GANs)~\\cite{Goodfellow2014-df} in that a complex discriminator\/loss function is trained using a comparatively simple analytical loss function.\nIt also bears resemblances to perceptual losses~\\cite{Johnson2016-lu}, where the loss function is derived from the feature activations of a network trained on a proxy task such as image classification.\n\n\\section{Learning Matchable Colorspace Transformations}\nOur goal in this work is to learn a nonlinear transformation $f: \\Real^3 \\rightarrow \\Real $ mapping the RGB colorspace onto a grayscale colorspace that explicitly maximizes a chosen performance metric of a vision-based localization pipeline.\nWe investigate two approaches to formulating such a mapping: 1) a single function to be applied as a pre-processing step to all incoming images, similarly to \\cite{Clement2017-gx,McManus2014-op,Paton2017-fi}; and 2) a parametrized function tailored to the specific image pair to be used for localization, where the parameters of this function are derived from the images themselves.\nAdditionally, the functional form of either mapping may be specified analytically (e.g., from physics) or learned from data using a function approximator such as a neural network.\n\nIn order to find an optimal colorspace transformation for a given application, we require an appropriate objective function to optimize, which should ideally be tied to the performance of the target localization pipeline.\nAn intuitive choice of objective could be to directly minimize pose estimation error for the entire pipeline relative to ground truth if it is available.\nIn the absence of accurate ground truth data, we might instead choose to maximize the number or quality of feature matches in the front-end of a feature-based localization pipeline.\nWe adopt the latter approach in this work, since high-quality 6-dof ground truth is difficult to obtain over long time scales.\n\nAlthough it is straightforward to choose a target performance metric to optimize, the most commonly used localization front-ends in robotics rely on non-differentiable components such as stereo matching, nearest-neighbors search, and RANSAC~\\cite{Fischler1981-ue}, which are incompatible with the gradient-based optimization schemes commonly used in deep learning.\nIn this work we \\emph{learn} an objective function by training a deep convolutional neural network (CNN) to act as a \\emph{differentiable proxy} to the localization front-end.\nSpecifically, we train a siamese CNN to predict the number of inlier feature matches for a given image pair, where the training targets are generated using a conventional non-differentiable feature detector\/matcher algorithm based on \\texttt{libviso2} features~\\cite{Geiger2011-xe}.\nThis proxy network can then be used to define a fully differentiable objective function, allowing us to train a nonlinear colorspace mapping using gradient-based methods.\nFinally, the trained image transformation can be used as a pre-processing step in a conventional visual localization pipeline, enabling it to operate more reliably under appearance change.\n\\Cref{fig:pipeline} summarizes our full data pipeline pictorially.\n\n\\subsection{Differentiable Matcher Proxy} \n\\label{sec:matcher}\n\nWe consider the task of training a CNN $\\MatcherNet$, with parameters $\\MatcherNetParams$, to predict the number of inlier feature matches returned by a non-differentiable feature detector\/matcher $\\Matcher$ for a given image pair $ (\\Image_1, \\Image_2) $.\nThis training objective is a convenient choice for our intended application as it is closely tied to the ability of our visual localization pipeline to operate across appearance change, however it is not by any means the only choice.\nFor example, we could also train a CNN to predict a measure of localization accuracy such as geodesic distance from ground truth on the $\\LieGroupSE{3}$ manifold, similarly to the estimator correction framework proposed in~\\cite{Peretroukhin2018-qc}.\nImportantly, this formulation admits a self-supervised training approach as training targets can be generated automatically by $\\Matcher$.\n\n\\Cref{fig:pipeline} (right-hand side) summarizes the training setup for this task.\nA pair of single-channel images is fed into a conventional feature detection\/matching algorithm (e.g., SURF~\\cite{Bay2008-zi}, ORB~\\cite{Rublee2011-hd}, or \\texttt{libviso2}~\\cite{Geiger2011-xe}), and a summary statistic is computed such as the quantity of RANSAC-filtered inlier feature matches.\nThis summary statistic forms the training target for a CNN whose task is to predict the statistic for the same image pair.\nGiven enough training pairs, the network should learn a set of convolutional filters that correspond to the types of features and patterns that best predict the performance of $\\Matcher$ in a given environment.\nCritically, the proxy network is fully differentiable and can provide a gradient signal to train a nonlinear image transformation.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\columnwidth]{matcher_net}\n \\caption{Network architecture for $\\MatcherNet$. Each block is denoted by stride, kernel size, input channels, and output channels. The left and right branches share weights. The final output is produced by a fully-connected layer (implemented as a convolution operator), which projects the feature map to a scalar value.}\n \\label{fig:matcher_net}\n \\vspace{-12pt}\n\\end{figure}\n\nOur matcher proxy network (\\Cref{fig:matcher_net}) is a siamese network built from convolutional and residual~\\cite{He2016-zg} blocks using batch normalization~\\cite{Ioffe2015-bs} and PReLU non-linearities~\\cite{He2015-zg}.\nEach image in the input pair is processed by one of two feature detection branches, which share weights to ensure that both images are mapped onto a common feature space.\nThe outputs of the feature detection branches are concatenated along the channel dimension to be further processed by the remainder of the network.\nEach non-residual convolution block downsamples the feature map by a factor of two, allowing for salient features to be learned at multiple scales.\nThe final output is produced by a fully-connected layer, which projects the feature map to a scalar value.\nWe train $\\MatcherNet$ to fit $\\Matcher$ in a least-squares sense by minimizing the mean squared error of the predicted match counts for a minibatch of $N$ image pairs:\n\\begin{align} \\label{eq:matcher_loss}\n \\CNNLoss(\\MatcherNetParams) & = \\frac{1}{N} \\sum_{i=1}^N \\left( \\MatcherNet(\\Image_1^i, \\Image_2^i) - \\Matcher(\\Image_1^i, \\Image_2^i) \\right)^2.\n\\end{align}\n\n\\subsection{Physically Motivated Transformations}\n\\label{sec:logrgb}\nPrior work in \\cite{Ratnasingam2010-cr} has shown that under the assumptions of a single black-body illuminant and an infinitely narrow sensor response function, an appropriately weighted linear combination of the log-responses of a three-channel (e.g., RGB) camera represents a projection onto an invariant one-dimensional chromaticity space that is independent of both the intensity and color temperature of the illuminant, and depends only on the imaging sensor and the materials in the scene:\n\\begin{align} \\label{eq:color_constant}\n \\InvariantImage^i_j = \\log \\Image^i_j({\\lambda_2}) - \\alpha \\log\\Image^i_j({\\lambda_1}) - \\beta \\log\\Image^i_j({\\lambda_3}),\n\\end{align}\nwhere $ \\Image^i_j({\\lambda_k}) $ is the image of sensor responses at wavelength $\\lambda_k$, the weights $(\\alpha, \\beta)$ are subject to the constraints\n\\begin{align} \\label{eq:color_constant_constraints}\n \\frac{1}{\\lambda_2} & = \\frac{\\alpha}{\\lambda_1} + \\frac{\\beta}{\\lambda_3} & \\mathrm{and} & & \\beta & = (1-\\alpha),\n\\end{align}\nand the indices $k$ are chosen such that $\\lambda_1 < \\lambda_2 < \\lambda_3$ (i.e., red, green and blue channels, respectively).\n\nThe image formed from this pixel-wise linear combination of log-responses can then be rescaled to produce a valid grayscale image that can be further processed by a localization pipeline.\nGrayscale images generated using this procedure are somewhat resistant to variations in lighting and shadow, and have been shown to improve stereo localization quality in the presence of shadows and changing daytime lighting conditions \\cite{Clement2017-gx,McManus2014-op,Paton2017-fi}, but have not been successful in adapting to nighttime navigation with headlights.\n\nGiven the constraints defined by~\\Cref{eq:color_constant_constraints}, the weights $(\\alpha, \\beta)$ are completely specified as a function of the imaging sensor.\nHowever, in practice, these constraints are relaxed and the parameters $(\\alpha, \\beta)$ are tuned to a specific environment, sensor, and feature matcher, where the theoretical values do not perform optimally.\nIndeed, \\cite{Clement2017-gx,Paton2017-fi} used two sets of parameters tuned to maximize the stability of SURF features~\\cite{Bay2008-zi} in regions where grassy or sandy materials dominate.\n\nWe argue that environmental appearance is best thought of as continuous rather than categorical, and that a better approach to selecting the transformation parameters should take into account the content of the specific scene being imaged, rather than using the same parameters at every location within a large and potentially heterogeneous operating environment.\nAccordingly, we train a second encoder network $\\EncoderNet$, with parameters $\\EncoderNetParams$, to predict the optimal values of the transformation parameters (i.e., which yield the most inlier feature matches) for a given RGB image pair.\n\nFurthermore, we relax the constraints in~\\Cref{eq:color_constant_constraints} and generalize~\\Cref{eq:color_constant} to be of the form\n\\begin{align} \\label{eq:color_constant_generalized}\n \\InvariantImage^i_j & = \\alpha^i \\log \\Image^i_j({\\lambda_1}) + \\beta^i \\log\\Image^i_j({\\lambda_2}) + \\gamma^i \\log\\Image^i_j({\\lambda_3}),\n\\end{align}\nwhere the parameters are computed for the $i^\\mathrm{th}$ image pair as\n\\begin{align} \\label{eq:encoder_output}\n \\TransformParams^i & = \\bbm \\alpha^i & \\beta^i & \\gamma^i \\ebm^T = \\EncoderNet(\\Image^i_1, \\Image^i_2),\n\\end{align}\nand~\\Cref{eq:color_constant_generalized} is applied to both $\\Image^i_1$ and $\\Image^i_2$ using the same set of parameters.\nDue to the need to rescale $\\InvariantImage^i_j$ to form a valid single-channel image, a degree of freedom in $\\TransformParams^i$ is lost and $(\\alpha, \\beta, \\gamma)$ represent the relative mixing proportions of the three color channels.\nWe enforce $\\Norm{ \\TransformParams^i }_1 = 1$ using a normalization layer to ensure a consistent range of outputs.\n\nOur encoder network $\\EncoderNet$ follows a similar siamese architecture to $\\MatcherNet$, but takes pairs of 3-channel images as inputs and outputs a 3-dimensional vector.\nWe train $\\EncoderNet$ to maximize the mean number of inlier feature matches as predicted by $\\MatcherNet$, or equivalently, to minimize its negation:\n\\begin{align} \\label{eq:encoder_loss}\n \\CNNLoss(\\EncoderNetParams) & = -\\frac{1}{N} \\sum_{i=1}^N \\MatcherNet(\\InvariantImage_1^i, \\InvariantImage_2^i),\n\\end{align}\nwhere $\\InvariantImage_1^i, \\InvariantImage_2^i$ are computed from input RGB images $\\Image_1^i, \\Image_2^i$ using \\Cref{eq:color_constant_generalized,eq:encoder_output}.\n\nRather than rescaling using the minimum and maximum response of each output image, we rescale by the joint mean $ \\mu^i $ and standard deviation $ \\sigma^i $ of the output pair and apply a clamping operation to map the output onto the range $ [0, 1] $:\n\\begin{align} \\label{eq:rescale_instancenorm}\n \\InvariantImage^i_j & \\leftarrow \\frac{1}{2} \\left[ \\frac{\\InvariantImage^i_j - \\mu^i }{3 \\sigma^i} \\right]_{-1,1} + \\frac{1}{2},\n\\end{align}\nwhere we have used the notation $\\left[ \\cdot \\right]_{a,b} = \\min(\\max(\\cdot, a), b)$.\nThis rescaling scheme allows the model to saturate parts of the output images while still using the full range of valid pixel values.\nMoreover, it avoids introducing significant sparsity in the gradients through the $\\min(\\cdot)$ and $\\max(\\cdot)$ operators, which improves the flow of gradient information during training.\n\n\n\\subsection{Learned Nonlinear Transformations}\n\\label{sec:learned}\nWhile the assumption of a single black-body illuminant in~\\cite{Ratnasingam2010-cr} is reasonable for daytime navigation where the dominant light source is the sun, it does not hold in many common navigation scenarios such as nighttime driving with headlights.\nMoreover, the assumption of an infinitely narrow sensor response is unrealistic for real cameras.\nAs an alternative to the physically motivated colorspace transformation outlined in \\Cref{sec:logrgb}, we investigate the possibility of learning a bespoke nonlinear mapping that maximizes matchability for a particular combination of imaging sensor, estimator and environment.\nWe parametrize this mapping using a small neural network $\\TransformerNet$, with parameters $\\TransformerNetParams$, operating independently on each pixel of each input RGB image.\nWe structure $\\TransformerNet$ as a multilayer perceptron (MLP) implemented using $1 \\times 1$ convolutions and PReLU nonlinearities.\n\nWe consider two versions of this MLP-based transformation, both with and without incorporating an additional pairwise context feature obtained from encoder network $\\EncoderNet$ using \\Cref{eq:encoder_output}.\nIn the case where $\\EncoderNet$ is used, the input to $\\TransformerNet$ becomes the concatenation of the input RGB image and the parameters $\\TransformParams^i$ along the channel dimension, and the first convolutional layer of $\\TransformerNet$ is modified accordingly.\nWe train $\\TransformerNet$ and $\\EncoderNet$ (if used) jointly by minimizing a similar loss function to \\Cref{eq:encoder_loss}, where in place of \\Cref{eq:color_constant_generalized}, we have\n\\begin{align} \\label{eq:learned_invariant_transform}\n \\InvariantImage_j^i & = \\begin{cases} \\TransformerNet(\\Image_j^i, \\TransformParams^i) & \\text{if $\\EncoderNet$ is used,} \\\\ \\TransformerNet(\\Image_j^i) & \\text{if $\\EncoderNet$ is not used}. \\end{cases}\n\\end{align}\nSimilarly to the physically motivated transformations described in~\\Cref{sec:logrgb}, we rescale $\\InvariantImage_j^i$ to fill the range of valid grayscale values by applying~\\Cref{eq:rescale_instancenorm}.\n\n\\begin{figure*}\n \\centering\n \n \n \n \n \n \n \\begin{subfigure}{0.32\\textwidth}\n \\includegraphics[width=\\textwidth]{vkitti\/morning-sunset\/all_matcher_target_est}\n \\caption{\\texttt{VKITTI} (Morning vs. Sunset)}\n \\label{fig:matcher_target_est:vkitti_morning-sunset}\n \\end{subfigure}\n ~\n \\begin{subfigure}{0.31\\textwidth}\n \\includegraphics[width=\\textwidth]{inthedark\/all_matcher_target_est}\n \\caption{\\texttt{InTheDark}}\n \\label{fig:matcher_target_est:inthedark}\n \\end{subfigure}\n ~\n \\begin{subfigure}{0.31\\textwidth}\n \\includegraphics[width=\\textwidth]{oxford\/all_matcher_target_est}\n \\caption{\\texttt{RobotCar}}\n \\label{fig:matcher_target_est:oxford}\n \\end{subfigure}\n \\caption{Estimated vs. actual match counts for $\\MatcherNet$ after ten epochs of pre-training on each dataset, colour-coded by relative density. Match counts are aggregated over all test sequences and include self-matches. The dashed red line corresponds to perfect agreement of $\\MatcherNet$ with \\texttt{libviso2}. In each case, the match count predictions produced by $\\MatcherNet$ are very strongly correlated with the true match counts.}\n \\label{fig:matcher_target_est}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\begin{subfigure}{0.95\\textwidth}\n \\includegraphics[width=\\textwidth]{vkitti\/morning-sunset\/0020_grid_horiz_cropped}\n \\caption{\\texttt{VKITTI\/0020} (Sunset to Morning, cropped)}\n \\label{fig:vkitti_0020_grid}\n \n \\end{subfigure}\n ~\n \\begin{subfigure}{0.95\\textwidth}\n \\includegraphics[width=\\textwidth]{inthedark\/inthedark_000041_grid}\n \\caption{\\texttt{InTheDark\/0041} (Night to Day)}\n \\label{fig:inthedark_000041_grid}\n \\end{subfigure}\n \\caption{Sample input RGB pairs and corresponding outputs of each RGB-to-grayscale transformation.}\n \\label{fig:output_grid}\n \\vspace{-12pt}\n\\end{figure*}\n\n\\section{Experiments}\nWe conducted experiments on synthetic and real-world long-term vision datasets to validate and compare each approach.\nSpecifically, we evaluated the ability of our matcher proxy network to capture the performance of \\texttt{libviso2} feature matching across viewpoint and appearance changes, as well as the effect of each image transformation on feature matching and localization performance.\nWhen evaluating feature matching, we assumed that we had a prior on the vehicle's topological location in the map, such that we could reliably identify the nearest vertex in the pose graph.\nThis is typical for autonomous visual route-following systems such as~\\cite{Paton2018-qa}, where the topological prior is derived by dead reckoning from a previous successful localization, or using place recognition or GNSS in the event that the system becomes lost.\n\nWe refer to the generalized color-constancy model of~\\cite{Ratnasingam2010-cr} (\\Cref{sec:logrgb}) as ``SumLog'' and ``SumLog-E'' , where the latter uses \\Cref{eq:encoder_output} to derive the parameters $\\TransformParams^i$ per image pair, and the former uses a constant $\\TransformParams$ that maximizes inlier feature matches over the training set (similarly to~\\cite{Clement2017-gx,Paton2017-fi}).\nAnalogously, we refer to the learned multilayer perceptron models (\\Cref{sec:learned}) as ``MLP'' and ``MLP-E'', where the latter incorporates $\\EncoderNet$ and the former does not .\nWe refer to the standard RGB-to-grayscale transformation as ``Gray''.\\footnote{We refer specifically to the ITU-R 601-2 luma transform implemented by the \\texttt{Pillow} library: $L = 0.299 R + 0.587 G + 0.114 B$.}\n\nTraining proceeds in two stages.\nFirst, we pre-train $\\MatcherNet$ using standard grayscale images.\nTraining labels are generated using the open-source \\texttt{libviso2} library~\\cite{Geiger2011-xe} to detect and match features, and the eight-point RANSAC algorithm to reject outlier matches.\nSecond, we train $\\EncoderNet$ and\/or $\\TransformerNet$ using the matchability loss defined in \\Cref{eq:encoder_loss}.\nTo ensure that $\\MatcherNet$ accurately predicts feature match counts for the output images, which differ significantly from standard grayscale images, we continue to train $\\MatcherNet$ in an alternating fashion using the output images at each iteration.\nAll models are implemented in \\mbox{PyTorch}~\\cite{paszke2017automatic} and trained for 10 epochs with a batch size of 8, using the Adam optimizer~\\cite{Kingma2015-wl} with default parameters and a learning rate of $10^{-4}$. \nWe rescale all images to a height of 192 pixels for both training and testing.\n\n\\subsection{Datasets}\nWe evaluated our approach using both synthetic and real-world datasets exhibiting severe illumination change.\n\n\\paragraph{Virtual KITTI} \\label{sec:vkitti_dataset}\nThe Virtual KITTI (\\texttt{VKITTI}) dataset~\\cite{Gaidon2016-by} is a synthetic reconstruction of a portion of the KITTI vision benchmark~\\cite{Geiger2013-ky}, consisting of five sets of non-overlapping trajectories with \\mbox{RGB-D} imagery rendered under a variety of simulated illumination conditions.\nThis dataset is a convenient validation tool as it provides perfect data association and a range of daytime illumination conditions.\nFor each trajectory, we train models using image pairs from the others.\nSince each trajectory is non-overlapping, this spatial split allows us to evaluate how well our method generalizes to unseen environments.\nFurther, since \\texttt{VKITTI} provides corresponding images from identical viewpoints, we augmented the training data to ensure generalizability to viewpoint changes and dynamic objects by associating each training image in one condition with a window of nearby images in the other.\n\n\\paragraph{UTIAS In The Dark}\nThe UTIAS In The Dark (\\texttt{InTheDark}) dataset~\\cite{Paton2016-bz} provides stereo imagery of a 250 m outdoor loop traversed repeatedly over a 30-hour period using on-board headlights to illuminate the scene at night.\nWe use the multi-experience localization system in \\cite{Paton2016-bz} to obtain corresponding image pairs with overlapping fields of view but non-identical poses.\nWe train our models using left-camera images from 66 traversals and test on 7 held-out traversals spanning a full day-night cycle (listed in \\Cref{tab:match_stats}).\nThis temporal split allows us to evaluate how well our method generalizes to multiple unseen illumination conditions.\n\n\\paragraph{Oxford RobotCar}\nWe further evaluate our method using three sequences from the Oxford RobotCar (\\texttt{RobotCar}) dataset~\\cite{Maddern2016-ng}, captured along the same 10~km route in overcast, nighttime, and sunny conditions.\\footnote{``Overcast'', ``Night'', and ``Sunny'' refer to \\texttt{2014-12-09-13-21-02}, \\texttt{2014-12-10-18-10-50}, and \\texttt{2014-12-16-09-14-09}, respectively.}\nWe find corresponding images using the GNSS\/INS poses, and split the trajectory into two non-overlapping segments: the first 70\\% of the trajectory is used for training and the remaining 30\\% is used for testing.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{vkitti\/morning-sunset\/0020_matches}\n \\caption{\\texttt{VKITTI\/0020} (Sunset to Morning)}\n \\label{fig:vkitti_0020_matches}\n \n \\end{subfigure}\n ~\n \\begin{subfigure}{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{inthedark\/run_000041_matches}\n \\caption{\\texttt{InTheDark\/0041} (Night to Day)}\n \\label{fig:inthedark_000041_matches}\n \\end{subfigure}\n \\caption{Box-and-whiskers plots of inlier \\texttt{libviso2} feature matches for corresponding image pairs with each RGB-to-grayscale transformation applied. Orange lines indicate the median values.}\n \\label{fig:matches_box_whiskers}\n \\vspace{-12pt}\n\\end{figure}\n\n\\begin{table}[]\n \\setlength{\\tabcolsep}{2.7pt}\n \\centering\n \\caption{Actual inlier feature matches using \\texttt{libviso2} and each RGB-to-grayscale transformation. The highest mean number of matches for each sequence is highlighted in bold.}\n\n \\begin{tabular}{@{}llccccc@{}}\n \\toprule\n \\multicolumn{2}{l}{} & \\multicolumn{5}{c}{\\textbf{Inlier Feature Matches $\\mu (\\sigma)$}} \\\\ \\cmidrule{3-7}\n \\multicolumn{2}{l}{\\textbf{Test Sequence}} & Gray & SumLog & SumLog-E & MLP & MLP-E \\\\ \\midrule\n \\multicolumn{2}{l}{\\texttt{VKITTI\/0001}} & & & & & \\\\\n & Sunset-Morning & 262 (82) & \\textbf{726 (136)} & 689 (157) & 661 (108) & 623 (116) \\\\\n & Overcast-Clone & 444 (58) & \\textbf{790 (129)} & 767 (125) & 747 (106) & 770 (107) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0002}} & & & & & \\\\\n & Sunset-Morning & 240 (22) & \\textbf{812 (67)} & 803 (70) & 774 (65) & 702 (62) \\\\\n & Overcast-Clone & 290 (41) & 739 (67) & 757 (76) & 755 (65) & \\textbf{764 (84)} \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0006}} & & & & & \\\\\n & Sunset-Morning & 125 (33) & 669 (44) & \\textbf{735 (43)} & 711 (37) & 642 (41) \\\\\n & Overcast-Clone & 142 (33) & 647 (39) & \\textbf{666 (43)} & 566 (47) & 546 (38) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0018}} & & & & & \\\\\n & Sunset-Morning & 234 (53) & \\textbf{817 (36)} & 816 (37) & 745 (37) & 731 (40) \\\\\n & Overcast-Clone & 311 (46) & 548 (52) & \\textbf{555 (50)} & 450 (42) & 486 (39) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{VKITTI\/0020}} & & & & & \\\\\n & Sunset-Morning & 210 (39) & \\textbf{762 (69)} & 758 (71) & 756 (62) & 675 (69) \\\\\n & Overcast-Clone & 287 (78) & 716 (71) & 716 (72) & \\textbf{718 (62)} & 708 (64) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{InTheDark}} & & & & & \\\\\n & Map~~(08:54) & - & - & - & - & - \\\\\n & \\texttt{0006} (09:46) & \\textbf{178 (48)} & 125 (56) & 165 (51) & 138 (52) & 158 (53) \\\\\n & \\texttt{0027} (18:36) & 9 (9) & 52 (29) & \\textbf{57 (26)} & 44 (25) & 48 (28) \\\\\n & \\texttt{0041} (21:48) & 10 (16) & 33 (25) & \\textbf{39 (31)} & 33 (29) & 35 (29) \\\\\n & \\texttt{0058} (05:48) & 99 (34) & 95 (47) & \\textbf{114 (43)} & 97 (44) & 113 (40) \\\\\n & \\texttt{0071} (09:18) & \\textbf{181 (44)} & 126 (52) & 167 (47) & 141 (48) & 161 (48) \\\\\n & \\texttt{0083} (14:01) & 53 (21) & 76 (39) & \\textbf{83 (37)} & 82 (37) & \\textbf{83 (37)} \\\\\n & \\texttt{0089} (16:43) & 45 (22) & 67 (35) & \\textbf{70 (36)} & 63 (31) & 68 (33) \\\\ \\addlinespace\n\\multicolumn{2}{l}{\\texttt{RobotCar}} & & & & & \\\\ \n& Overcast-Night & 11(3) & \\textbf{12 (2)} & \\textbf{12 (3)} & 11 (2) & 11 (2) \\\\\n& Overcast-Sunny & 26 (12) & 25 (12) & 25 (12) & 24 (9) & \\textbf{71 (41)} \\\\ \\bottomrule\n \\end{tabular}\n \\label{tab:match_stats}\n \\vspace{-12pt}\n\\end{table}\n\n\\subsection{Feature Matcher Approximation}\nWe train $\\MatcherNet$ in a self-supervised manner to predict the number of inlier feature matches for overlapping image pairs captured under different illumination conditions from nearby but different poses.\nTraining labels are generated on the fly for each image pair using the open-source \\texttt{libviso2} library~\\cite{Geiger2011-xe} in monocular flow matching mode with default parameters, using the eight-point RANSAC algorithm to reject outlier matches.\nIn practice we train $\\MatcherNet$ to minimize \\Cref{eq:matcher_loss} over all combinations of input images.\n\n\\Cref{fig:matcher_target_est} plots actual and estimated match counts for each dataset after ten epochs of pre-training on Gray images, aggregated over all test sequences.\nThese include self-matches (same viewpoint, same appearance) and non-self-matches (different viewpoint, different appearance), which appear as clusters.\nIn each case the test-time match counts predicted by $\\MatcherNet$ are strongly correlated with the true performance of \\texttt{libviso2}.\nThis indicates that our approach generalizes well and that $\\MatcherNet$ is capturing salient properties of feature matching rather than memorizing training examples.\n\n\\begin{table*}[]\n \\centering\n \\caption{Maximum distances travelled on dead reckoning for each test sequence of the UTIAS In The Dark dataset, based on various thresholds of inlier feature matches against the ``Map'' sequence. The best results are highlighted in bold.}\n \\begin{threeparttable} \n \\begin{tabular}{@{}ll*{17}c@{}}\n \\toprule\n & & \\multicolumn{17}{c}{\\textbf{Maximum Distance on Dead Reckoning (m)}} \\\\ \\cmidrule{3-19}\n & & \\multicolumn{5}{c}{\\textbf{$\\ge$ 10 Inliers}} & & \\multicolumn{5}{c}{\\textbf{$\\ge$ 20 Inliers}} & & \\multicolumn{5}{c}{\\textbf{$\\ge$ 30 Inliers}} \\\\ \n \\multicolumn{2}{l}{} & G\\tnote{1} & S\\tnote{2} & S-E\\tnote{2} & M\\tnote{3} & M-E\\tnote{3} & & G\\tnote{1} & S\\tnote{2} & S-E\\tnote{2} & M\\tnote{3} & M-E\\tnote{3} & & G\\tnote{1} & S\\tnote{2} & S-E\\tnote{2} & M\\tnote{3} & M-E\\tnote{3} \\\\ \\cmidrule{3-7} \\cmidrule{9-13} \\cmidrule{15-19}\n \\multicolumn{2}{l}{\\texttt{InTheDark}} & & & & & & & & & & & & & & & & & \\\\\n & Map~~(08:54) & - & - & - & - & - & & - & - & - & - & - & & - & - & - & - & - \\\\\n & \\texttt{0006} (09:46) & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & 0.1 & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} \\\\\n & \\texttt{0027} (18:36) & 7.5 & 0.7 & \\textbf{0.0} & 0.3 & 0.3 & & 25.9 & 3.6 & \\textbf{0.7} & 2.5 & 2.2 & & 101.1 & 10.5 & \\textbf{0.7} & 8.3 & 7.8 \\\\\n & \\texttt{0041} (21:48) & 14.9 & 0.5 & \\textbf{0.2} & 0.9 & 1.4 & & 46.3 & 8.4 & \\textbf{3.2} & 7.2 & 7.4 & & 104.5 & 15.1 & \\textbf{10.0} & 18.6 & 15.7 \\\\\n & \\texttt{0058} (05:48) & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & 0.2 & \\textbf{0.0} & 0.2 & \\textbf{0.0} & & \\textbf{0.0} & 0.6 & 0.3 & 0.6 & \\textbf{0.0} \\\\\n & \\texttt{0071} (09:18) & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} \\\\\n & \\texttt{0083} (14:01) & \\textbf{0.0} & 0.2 & \\textbf{0.0} & \\textbf{0.0} & \\textbf{0.0} & & 0.4 & 0.5 & 0.3 & 0.2 & \\textbf{0.0} & & 2.2 & 4.0 & \\textbf{0.3} & 0.4 & 0.5 \\\\\n & \\texttt{0089} (16:43) & 0.3 & 0.3 & 0.2 & \\textbf{0.0} & \\textbf{0.0} & & 4.8 & 1.0 & 3.7 & 1.2 & \\textbf{0.8} & & 6.1 & 6.1 & 4.7 & \\textbf{2.9} & \\textbf{2.9} \\\\ \\addlinespace\n \\multicolumn{2}{l}{\\texttt{RobotCar}} & & & & & & & & & & & & & & & & & \\\\ \n& Map (Overcast) & - & - & - & - & - & & - & - & - & - & - & & - & - & - & - & - \\\\\n& Night & 27.0 & 7.6 & \\textbf{3.7} & 27.3 & 27.3 & & 307.1 & 307.1 & \\textbf{245.9} & 400.8 & 398.9 & & 740.5 & \\textbf{541.5} & 740.5 & 697.3 & 740.5 \\\\\n& Sunny & 1.3 & 1.3 & 1.5 & 1.8 & \\textbf{0.9} & & 36.1 & 65.1 & 48.0 & 39.4 & \\textbf{6.3} & & 129.1 & 139.8 & 109.2 & 124.3 & \\textbf{16.5} \\\\ \\bottomrule\n \\end{tabular}\n \\begin{tablenotes}\n \\item[1] G: Gray\n \\item[2] S: SumLog \n \\item[3] M: MLP \n \\end{tablenotes}\n \\label{tab:inthedark:loc_succ}\n \\end{threeparttable}\n \\vspace{-12pt}\n\\end{table*}\n\n\\subsection{Feature Matching Across Appearance Change}\n\\Cref{fig:output_grid} shows the outputs of each image transformation for sample RGB image pairs in the \\texttt{VKITTI\/0020} Morning and Sunset sequences (\\Cref{fig:vkitti_0020_grid}) and the challenging sequence \\texttt{InTheDark\/0041} (\\Cref{fig:inthedark_000041_grid}).\nWe see that each model produced image pairs that are visually more consistent than standard Gray images, and that local illumination variations such as shadows, uneven lighting, and specular reflections were minimized by optimizing \\Cref{eq:encoder_loss}.\n\n\\Cref{fig:matches_box_whiskers} visually compares the distributions of actual \\texttt{libviso2} feature matches for each transformation, while \\Cref{tab:match_stats} summarizes the results numerically.\nEach model significantly increased the mean number of inlier matches across most test sequences, with the greatest improvements generally obtained from the SumLog and SumLog-E transformations.\nSequences \\texttt{InTheDark\/0006} and \\texttt{InTheDark\/0071} are exceptions in that the standard Gray transformation performed best.\nThese sequences were recorded under similar conditions to the ``Map'' sequence, so feature matching can be expected to perform optimally on Gray images.\nWe saw little improvement in match counts on the \\texttt{RobotCar\/}Overcast-Night experiment, which we attribute to motion blur in the nighttime images making feature matching exceptionally difficult.\nIn contrast, the {MLP-E} method more than doubled the mean number of feature matches in the Sunny experiment.\n\nWe note that the pairwise encoder did not confer any significant benefit on the \\texttt{VKITTI} sequences.\nThese results are consistent with \\cite{Clement2017-gx,Corke2013-hl,McManus2014-op,Paton2017-fi}, where one or two sets of parameter values were sufficient to achieve good performance across varying daytime conditions.\nIn contrast, the encoder network provided a noticeable performance boost on most \\texttt{InTheDark} and \\texttt{RobotCar} sequences.\nWe attribute this difference to more variation in illumination and terrain, as a single transformation is less likely to perform well under more varied conditions.\nWe also note that the MLP-E transformation frequently performed similarly to the SumLog-E transformation, suggesting that, in spite of key assumptions being broken, a linear combination of log-responses as proposed by~\\cite{Ratnasingam2010-cr} may in fact be an optimal solution for this problem, and that a careful choice of weights is the key to obtaining good cross-appearance feature matching over day-night cycles.\n\n\\subsection{Impact on Localization Performance}\nWe evaluate localization performance in an autonomous route-following context by examining the maximum distances in each sequence that would have been navigated using dead reckoning (e.g., visual odometry) as a result of failing to localize against the map.\nThese results are summarized in \\Cref{tab:inthedark:loc_succ} for thresholds of 10, 20, and 30 inlier feature matches against the ``Map'' sequence.\nA typical criterion for requiring manual intervention is dead reckoning in excess of 10 meters, depending on the accuracy of the underlying dead reckoning system.\nBased on a relatively conservative threshold of 20 inlier feature matches against the ``Map'' sequence, we see that \\texttt{InTheDark\/0027} (evening) and \\texttt{InTheDark\/0041} (night) presented significant difficulty for localization, which was substantially alleviated using any of the four image transformations.\nIn particular, the SumLog-E transformation yielded maximum dead reckoning distances below four meters across all illumination conditions.\nWe see similar improvements using the SumLog-E method with a threshold of 10 inliers on the \\texttt{RobotCar} dataset.\nTogether, these results imply that near-continuous 6-dof visual localization over a full day-night cycle is achievable using only a single mapping experience and a simple image pre-processing step, representing a dramatic reduction in data requirements to scale experience-based localization to long deployments.\n\nWith more conservative thresholds (e.g., 30 inliers), localization failures became more common, as expected.\nHowever, the proposed image transformations continued to provide substantially more robust localization performance compared to standard grayscale images.\nFor example, we achieved a maximum distance on dead reckoning of 10.0 m using the SumLog-E method on sequence \\texttt{InTheDark\/0041} with a threshold of 30 inliers.\nIn contrast, using the standard Gray method would have required the system to rely continuously on dead reckoning for approximately 40\\% of the route.\n\n\\section{Conclusions and Future Work}\nThis paper presented a method for learning pixel-wise RGB-to-grayscale colorspace mappings that explicitly maximize the number of inlier feature matches for a given input image pair, feature detector\/matcher and operating environment.\nOur key insight is that by training a deep neural network to predict the performance of a conventional non-differentiable feature detector\/matcher, we can define a fully differentiable loss function that can be used to learn image transformations optimized for localization performance.\nWe evaluated our approach using both physically motivated and data-driven transformations and demonstrated substantially improved feature matching and localization performance on synthetic and real long-term vision datasets exhibiting severe illumination change, allowing experience-based localization to scale to long deployments with dramatically fewer bridging experiences.\nWe consistently achieved the best performance using a physically motivated weighted sum of log-responses with weights derived from a pairwise context encoder network.\nIn future work we plan to explore alternative loss functions such as pose estimation error, the use of feature locations or photometric consistency as a more granular supervision signal, and the impact of context feature dimension on MLP-based transformations.\nWe also intend to investigate the impact of our method on feature matching across seasonal appearance change.\n\n\\bibliographystyle{ieeetr}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nZero-range processes \\cite{cc1,cc2,cc3,cc4,cc5,cc6} have attracted attention of many researchers\nsince they provide an exactly solvable example of \nfar-from-equilibrium dynamics and of condensate formation.\nMany questions concerning the dynamics of the model\ncan be addressed and solved analytically. It is known that a \nzero-range process has a steady state and that this state \nis described by the partition function of the balls-in-boxes (B-in-B) \nmodel \\cite{bbj,bbj2}, also called the Backgammon model. \nThe B-in-B model has two phases: a fluid and a condensed \none, separated by a critical point at which the system \nundergoes a phase transition and the condensate is formed. \nUnlike the Bose-Einstein condensation,\nthe B-in-B condensation takes place in real space\nrather than in momentum space. \nTherefore it mimics such processes like mass transport \\cite{cc1}, \ncondensation of links in complex networks \\cite{cc2,bbw} or phase separation \\cite{ph1,ph2}.\n\nThe zero-range process (ZRP) \\cite{evans} discussed in this paper describes\na gas of identical, indistinguishable \nparticles hopping between the neighboring nodes of a network.\nThe state of such a system is characterized by the topology of \nthe network, which is fixed during the process, and\nby the particle distribution which is given by\nthe occupation numbers of particles $\\{m_i\\}$ on all nodes \n$i=1,\\ldots, N$ of the network.\nThe total number of particles $M=m_1+m_2+\\ldots + m_N$ \nis conserved during the process. \nThe zero-range dynamics is characterized by the outflow rates $u(m)$ of particles from network nodes,\nwhich depend only on the occupation number $m$ of the node from which the particle hops.\nWe shall assume that these ultra local hopping rates\nare identical for each node. The stream of particles \noutgoing from a node is equally distributed among all its\nneighbors. So if we denote by $k_i$ the number of neighbors of the node $i$, called also its degree, \nthen the hopping rate from the $i$-th node to each of its neighbors is equal to $\\frac{1}{k_i} u(m_i)$.\nThe outflow rate $u(m)$ is a semi-positive function which is \nequal to zero for $m=0$. For a given graph the function\n$u(m)$ entirely defines the dynamics of the system. \n\nWe consider here the ZRP on a network being a connected simple\ngraph. In this case, the ZRP has a unique steady state,\nin which the probability $P(m_1,\\dots,m_N)$ \nof finding the distribution of balls $\\{m_1,\\ldots,m_N\\}$ \nfactorizes into a product of some weight functions\n$p_i(m_i)$ for individual nodes, except that there is a global constraint reflecting the conservation of particles. \n\nOn a $k$-regular network, that is when all node degrees are equal to $k$, \nall weight functions $p_i(m)$ are identical, $p_i(m)\\equiv p(m)$, \nand thus the probability $P$ is invariant under permutations \nof the occupation numbers. When the density\n$\\rho=M\/N$ of balls per node exceeds a certain critical value $\\rho_c$ depending on the functional form of $p(m)$, \na single node attracts an extensive number of balls called the condensate. The relative occupation of that node does not disappear\nin the thermodynamic limit $N,M\\to\\infty$, with fixed $\\rho$. The larger the density $\\rho$, the larger is the\nnumber of balls in the condensate. In other words,\nthe system undergoes a phase transition at $\\rho=\\rho_c$ \nbetween the fluid (low density) and the condensed (high density)\nphase. The permutation symmetry of $\\{m_i\\}$ is respected in the\nfluid phase while it is broken in the condensed phase where one node\nbecomes evidently distinct from the $N\\!-\\!1$ remaining ones.\nThe symmetry of the partition function reduces to the subgroup of permutations\nof $N\\!-\\!1$ occupation numbers. This mechanism \nhas been extensively studied in the B-in-B model \\cite{bbj,bbj2,bbjw}.\nThe value of the critical\ndensity depends on the weight function\n$p(m)$ which can be translated to the asymptotic properties of $u(m)$ for $m\\to\\infty$.\nIf $u(m)$ tends to infinity\nthen also $\\rho_c=\\infty$ and the condensation does not occur regardless\nof the density of balls. The system is\nin the fluid phase for any finite density $\\rho$. Intuitively,\nthis means that there exists an effective repulsive force \npreventing a node from being occupied by many balls\nand they distribute uniformly on the whole graph.\nOn the contrary, if $u(m) \\to 0$, \nthe critical density is $\\rho_c=0$ and therefore the system is in the condensed phase\nfor any $\\rho>0$. The larger the number of particles on a node, the smaller\nare the chances for balls to escape from it since $u(m)$ becomes very small\nfor large $m$. This can be seen as the existence of an effective attraction between particles.\n\nThe most interesting case is when $u(m)$ goes to some positive constant $u_\\infty$ with $m\\to\\infty$. \nOne can show that the probability distribution\n$P(m_1,\\ldots,m_N)$ does not depend on $u_\\infty$ \\cite{bbjw}\nbut on how fast $u(m)$ approaches the constant value.\nTherefore without loss of generality we can choose $u_\\infty=1$ and concentrate on the asymptotic behavior\nhaving the form: $u(m)=1+b\/m$, with $b$ being some positive number.\nIf $0\\le b \\le 2$, then the critical density $\\rho_c$ is infinite.\nThe effective attraction between balls is too weak to form the condensate.\nHowever, if $b>2$, then $\\rho_c$ has a finite value. \nIn this case the attraction is strong enough to trigger \nthe condensation above the critical density $\\rho_c$. \n\nSo far we have discussed the criteria of the condensation on $k$-regular networks, where it arises\nas a result of the spontaneous symmetry breaking. All those facts are well known \\cite{evans,evans2}.\nOn the other hand, the permutation symmetry can \nbe explicitly broken if the weight functions $p_i(m)$ \nare not identical for all nodes. For instance, this happens when the network \non which the process takes place is inhomogeneous, that is when\nthe degrees $k_1,\\dots,k_N$ vary. A particularly important example are complex networks \\cite{cn},\nfor which the distribution of degrees has usually a long tail, and thus there are\nmany nodes with relatively high degree. They are, however, not easy for analytical studies although \nsome predictions are possible \\cite{jdn}.\nBelow we shall argue that to gain some insight into the static and dynamical properties\nof the ZRP on such networks it is sufficient to study some simplified models.\n\nIn the remaining part of the paper we \nconsider only the most favorable situation when\nthe weight of only one node is different from \nthe remaining ones. Such an inhomogeneity of the weights\ncan be introduced either by an inhomogeneity of the outflow\nrates $u_i(m)$ or by an inhomogeneity of the degree distribution.\nWe focus here on the latter situation, when a graph has one node of \ndegree $k_1$ which differs from all\nremaining degrees $k_2=\\ldots =k_N\\equiv k$. As we shall see,\nin this case the quantity $\\mu = \\log(k_1\/k)$ plays the role\nof an external field breaking the permutation symmetry.\n\nIn particular, we discuss the dynamics of the condensate on such inhomogeneous networks.\nThis is a relatively new topic and, in contrast to the stationary properties, less understood.\nAlthough the emergence of the condensate has been studied for homogeneous and inhomogeneous systems \\cite{evans} and numerically for scale-free networks in \\cite{jdn}, studies of the dynamics of an existing condensate\nare rare \\cite{god}.\nFor instance, one question which may be asked is what is the typical life time of the condensate,\nthat is how much time does it take to ``melt'' the condensate at one node and rebuild it at another node.\nTo provide an answer to this problem is the main goal of the present article.\n\nThe rest of the paper is organized as follows. In Sec.~II we \ndiscuss the static properties of the ZRP on inhomogeneous networks. \nWe consider some particular graph topologies: $k$-regular graphs,\nstar graphs and $k$-regular graphs with a single inhomogeneity\nintroduced by a vertex of degree $k_1>k$. Their advantage is that\nall calculations can be done exactly or at least with an excellent approximation.\nIn all cases we calculate the effective occupation number distribution\n$\\pi(m)$ and use it to derive information about the condensate\ndynamics. In Sec.~III we derive\nanalytic expressions for the life time of the condensate.\nWe concentrate on the role of\ninhomogeneity and typical scales at which it becomes relevant.\nAll analytical results are cross-checked by Monte Carlo simulations.\nThe last section is devoted to a summary of our results.\n\n\\section{Steady state -- statics}\nA zero-range process on a connected simple graph has a steady state\nwith the following partition function $Z(N,M,\\{k_i\\})$ \n\\cite{evans}:\n\\begin{equation}\nZ(N,M,\\{k_i\\}) = \n\\sum_{m_1=0}^M \\cdots \\sum_{m_N=0}^M \\delta_{ \\sum_{i=1}^N m_i, M}\n\\prod_{i=1}^N p(m_i) k_i^{m_i},\t\\label{part}\n\\end{equation}\nwhere $\\delta_{i,j}$ denotes the discrete delta function and the weight function $p(m)$ is related to the hop rate $u(m)$ through the formula:\n\\begin{equation}\np(m)= \\prod_{n=1}^m \\frac{1}{u(n)}, \\;\\;\\; p(0)=1.\t\n\\label{pbyu}\n\\end{equation}\nWe shall denote $Z(N,M,\\{k_i\\})$ in short by $Z(N,M)$, having in mind its dependence on the degrees. \nThe partition function (\\ref{part}) contains the whole \ninformation about the static properties of the system in the steady state.\nThe only trace of graph topology in the formula is through\nthe nodes degrees. The dynamics, however, depends also on other topological \ncharacteristics but they become important only in refined treatments.\nIn a sense, the degree sequence is the first-order approximation also for the dynamics. \n\nThe probability $P(m_1,\\dots,m_N)$ of a given configuration \n$\\{m_i\\}$ reads:\n\\begin{eqnarray}\nP(m_1,\\dots,m_N) &=& \\frac{1}{Z(N,M)} \n\\prod_{i=1}^N p(m_i) k_i^{m_i} \\nonumber \\\\\n&=& \\frac{1}{Z(N,M)} \\prod_{i=1}^N \\tilde{p}_i(m_i),\n\\end{eqnarray}\nwhere we have defined ``renormalized'' weights\n$\\tilde{p}_i(m_i) = p(m_i)k_i^{m_i}$, being now node-dependent.\nThe most important quantity characterizing the steady state is\nthe probability $\\pi_i(m)$ that the\n$i$-th node is occupied by $m$ particles:\n\\begin{eqnarray}\n\\pi_i(m_i) = \\sum_{m_1} \\cdots \\sum_{m_{i-1}} \n\\sum_{m_{i+1}} \\cdots \\sum_{m_N} P(m_1,\\dots,m_N) \\times \\nonumber \\\\\n\\times \\delta_{\\sum_{j=1}^N m_j, M} = \\frac{Z_i(N-1,M-m_i)}{Z(N,M)} \\tilde{p}_i(m_i),\n\\label{pigeneral}\n\\end{eqnarray}\nwhere $Z_i(N-1,M-m)$ denotes the partition function \nfor $M-m$ particles occupying a graph consisting \nof $N-1$ nodes with degrees $\\{k_1,\\dots,k_{i-1},k_{i+1},\\dots,k_N\\}$. \nWe shall call $\\tilde{p}_i(m)$ ``bare'' occupation probability\nwhile $\\pi_i(m)$ ``dressed'' or effective occupation probability\nof the node $i$. We also define the average occupation probability:\n\\begin{equation}\n\\pi (m) = (1\/N) \\sum_i \\pi_i(m).\n\\end{equation}\nFor a $k$-regular graph, that is for $k_i\\equiv k$,\nthe occupation probability $\\pi_i(m)=\\pi(m)$ is the same for every\nnode and all the formulas above reduce to those discussed \nin \\cite{bbj}. In general, the partition function can be calculated recursively:\n\\begin{eqnarray}\n& & Z(N,M,\\{k_1,\\dots,k_N\\}) = \\nonumber \\\\\n&=& \\sum_{m_N} \\tilde{p}_N(m_N) \\sum_{m_1,\\dots,m_{N-1}} \\delta_{\\sum_{i=1}^{N-1} m_i, M-m_N} \\prod_{i=1}^{N-1} \\tilde{p}_i(m_i) \\nonumber \\\\\n& =& \\sum_{m_N=0}^{M} \\tilde{p}_N(m_N) Z(N-1,M-m_N,\\{k_1,\\dots,k_{N-1}\\}). \\nonumber \\\\\n\\label{znmrec}\n\\end{eqnarray}\nFor $N=1$ the partition function simply reads \n$Z(1,M,k_1) = \\tilde{p}_1(M)$. The recursive use of the\nformula (\\ref{znmrec}) allows one to compute \nthe partition function within a given numerical accuracy.\nUsing this method we were able to push the computation as far as to\n$N$ of order $500$. For identical weights, one can find\na more efficient recursion relation by splitting the system into\ntwo having similar size, which allows to study much larger systems.\nThe computation of the partition function can be used together\nwith Eq.~(\\ref{pigeneral}) to determine \nnumerically the node occupation distribution $\\pi_i(m)$. \nThis gives an exact result with a given accuracy and is more efficient than the corresponding \nMonte Carlo simulations of the ZRP. The dynamics, however, is not accessible in this way.\n\nAs mentioned in the introduction we are going to examine the effect of\ntopological inhomogeneity on the properties of the ZRP.\nWe shall consider an almost $k$-regular graph \nwith one node, say number one, having a degree bigger than the rest of nodes:\n$k_1>k$ and $k_2=\\dots=k_N=k$. \nThe simplest realization of such a graph is a star or a wheel graph.\nIn general, such a graph can be constructed from any $k$-regular\ngraph by a local modification. We proceed as follows. First we\n derive exact formulas for the particle distribution\nin a steady state on a $k$-regular graph which shall later\nserve us as a reference point. Then we consider the particular\nexample of a star topology as the simplest example\nof a single defect and finally an arbitrary $k$-regular graph with a singular node $k_1>k$.\nThe system has now the following weights: \n$\\tilde{p}_1(m)= k_1^m p(m)$ for the singular and $\\tilde{p}_i(m)= k^m p(m)$ for the regular nodes. \nThey differ by an exponential factor\n$\\tilde{p}_1(m)\/\\tilde{p}_i(m)= (k_1\/k)^m = e^{\\mu m}$\nwhere $\\mu =\\log(k_1\/k)>0$, which clearly favors the situation\nin which the singular node has much more particles\nthan the regular ones. To make things as simple as possible, \nand to concentrate on the effect of inhomogeneity\nwe assume that the outflow rate $u(m)=1$ is constant\nand independent of $m$. In this case $p(m)=1$ is also constant,\nsimplifying calculations.\nAll other functions with the asymptotic behavior \n$u(m)\\rightarrow 1$ would lead basically to the same qualitative \nbehavior. This is because\nin this case $p(m)$ would have a power-law tail which is much\nless important for the large $m$-behavior than the exponential\nfactor $e^{\\mu m}$ introduced by the inhomogeneity.\nWe shall briefly comment on this towards the end of the paper.\n\n\\subsection{$k$-regular graph}\nWith the assumption $p(m)=1$,\nthe partition function $Z(N,M)$ from Eq.~(\\ref{part}) for the steady state of the ZRP on a $k$-regular graph reads:\n\\begin{eqnarray}\nZ_{\\rm reg}(N,M) &=& \\sum_{m_1=0}^M \\cdots \\sum_{m_N=0}^M \n\\delta_{\\sum_{i} m_i, M} \\;\nk^{\\sum_i m_i} \\nonumber \\\\\n&=& k^M \\frac{1}{2\\pi i} \\oint {\\rm d} z \\, \nz^{-M-1} \\left(\\sum_{m=0}^M z^m\\right)^N.\n\\label{cint}\n\\end{eqnarray}\nWe used an integral representation of the discrete delta function\nwhich allowed us to decouple the sums over $m_1,\\dots, m_N$ for\nthe price of having the integration over $z$.\nThe sum over $m$ can be done yielding $1\/(1-z)$. Using\nthe expansion:\n\\begin{equation}\n\\left(\\frac{1}{1-z}\\right)^N = \n\\sum_{m=0}^\\infty \\binom{-N}{m} (-z)^m \n= \\sum_{m=0}^\\infty \\binom{N+m-1}{m} z^m\n\\end{equation}\nand Cauchy's theorem we see that the contour\nintegration over $z$ selects only the term with $m=M$\nfrom the integrand in (\\ref{cint}), so we obtain:\n\\begin{equation}\nZ_{\\rm reg}(N,M) = k^M \\binom{N+M-1}{M}. \\label{zreg}\n\\end{equation}\nInserting this into Eq.~(\\ref{pigeneral})\nwe find the occupation number distribution\n\\begin{eqnarray}\n\\pi(m) &=& \\pi_i(m) = \\binom{M+N-m-2}{M-m} \/ \\binom{M+N-1}{M} \\nonumber \\\\\n&\\propto & \\frac{(M+N-m-2)!}{(M-m)!}. \\label{pik}\n\\end{eqnarray}\nThe distribution $\\pi(m)$ is identical for all nodes and independent on $k$. \nIt falls faster than exponentially for large $m$, therefore the condensate\nnever appears.\n\nIn particular one can apply these formulas to the complete \ngraph which is just a $k$-regular graph with $k=N-1$.\nIn Fig.~\\ref{f1} we see the comparison between the\ntheoretical expression (\\ref{pik}) and results of\nnumerical Monte Carlo simulations for a $4$-regular graph with $N=20$ nodes and \ntwo different numbers of balls, $M$. \nThe simulations of the\nZRP were organized in sweeps consisting on $N$ steps each. In a single\nstep a node was chosen at random and if it was non-empty\na particle was picked and moved to a neighboring node.\nFor each graph the process was initiated from a random\ndistribution of particles. After some thermalization,\n measurements of $\\pi(m)$\nwere done on $10^4$ configurations generated every sweep.\nThe results were averaged over these configurations and then\nover $5\\times 10^4$ independent graphs drawn at random\nfrom the ensemble of $k$-regular graphs. \n\n\\begin{figure}\n\\psfrag{x}{$m$}\n\\psfrag{y}{$\\pi(m)$}\n\\includegraphics[width=8cm]{f1.eps}\n\\caption{The ``experimental'' distribution $\\pi(m)$ compared to \nthe theoretical prediction (\\ref{pik}) for regular graphs \nwith $k=4$ and $N=20$ and for $M=20$ (+) and $M=40$ (x) balls.}\n\\label{f1}\n\\end{figure}\n\n\\subsection{Star graph}\nWe consider first a special case of a single \ninhomogeneity graph, namely the star graph having $N-1$ nodes of \ndegree $k_2=\\ldots=k_N=1$ connected to the central node with $k_1=N-1$.\nThe partition function $Z_{\\rm star}(N,M)$ is\n\\begin{eqnarray}\nZ_{\\rm star}(N,M) &=& \n\\sum_{m_1=0}^\\infty \\cdots \\sum_{m_N=0}^\\infty \\delta_{\\sum_{i=1}^N m_i , M}\n(N-1)^{m_1} \\hspace{5mm} \\nonumber \\\\\n&=& \\sum_{m=0}^M (N-1)^m \\binom{M+N-m-2}{M-m}, \\label{zstar}\n\\end{eqnarray}\nas follows from Eq.~(\\ref{zreg}). It is convenient to change\nthe summation index from $m$ to $j = M-m$ which can be interpreted as\na deficit of particles counted relatively to the full occupation:\n\\begin{eqnarray}\nZ_{\\rm star}(N,M) = (N-1)^M \\sum_{j=0}^M (N-1)^{-j} \\binom{N+j-2}{j} . \\nonumber \\\\\n\\end{eqnarray}\nLet us assume that $N\\gg 1$. The summands in the last expression are strongly\nsuppressed when $j$ increases so the sum can \nbe approximated by changing the upper limit from $M$ to $\\infty$. We obtain\n\\begin{eqnarray}\nZ_{\\rm star}(N,M) \n&\\cong & (N-1)^M \\sum_{j=0}^\\infty \\left(\\frac{-1}{1-N}\\right)^j \n\\binom{-(N-1)}{j} \\nonumber \\\\\n&=& (N-1)^M \\left(1-\\frac{1}{N-1}\\right)^{1-N} \\nonumber \\\\\n&=& (N-1)^M \\left( \\frac{N-1}{N-2} \\right)^{N-1}.\n\\end{eqnarray}\nUsing Eq.~(\\ref{pigeneral}) \nand the partition function $Z_{\\rm reg}$ \ncalculated in Eq. (\\ref{zreg}) of the previous subsection we can determine the distribution of particles\n$m$ at the central (singular) node,\n\\begin{eqnarray}\n&\\pi_1(m) &= \\frac{Z_{\\rm reg}(N-1,M-m)}{Z_{\\rm star}(N,M)} (N-1)^m \\nonumber \\\\\n&=& (N-1)^{m-M} \\binom{M+N-m-2}{M-m} \\left( \\frac{N-2}{N-1} \\right)^{N-1}. \\nonumber \\\\\n\\label{pi1star}\n\\end{eqnarray}\nSimilarly, we can determine the distribution of particles\non any external (regular) node $i$:\n\\begin{equation}\n\\pi_{i}(m) = \\frac{(N-2)(N-1)^{-m}}{N-1-(N-1)^{-M}} \\approx \\frac{N-2}{(N-1)^{m+1}}. \\label{pistext}\n\\end{equation}\nWe see that $\\pi_{i}(m)$ decays exponentially \nwith $m$ while $\\pi_1(m)$ grows exponentially for $m\\ll M$: \n\\begin{equation}\n\\pi_1(m) \\propto e^{ m\\left( \n-\\frac{1}{2M}+\\frac{1}{2(M+N-2)} +\\log\\frac{M(N-1)}{M+N-2} \\right)}.\n\\end{equation}\nThe growth slows down for $m$ approaching $M$. \nAt $m=M$, $\\pi_1(m)$ reaches its maximal value:\n\\begin{equation}\n\\pi_1(M) = \\left(\\frac{N-2}{N-1}\\right)^{N-1},\n\\end{equation}\nwhich tends to $e^{-1}$ when $N$ goes to infinity.\nIn Fig.~\\ref{f2} we compare the theoretical distributions \n(\\ref{pi1star}) and (\\ref{pistext}) with Monte Carlo \nsimulations of the ZRP for $N=20$ nodes and $M=20, 30, 40$. \nThe agreement is very good.\n\\begin{figure}\n\\psfrag{x}{$m$}\n\\psfrag{y}{$\\pi(m)$}\n\\includegraphics[width=8cm]{f2.eps}\n\\caption{The ``experimental'' and the theoretical (solid lines) \nparticle number distributions for the star graph, \nfor $N=20$, and $M=20$ (crosses), $30$ (empty squares) and $40$ (filled squares). \nThe theoretical distributions were calculated according to the\nformula (\\ref{pi1star}) for the central node (rising curves) \nand according to Eq.~(\\ref{pistext}) for external nodes (falling curve, the Monte Carlo data (points) plotted for $N=20$ and $M=30$). \n}\n\\label{f2}\n\\end{figure}\n\nIt is instructive to calculate the mean number \nof particles at the central node. To simplify calculations\nwe make use of the distribution $\\widehat{\\pi}_1(j)\\equiv \\pi_1(M-j)$ of\nthe deficit of particles defined above:\n\\begin{equation}\n\\widehat{\\pi}_1(j) = \n(1-\\alpha)^{1-N} (-\\alpha)^j \\binom{-(N-1)}{j}.\n\\label{remain}\n\\end{equation}\nThe parameter $\\alpha=1\/(N-1)$ is just the ratio of any external node degree to the degree of the central node and measures\nthe level of inhomogeneity. \nThe overall prefactor $(1-\\alpha)^{1-N}$ is independent of $j$. It is just a normalization constant\nthat results from summing the $j$-dependent part of the expressions (\\ref{remain}),\n\\begin{equation}\nS(\\alpha) = \\sum_{j=0}^\\infty \n(-\\alpha)^j \\binom{-(N-1)}{j} = \n\\frac{1}{(1-\\alpha)^{N-1}} .\n\\label{auxs}\n\\end{equation}\nAs before we changed the upper limit from\n$M$ to infinity because $\\alpha \\ll 1$ for the star graph and hence\nthe summands are strongly suppressed for large $j$.\nThe average deficit at the central node is\n\\begin{equation}\n\\left = \n\\sum_{j=0}^M \\widehat{\\pi}_1(j) j = \n\\alpha \\frac{{\\rm d} \\log S(\\alpha)}{{\\rm d}\\alpha} = \\frac{N-1}{N-2}, \n\\label{m1star}\n\\end{equation}\nas follows from Eq.~(\\ref{auxs}). For large $N$ it tends to \none, so we have\n\\begin{equation}\n\\left = M - \\left \\cong M-1.\n\\end{equation}\nWe see that on average almost all balls are concentrated at the central\nnode and only one ball is in the rest of the system.\nWe can also determine the range of fluctuations around $\\left$\nby calculating the variance. Taking again advantage of the generating function (\\ref{auxs}) one finds\n\\begin{eqnarray}\n\\left<(m_1-\\left)^2\\right> = \\left<(j_1-\\left)^2\\right> \\nonumber \\\\\n= \\frac{{\\rm d}^2 \\log S(e^{-\\mu})}{{\\rm d}\\mu^2} = \\left(\\frac{N-1}{N-2}\\right)^2,\n\\end{eqnarray}\nwhere we used the parameter $\\mu=-\\log\\alpha=\\log (N-1)$.\nFor large $N$ the result tends to one, so\nwe can draw the following picture. \nFor any $N\\gg 1$ we observe a condensation of \nparticles at the central node regardless of their density $\\rho=M\/N$. \nThe critical density is equal to zero and the system is always in the condensed phase. \nThe condensate residing at the central node contains $M-1$ particles, with very small fluctuations,\nwhile the other nodes are almost empty. \n\n\\subsection{Single inhomogeneity}\nA very particular property of the star graph is that the\ninhomogeneity increases with its size.\nTherefore, it is interesting to consider the situation\nwhen the inhomogeneity $\\alpha = k\/k_1$ is arbitrary and independent of $N$. \nThe single inhomogeneity graph we consider here has one node of degree \n$k_1$ and $N-1$ nodes of degree $k$. Again, we assume that $k_1>k$.\nThe partition function (\\ref{part}) takes now the form\n\\begin{equation}\nZ_{\\rm inh}(N,M) = \n\\sum_{m_1=0}^M k_1^{m_1} \n\\sum_{m_2,\\dots,m_N=0}^M \\delta_{\\sum_i m_i,M} \n\\; k^{\\sum_{i=2}^N m_i}.\n\\label{zsh1}\n\\end{equation}\nThe sum over $m_2,\\dots,m_N$ is equal to the partition function \n$Z_{\\rm reg}(N-1,M-m_1)$ given by Eq.~(\\ref{zreg}).\nThe whole formula looks almost identical to that\nfor the star graph except that\nnow the degree $k_1$ does not need to be much greater than \n$k$ and therefore the substitution of $M$\nby $\\infty$ has to be done carefully in a manner\nincorporating finite-size corrections. As before, we first\nchange variables from $m_1$ to $j = M-m_1$. Using\nEq.~(\\ref{auxs}) we can cast the formula (\\ref{zsh1}) into the \nfollowing form:\n\\begin{eqnarray}\nZ_{\\rm inh}(N,M) &=& \nk_1^M \\sum_{j=0}^M \\alpha^j \n\\binom{N+j-2}{j} \\nonumber \\\\\n&=& k_1^M \\left[ (1-\\alpha)^{1-N} - c(M) \\right].\n\\label{zinh}\n\\end{eqnarray}\nThe correction $c(M)$ is equal to the sum over $j$ from $M+1$ to infinity.\nIt corresponds to the surplus which has to be\nsubtracted from the infinite sum represented by the first\nterm in square brackets. In the limit $M\\to\\infty$ it can be estimated\nas follows:\n\\begin{equation}\nc(M) = \\sum_{j=M+1}^\\infty \\alpha^j \\binom{N+j-2}{j} \n\\approx \\frac{1}{(N-2)!} \\int_M^\\infty {\\rm d} j \\, e^{F(j)},\n\\label{intc}\n\\end{equation}\nwhere \n\\begin{equation}\nF(j) = j \\log\\alpha + \\log\\left((N+j-2)!\\right) - \\log(j!).\n\\end{equation}\nUsing Stirling's formula we can calculate the \nintegral (\\ref{intc}) by the saddle-point method.\nTaking into account only leading terms we have\n\\begin{eqnarray}\n& &\\int_M^\\infty {\\rm d} j e^{F(j)} \\approx e^{F(j_*)} \\times \\nonumber \\\\\n& &\\times \\sqrt{\\frac{-\\pi}{2F''(j_*)}} \\mbox{erfc}\\left( (M-j_*)\n\\sqrt{-F''(j_*)} \\right),\n\\end{eqnarray}\nwhere erfc denotes the complementary error function,\n\\begin{equation}\n\\mbox{erfc}(x) = \\frac{2}{\\sqrt{\\pi}} \\int_x^\\infty {\\rm d} y e^{-y^2} ,\n\\end{equation}\nand $j_*$ is determined from the saddle equation $F'(j_*)=0$:\n\\begin{equation}\nj_* \\approx \\frac{\\alpha (N-2)}{1-\\alpha}.\n\\end{equation}\nCollecting all terms we eventually find\n\\begin{eqnarray}\nc(M) &\\approx &\n\\frac{\\alpha^{\\frac{\\alpha(N-2)}{1-\\alpha}}}{1-\\alpha} \\frac{((N-2)\/(1-\\alpha))!}\n{(\\alpha(N-2)\/(1-\\alpha))!}\\sqrt{\\frac{\\pi \\alpha (N-2)}{2}} \\times \\nonumber \\\\\n&\\times & \\frac{1}{(N-2)!} \\mbox{erfc}\\left( \\frac{M(1-\\alpha)-\\alpha(N-2)}{\\sqrt{\\alpha(N-2)}}\\right)\t.\n\\label{cmfin}\n\\end{eqnarray}\nIn order to keep formulas shorter we used here the notation\n$x! \\equiv \\Gamma(1+x)$ also for non-integer arguments.\nThe complete partition function \n$Z_{\\rm inh}(N,M)$ is given by the right-hand side of Eq.~(\\ref{zinh}) \nwith $c(M)$ given by Eq.~(\\ref{cmfin}). \nWe can now calculate $\\pi_1(m)$, that is the distribution\nof balls at the singular node,\n\\begin{equation}\n\\pi_1(m) = \\frac{Z_{\\rm reg}(N-1,M-m)}{Z_{\\rm inh}(N,M)} k_1^m,\n\\end{equation}\nwhere $Z_{\\rm reg}$ is the partition function (\\ref{zreg}) for a \nregular graph with degree $k$. Using Eqs.~(\\ref{zreg}) and (\\ref{zinh})\nwe obtain\n\\begin{equation}\n\\pi_1(m) = \\binom{M+N-m-2}{M-m} \\frac{\\alpha^{M-m}}{(1-\\alpha)^{1-N} - c(M)}. \\label{pi1inh}\n\\end{equation}\nIn Fig.~\\ref{f3} we show the theoretical ball distributions for graphs \nwith $k=4$, $N=20$ and various $M$ for singular nodes with $k_1=8$ and $k_1=16$, respectively,\nand compare them with the corresponding results obtained\nby Monte Carlo simulations. The agreement, although very good for the presented plots, is the better, the smaller is the ratio $\\alpha=k\/k_1$.\n\\begin{figure}\n\\psfrag{x}{$m$}\n\\psfrag{y}{$\\pi(m)$}\n\\psfrag{a}{$\\alpha=\\frac{1}{2}$}\n\\psfrag{a2}{$\\alpha=\\frac{1}{4}$}\n\\includegraphics[width=8cm]{f3a.eps}\n\\includegraphics[width=8cm]{f3b.eps}\n\\caption{The distribution of balls at the singular node for graphs \nwith $k=4$, $N=20$, $k_1=8$ (top) and $k_1=16$ (bottom). \nThe total number of balls is $M=20,40$ and $80$ from left to \nright curve, respectively. Points represent numerical data while solid \nlines show Eq.~(\\ref{pi1inh}).\n}\n\\label{f3}\n\\end{figure}\nNeglecting an inessential normalization, we see that\nEq.~(\\ref{pi1inh}) has the asymptotic behavior\n\\begin{equation}\n\\pi_1(m) \\propto \\left(\\frac{k_1}{k}\\right)^m \n\\binom{M+N-m-2}{M-m} \\sim \\exp(G(m)),\n\\end{equation}\nwhere \n\\begin{eqnarray}\nG(m) &=& \\left(M+N-m-\\frac{3}{2}\\right)\\log(M+N-m-2)- \\nonumber \\\\\n&-& m \\log\\alpha - \\left(M-m+\\frac{1}{2}\\right) \\log(M-m).\n\\label{gfunc}\n\\end{eqnarray}\nThe number of particles of the condensate can be \nestimated using the saddle point equation\n$G'(m_*)=0$ for $m_*>0$. Neglecting terms \nof order $1\/M^2$ we find\n\\begin{equation}\nm_* \\cong M - \\frac{\\alpha}{1-\\alpha} (N-2).\n\\end{equation}\nAlternatively one can calculate the number of particles\nof the condensate as the mean of the distribution $\\pi_1(m)$.\nAdapting the same trick as in the previous section,\n\\begin{eqnarray}\n\\left &=& M - \\left = \nM - \\alpha \\frac{{\\rm d} \\log S(\\alpha)}{{\\rm d}\\alpha} \\nonumber \\\\\n&=& M - \\frac{\\alpha}{1-\\alpha} (N-1) \\approx m_*,\n\\label{m1inh}\n\\end{eqnarray}\nas follows from Eqs.~(\\ref{auxs}) and (\\ref{m1star}). The criterion for\ncondensation is that the central node contains an extensive\nnumber of balls. In the limit $N,M\\to \\infty$ and fixed density\n$\\rho=M\/N$ it amounts to the condition $\\left>0$ \nleading to the critical density\n\\begin{equation}\n\\rho_c = \\frac{\\alpha}{1-\\alpha}.\n\\end{equation}\nThe condensation takes place when $\\rho>\\rho_c$ exactly like \nin the Single Defect Site model \\cite{evans}. \nThe critical density decreases with decreasing ratio $\\alpha=k\/k_1$ or, equivalently, \nwith increasing ``external field'' $\\mu=\\log(k_1\/k)$. The singular node attracts\n$N(\\rho-\\rho_c)+\\rho_c$ balls on average as \nfollows from Eq.~(\\ref{m1inh}).\nIt is also easy to find that the distribution of balls \n$\\pi_i(m)$ at any regular node falls exponentially,\n\\begin{equation}\n\\pi_i(m) \\sim \\left(\\frac{k}{k_1}\\right)^m = \\alpha^m = e^{-\\mu m},\n\\end{equation}\nthus the condensate never appears on it. \nA regular node contains on average $\\left = \\rho_c$ balls independently of the total density $\\rho$ of balls in the system\nas long as it exceeds $\\rho_c$.\n\n\\section{Dynamics of the condensate}\n\nLet us now turn to a discussion of the dynamics of the condensate.\nFrom the previous section we know that the condensate\nspends almost all time at the node with highest degree. However,\n occasionally it ``melts'' and disappears from the singular node for a short while.\nWe know that the probability of such an event is very small, so we expect \nthe life-time of the condensate to be very large. \nFollowing the ideas of \\cite{god}, let us imagine that we monitor only\nthe number of particles at the singular node, which fluctuates in time. \nThe temporal sequence of occupation numbers at this node\nperforms a sort of one-dimensional random walk \nand can be viewed as a Markov chain. Using a mean-field approximation\none can derive effective detailed balance equations for \nthe incoming and the outgoing flow of particles for this node.\nThe approximation is based on the assumption that the\nremaining part of the system is quickly thermalized, much faster\nthan the typical time scale of the melting process on the monitored node.\nThus the balance equations are written for the singular node and a single mean-field node having \nsome typical properties. \nFor this mean-field dynamics one can derive many quantities of interest.\nIn particular it is convenient to calculate the average \ntime $\\tau_{mn}$ it takes to decrease the occupation number of\nthe monitored node from $m$ to $n$,\nor more precisely, the average first passage time for the Markov process\ninitiated at $m$ to pass $n$. This quantity was first derived \nin \\cite{god} for the ZRP on a complete graph with outflow rates \n$u(m) = 1+b\/m$. The formula derived there,\n\\begin{equation}\n\\tau_{mn} = \\sum_{p=n+1}^{m} \\frac{1}{u(p)\\pi(p)} \\sum_{l=p}^{M} \\pi(l),\n\\label{Tmngen}\n\\end{equation}\ncan be easily adapted to the case discussed in our paper by \nsetting $u(p)=1$ and using the distribution $\\pi_1(p)$ of the singular node\nin place of $\\pi(p)$ in the original formula.\nEquipped with the formula for $\\tau_{mn}$ we are in principle able to calculate the \ntypical melting time $\\tau$. What is yet missing is the condition for $n$ at\nwhich the condensate can be considered as completely melted. \nWe shall choose the simplest possible criterion and define the\n``typical'' melting time $\\tau$ as $\\tau_{m0}$, that is the time needed to completely empty \nthe monitored node beginning from $m$ equal to the average occupation \nof the node in the steady state. \n\nIt was shown \\cite{god} that for the complete graph and $u(m) = 1+b\/m$, the melting time\nis approximately given by\n\\begin{equation}\n\\tau\\propto (\\rho-\\rho_c)^{b+1} M^b,\n\\end{equation}\nwhere $\\rho_c$ is the critical density above which the condensate\nis formed. The power-law increase with $M$ can be attributed\nto the power-law fall of $\\pi(m)\\sim m^{-b}$, characteristic for homogeneous systems\nwith $u(m)=1+b\/m$.\nThe key point of our paper is that for inhomogeneous networks the melting time does no longer \nfollow a power-law but instead increases {\\em exponentially} with $M$ due to the occurrence\nof the inhomogeneity which can be regarded as an external field $\\mu = \\log (k_1\/k)$, breaking the symmetry.\n\nBefore we do the calculations let\nus make a general remark about the dependence of $\\tau_{mn}$ on $m$ and $n$. A quick inspection of Eq.~(\\ref{Tmngen}) \ntells us that significant contribution to the sum over $p$ comes from terms \nfor which $\\pi(p)$, respectively $\\pi_1(p)$, is small. As we know from the previous section, in the condensed phase\n$\\pi_1(p)$ is many orders of magnitude greater for large $p$ than for\nsmall $p$. Therefore when $m$ is of order $M$, and $n$ is of order $1$, \nthe time $\\tau_{mn}$ varies very slowly with $m$ and, on the other hand, it is\nvery sensitive to $n$. We thus put $m=M$ for simplicity and concentrate on\n$\\tau_{Mn}$.\n\n\n\\subsection{Star graph}\nWe assume $u(p)=1$ as before. \nInserting the expression (\\ref{pi1star}) for the particle occupation distribution $\\pi_1(m)$ for the central node \nof the star into Eq.~(\\ref{Tmngen}) in place of $\\pi(m)$ we obtain\n\\begin{equation}\n\\tau_{mn} = \\sum_{p=n+1}^{m} \\sum_{l=p}^{M} (N-1)^{l-p} \n\\frac{(M+N-l-2)!(M-p)!}{(M+N-p-2)!(M-l)!} .\n\\label{tmnstar}\n\\end{equation}\nFrom Sec.~II~B we know that the condensate contains \n$m =\\left \\approx M$ balls in the steady state and that fluctuations \nare very small. This justifies the choice $m=M$ we made above.\nChanging the summation variables similarly as in the previous section\nwe find:\n\\begin{eqnarray}\n\\tau_{Mn} = (N-2)! \\sum_{r=0}^{M-n-1} \\frac{r!}{(N-2+r)!} (N-1)^r \\times \\nonumber \\\\\n\t\\times \\sum_{q=0}^r (N-1)^{-q} \\frac{(N-2+q)!}{q!(N-2)!}.\n\\end{eqnarray}\nIn the second sum we can move the upper limit to infinity using exactly\nthe same approximation as in Sec.~II~B:\n\\begin{equation}\n\\tau_{Mn} \\approx \\left(\\frac{N-1}{N-2}\\right)^{N-1}(N-2)! \n\\sum_{r=0}^{M-n-1} \\frac{r!(N-1)^r}{(N-2+r)!} .\n\\end{equation}\nAfter a variable change $r\\to M-n-1-r$, the remaining sum can be approximated as\n\\begin{eqnarray}\n\t\\sum_{r=0}^{M-n-1} \\frac{(M-n-1-r)!}{(M+N-n-3-r)!} (N-1)^{-r} \\approx \\nonumber \\\\\n\t\\approx \\frac{(M-n-1)!}{(M+N-n-3)!}\t \\sum_{r=0}^\\infty \\left(\\frac{M+N-n-3}{(M-n-1)(N-1)}\\right)^r, \\hspace{-5mm} \\nonumber \\\\\n\t\\label{eq:sumr0}\n\\end{eqnarray}\nsuch that we finally arrive at the formula:\n\\begin{eqnarray}\n\\tau_{Mn} \\approx \\left(\\frac{N-1}{N-2}\\right)^{N}(N-2)! (N-1)^{M-n-1} \\times \\nonumber \\\\\n\\times \\frac{M-n-1}{M-n-2} \\frac{(M-n-1)!}{(M+N-n-3)!}.\n\\label{TMnstar}\n\\end{eqnarray}\nWe see that the presence of $(N-1)^{-n}$ makes the time $\\tau_{Mn}$\nindeed very sensitive to $n$.\nIn Fig.~\\ref{f4} we see $\\tau_{M0}$ compared to computer simulations.\nThis complicated formula has a simple behavior in the limit of\nvery large systems and for $n=0$. In the limit of large $M$ and \nfor $N$ being fixed, the time $\\tau_{Mn}$ grows exponentially with $M$,\n\\begin{equation}\n\\tau_{M0} \\sim (N-1)^M = e^{\\mu M},\t\\label{tm0ap1}\n\\end{equation}\nwith $\\mu=\\log(k_1\/k)=\\log(N-1)$, while for fixed density $\\rho=M\/N$ \nand $N\\to\\infty$ it increases faster than exponentially,\n\\begin{equation}\n\\tau_{M0} \\sim e^{\\rho N \\log N}. \\label{tm0ap2}\n\\end{equation}\nThe approximate formulas (\\ref{tm0ap1}) and (\\ref{tm0ap2}) \ncan be alternatively obtained using a kind of Arrhenius law \\cite{arh,god},\nwhich states that the average life time is inversely proportional to the \nminimal value of the occupation number distribution:\n\\begin{equation}\n\\tau_{m0} \\sim 1\/\\pi_1({\\rm min}),\n\\end{equation}\nwhere one thinks about the condensate's melting as of tunneling \nthrough the potential barrier in a potential $V(m)=-\\log \\pi_1(m)$. In our case the potential $V(m)$ grows monotonically with $m$ going to zero, so the ball rather bounces from the wall at $m=0$ than tunnels through it, but the reasoning is the same.\nFrom Eq.~(\\ref{pi1star}) we have $\\pi_1({\\rm min})\\sim (N-1)^{-M}$ for \nfixed $N$ and large $M$ and we thus get again Eq.~(\\ref{tm0ap1}), while \nfor fixed density $\\pi_1({\\rm min})$ falls over-exponentially \nwhich results in Eq.~(\\ref{tm0ap2}).\n\\begin{figure}\n\\psfrag{x}{$M$}\n\\psfrag{y}{$\\tau_{M0}$}\n\\includegraphics[width=8cm]{f4.eps}\n\\caption{The average lifetime $\\tau_{M0}$ for the star graph with \n$N=10$ calculated from Eq.~(\\ref{TMnstar}) (solid line) and found in \ncomputer simulations (points).}\n\\label{f4}\n\\end{figure}\n\nSo far we have discussed the singular node. It is quite surprising that\nthe formula (\\ref{Tmngen}) works also well for regular nodes.\n If we blindly substitute $\\pi_1(k)$ by $\\pi_i(k)$ we get the following expression for $\\tau_{i,mn}$:\n\\begin{eqnarray}\n& &\\tau_{i,mn} = \\nonumber \\\\\n& &\\frac{(N-1)^n-(N-1)^m+(m-n)(N-2)(N-1)^M}{(N-2)^2(N-1)^{M-1}}. \\nonumber \\\\\n\\end{eqnarray}\nFor $m$ fixed, the transition time decreases \nalmost linearly with $n$. The typical occupation of the regular node\nis much smaller than $M$, so we concentrate on $m\\ll M,n=0$ and $N\\gg 1$.\nThe approximate formula reads:\n\\begin{equation}\n\\tau_{i,m0} \\approx m \\frac{N-1}{N-2}\t\\label{tm0ext},\n\\end{equation}\nand grows very slowly in comparison to the life time of the condensate \nat the singular node. This linear growth can be easily understood as \nthe minimal time needed for $m$ particles to hop out from a regular node.\nOne must remember that in practice we cannot observe transitions for large $m$, because the \nprobability of having such states is extremely small as it stems from \nEq.~(\\ref{pistext}) for $\\pi_i(m)$.\n\n\\subsection{Single inhomogeneity}\nNext we consider the single inhomogeneity graph from Sec.~II~C. \nIn the condensed phase the occupation $m$ fluctuates quickly \naround $\\left$ and even if it is smaller than $M$ we can\nassume that $\\tau_{mn}\\approx \\tau_{Mn}$ because the transition time $\\tau_{Mm}$\nfrom $M$ to $m$ balls is very small in comparison to $\\tau_{Mn}$ .\nTherefore we shall concentrate again on $\\tau_{Mn}$. \nFrom Eqs.~(\\ref{pi1inh}) and (\\ref{Tmngen}) we have\n\\begin{equation}\n\\tau_{Mn} = \\sum_{p=n+1}^M \\sum_{l=p}^M \\alpha^{p-l} \n\\frac{\\binom{M+N-l-2}{M-l}}{\\binom{M+N-p-2}{M-p}}.\n\\end{equation}\nChanging variables we get\n\\begin{eqnarray}\n\\tau_{Mn} = \\sum_{p=n+1}^M \\frac{(M-p)!}{(M+N-p-2)!} \\times \\nonumber \\\\\n\\times \\sum_{q=0}^{M-p}\\alpha^{-q} \\frac{(M+N-p-q-2)!}{(M-p-q)!}.\n\\end{eqnarray}\nThe sum over $q$ can be approximated by an\nintegral which can then be estimated by the saddle-point method.\nThe saddle point is $q_*=\\alpha (N-2)\/(1-\\alpha)$ as in \nSec.~II~C and therefore all calculations are almost identical. In \nthis way we obtain\n\\begin{equation}\n\\sum_q \\dots \\approx \\alpha^p \\times \\alpha^{\\alpha\\frac{N-2}{1-\\alpha}-M}\n\\frac{\\left(\\frac{N-2}{1-\\alpha}\\right)!}{\\left(\\frac{\\alpha(N-2)}{1-\\alpha}\\right)!}\n\\sqrt{\\frac{2\\pi \\alpha (N-2)}{(1-\\alpha)^2}},\n\\label{factor}\n\\end{equation}\nwhere we used again the notation \n$x!\\equiv \\Gamma(1+x)$.\nThe only dependence on $p$ in this expression is through the factor $\\alpha^p$.\nThus to calculate $\\tau_{Mn}$ it suffices to evaluate the sum\n\\begin{equation}\n \\sum_{p=n+1}^M \\alpha^p \\frac{(M-p)!}{(M+N-p-2)!}. \\label{eq:sumpn}\n\\end{equation}\nBecause every term in the sum is proportional to $1\/\\pi_1(p)$ from Eq.~(\\ref{pi1inh}), in the condensed phase the function under the sum has a minimum at the saddle point $p_*=m_*$. As this minimum is very deep, the effective contribution to the sum can be split into two terms: for small $p\\ll m_*$ and for $p\\gg m_*$. The ``small-$p$'' part can be evaluated like in Eq.~(\\ref{eq:sumr0}) by pushing the upper limit to infinity and approximating the ratio of factorials by some number to power $p$. To calculate the ``large-$p$'' part it is sufficient to take the last two terms in Eq.~(\\ref{eq:sumpn}), namely for $p=M$ and $p=M-1$, because they decrease quickly. The complete formula for $\\tau_{Mn}$ is finally given by\n\\begin{eqnarray}\n\t\\tau_{Mn} \\approx \\alpha^{\\alpha\\frac{N-2}{1-\\alpha}-M} \\frac{\\left(\\frac{N-2}{1-\\alpha}\\right)!}{\\left(\\frac{\\alpha(N-2)}{1-\\alpha}\\right)!}\n\t\\sqrt{\\frac{2\\pi \\alpha (N-2)}{(1-\\alpha)^2}} \\times \\nonumber \\\\ \n\t\\times \\left[ \\frac{M!}{(M+N-2)!} \\left(\\alpha\\frac{M+N-2}{M}\\right)^{n+1} \\times \\right. \\nonumber \\\\\n\t\\left. \\times \\left( 1-\\alpha\\frac{M+N-2}{M}\\right)^{-1} + \\frac{\\alpha^{M-1}(\\alpha(N-1)+1)}{(N-1)!} \\right]. \\nonumber \\\\\n\t\\label{compl}\n\\end{eqnarray}\nIn Fig.~\\ref{f5} we compare this theoretical formula with $\\tau_{M0}$ from numerical simulations. \nEquation (\\ref{compl})\nsimplifies in the limit of large systems. When \none allows $M\\to\\infty$ while keeping $N$ and $\\alpha$ fixed,\nthen the life time grows exponentially,\n\\begin{equation}\n\\tau_{M0} \\sim \\left(\\frac{1}{\\alpha}\\right)^M = e^{\\mu M}. \\label{tm0inh}\n\\end{equation}\nFor $\\mu=\\log(N-1)$, that is for a star graph, it reduces to the formula (\\ref{tm0ap1}).\nIn the limit of fixed density $\\rho=M\/N>\\rho_c$ \nand for $N,M\\to\\infty$:\n\\begin{equation}\n\\tau_{M0} \\sim e^{N\\left[ -\\log(1-\\alpha)+\n\\rho\\log(\\rho\/\\alpha)-(1+\\rho)\\log(1+\\rho)\\right]}. \\label{last}\n\\end{equation}\nWe see that the life time grows exponentially only if \\mbox{$k_1>k$}, that is for positive external field $\\mu=\\log(k_1\/k)$.\nAs before, we can explore the limit when the single inhomogeneity graph reduces to a star graph. \nInserting $\\mu=-\\log\\alpha=\\log(N-1)$ into Eq.~(\\ref{last}) we recover Eq.~(\\ref{tm0ap2}) as the leading term\nfor large $N$.\n\\begin{figure}\n\\psfrag{x}{$M$}\n\\psfrag{y}{$\\tau_{M0}$}\n\\psfrag{y2}{\\hspace{-5mm}$\\log \\tau_{M0}$}\n\n\\includegraphics[width=8cm]{f5.eps}\n\\caption{Comparison between the ``experimental'' results (points) and the\ntheoretical prediction (\\ref{compl}) (solid line) for $\\tau_{M0}$ of a $4$-regular graph\nwith single inhomogeneity $k_1=16$. The graph size is $N=20$. The inset compares $\\tau_{M0}$ calculated from Eq.~(\\ref{compl})\n(solid line) with that from Eq.~(\\ref{tm0inh}) (dashed line), valid in the large-$M$ limit. The prefactor\nin Eq.~(\\ref{tm0inh}) is chosen to match the $M\\to\\infty$ limit of the two formulas.}\n\\label{f5}\n\\end{figure}\n\n\\section{Conclusions}\nIn this paper we have studied static and dynamical properties of the condensation\nin zero-range processes on inhomogeneous networks. We have focused on the case where the network\nis almost a $k$-regular graph except that it has a single node of degree $k_1$\nlarger than $k$. This type of network could be a rude prototype \nof inhomogeneities encountered in scale-free networks having\na single hub with very high degree and many nodes of much smaller degrees.\nIndeed, from the point of view of the hub, \nthe remaining nodes look as if they formed an almost homogeneous system.\nWe have shown that the distribution of balls $\\pi_1(m)$ at the singular node has a maximum at $m\\approx N(\\rho-\\rho_c)$ where $\\rho_c$ is the critical density above which the condensate is formed. The average occupation of regular nodes is equal to $\\rho_c$ and the condensate never appears on them. However, the condensate is not a static phenomenon. It fluctuates and it sometimes\nmelts and disappears from the singular node. Then the surplus of balls distributes uniformly on all other nodes. After a while the condensate reappears and its typical life time $\\tau$ grows exponentially like $e^{\\mu M}$, where $M$ is the number of balls and $\\mu=\\log (k_1\/k)$ plays the role of an external field explicitly breaking the permutational symmetry of the system. This behavior is qualitatively distinct from that observed in homogeneous systems with a power-law distribution of balls, where $\\tau$ grows only like a power of $M$, and the symmetry is spontaneously broken. Thus the transition $\\mu=0\\to \\mu> 0$ changes dramatically all properties of the system.\n\nIn all above calculations we assumed for simplicity that the hop rate $u(m)=1$ and thus for the homogeneous system there would be no condensation. However, in the case $u(m)=1+b\/m$ which produces the power-law in $\\pi(m)$ for regular graphs, in an inhomogeneous system, apart from the condensation on the singular node, we would expect a second condensate on some regular node if the number of particles would be large enough to exceed the critical density for the homogeneous sub-system. Thus we could expect the presence of two critical densities $\\rho_{1c}$ and $\\rho_{2c}$ and two condensates having completely distinct properties. We leave this interesting question for future research.\n\n\\section{Acknowledgments}\nWe acknowledge support by the EC-RTN Network ``ENRAGE'' under \ngrant No. MRTN-CT-2004-005616, an Institute Partnership grant \nof the Alexander von Humboldt Foundation,\na Marie Curie Host Development Fellowship under \nGrant No. IHP-HPMD-CT-2001-00108 (L.B. and W.J.), \na Marie Curie Actions Transfer of Knowledge project ``COCOS'',\nGrant No. MTKD-CT-2004-517186 and a Polish Ministry of Science\nand Information Society Technologies Grant 1P03B-04029 (2005-2008)\n(Z.B.). B.W. thanks the German Academic Exchange Service (DAAD) for a fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nBy comparing the black hole physics with the thermodynamics \nand discovering of the black hole evaporation by Hawking, \nit was shown that the black hole entropy is proportional to the\nhorizon area\\cite{Bekenstein1,Hawking}.\n\\be\nS_{BH} = \\frac{A_H}{4}\n\\ee\nin unit $\\hbar = c = G = 1$.\nIn Euclidean path integration approach it was shown the tree level \ncontribution of the gravitation action gives the black hole \nentropy\\cite{Gibbons}.\nHowever the exact statistical origin of the Bekenstein-Hawking \nblack hole entropy is unclear.\n\nRecently many efforts have been concentrated to understand \nthe statistical origin of black hole thermodynamics, specially the \nblack hole entropy by various methods (for review \nsee \\cite{Bekenstein2}): \n't Hooft was calculated the entropy of a quantum field propagating\nthe outside of the black hole. After the regularization he obtained\n $S = 1\/4 A_H$ (the brick wall method)\n \\cite{tHooft,Pa1,uglum,barbon}.\nAnother approach is to identify the black hole entropy \nwith the entanglement entropy $S_{ent}$.\nEntanglement entropy arises from ignoring the degree of freedom\nof a proper region of space: $S = - Tr \\rho \\ln \\rho$.\nIt is found that the entropy is proportional to the area of the \nboundary\\cite{Ent}.\nIn fact the entanglement entropy and the brick wall method are \nequivalent.\nFrolov and Novikov argued that the black hole entropy can be \nobtained by identifying the dynamical degrees of freedom \nwith the states of all fields which are located inside the \nblack hole\\cite{Frolov}. \nThe leading term of the entropy obtained by those methods \nis proportional to the surface area of the horizon.\nHowever the proportional coefficient diverges as the cut off goes to \nzero.\nThe conical approach also gives similar result with \nothers\\cite{conical}. \nThe divergence is because of an \ninfinite number of states near the horizon,\nwhich can be explained by the equivalence principle \\cite{Barbon}.\nAn alternative approach by Frolov is to identify the black hole \nentropy with the thermodynamic one. In this approach the entropy\nis finite\\cite{Frolov2}.\nHowever they all treat the only spherical symmetrical black hole.\n\nIf the black hole has a rotation, what is changed ?\nIt is well known that in a rotating black hole spacetime a particle\nwith a zero angular momentum dropped from infinity is dragged \njust by the influence of gravity so that it acquires an angular \nvelocity in the same direction in which the black hole rotates. \nThe dragging becomes more and more extreme the nearer one approaches\nthe horizon of the black hole.\nThis effect is called the dragging of inertial frames\\cite{Misner}. \n\nThus the field at equilibrium with the rotating black hole\nmust also be rotating. The rotation is not rigid but\nlocally is different.\nSo the velocity of the radiation does not exceed the velocity of light. \nHowever we do not know how to treat the equilibrium state with \na locally different angular velocity. More precisely there \nis no global static coordinates. So we assume that \nthe radiation has a rigid rotation $\\Omega_0$ small than or equal to\nthe extremum value of the local rotation. In a rotating black hole\nthe extremum value of it is $\\Omega_H$, which is the \nangular velocity of the event horizon.\n\nRecently we considered the black hole entropy by the \nbrick wall method in the charged Kerr black hole in\\cite{minho} \nand showed the entropy is proportional to the event horizon \nin Hartle-Hawking states.\nIn this paper to more deep understand the black hole entropy we shall\ninvestigate the black hole entropy by the brick wall method\nin various stationary black holes: \nthe Kaluza-Klein black hole \\cite{kaluza} which is the solution of \nthe 4-dimensional effective theory reduced from the 5-dimensional \nKaluza-Klein theory, and the Sen black hole\\cite{sen} which \nis the solution of the Einstein-Maxwell-dilaton-antisymmetric tensor\ngauge field theory came from the heteroitic string theory, \nand the Kerr-Newman black hole\\cite{kerr} which is the solution of \nthe Einstein-Maxwell theory.\n\nIn order to understand the equilibrium state of the radiation \n(the field) in the rotating black hole spacetime in Sec.2 we will \nfirst consider the rotating heat bath in the flat spacetime.\nIn Sec.3 we will consider the radiation in equilibrium\nstate in Rindler spacetime with rotation, which is the most simple\nspacetime having the event horizon and a rotation. \nIn Sec.4 we will investigate the entropy of the quantum field\nin the stationary black hole background.\nWe find the condition to give the finite value to the free energy\nand the entropy. \nIn Sec.5 we calculate the entropy in Hartle-Hawking state for the \nrotating black holes.\nFinal section is devoted to the summary.\n\n\n\\section{A Rotating Heat Bath }\n\nLet us consider a massless scalar field with a constant angular \nvelocity $\\Omega_0$ about $z-$axis at thermal equilibrium \nwith a temperature $T = 1\/\\beta$ in Minkowski \nspacetime, of which line element in cylindrical coordinate is given by\n\\begin{equation}\nds^2 = -dt^2 + r^2 d \\phi^2 + dr^2 + dz^2. \\label{metric1}\n\\end{equation}\nIn this spacetime the positive frequency field mode can be written as\n $\\Phi_{q,m} (x) = f_{q m } (r,z) e^{- i \\omega t + i m \\phi } $,\n where $q$ denotes a quantum number and $m$ is the \n azimuthal quantum number.\n\nFor such a equilibrium ensemble of the states of the scalar \nfield the partition function is given by\n\\begin{equation}\nZ = \\sum_{n_q, m} e^{- n_q ( \\omega_q - m \\Omega_0 ) \\beta } \n\\end{equation}\nand the free energy is given by \n\\begin{equation}\n \\beta F = \\sum_m \\int_0^\\infty d \\omega g(\\omega, m) \n \\ln \\left( 1 - e^{- \\beta (\\omega - m \\Omega_0)} \\right),\n\\end{equation}\nwhere $g(\\omega ,m)$ is the density of state for \na fixed $\\omega$ and $m$.\n\nFollowing 't Hooft we assume that all possible modes of \na scalar field vanish\nat $r = r_1 $ ( $r_1$ is very small.)\nand at $r = L$. In the WKB approximation with \n$\\Phi = e^{i S(r) - i \\omega t + i m \\phi + i k z }$ the radial wave \nnumber $K( x, \\omega,m) = \\partial_r S $ is given by\n\\begin{equation}\nK^2 ( x, \\omega, m) = \\omega^2 - \\frac{m^2}{r^2} - k^2. \\label{con1}\n\\end{equation}\nThis expression denotes the ellipsoid in momentum phase space at a \nfixed frequency $\\omega$.\nThe total number of modes with energy less than $\\omega$ and a fixed $m$\nis obtained by integrating over the volume of phase space, which is \ndetermined by (\\ref{con1})\n\\begin{eqnarray}\n\\nonumber\n\\Gamma (\\omega,m) &=& \n \\sum_m \\int d \\phi dz \\int_{r_1}^L dr \\frac{1}{\\pi} \n \\int dk K(x,\\omega,m) \\\\\n &=& \\frac{1}{\\pi} \\sum_m \\int d \\phi dz \\int_{r_1}^L dr \n \\int dk \\left( \\omega^2 - \\frac{m^2}{r^2} - k^2 \\right)^{1\/2}.\n \\label{Pvol}\n\\end{eqnarray}\nThe integration over $k$ must be carried out over the phase space that\nsatisfies $ K^2 \\ge 0$. \n$\\Gamma ( \\omega, m)$ can be obtained by investigating\nthe shape of the expression (\\ref{con1}) in momentum phase space.\nThus the free energy, after the integration by parts, becomes\n\\begin{eqnarray}\n\\nonumber\n\\beta F &=& - \\beta \\sum_m \\int_0^\\infty d \\omega \\Gamma ( \\omega,m)\n \\frac{1}{e^{\\beta ( \\omega - m \\Omega_0 )} - 1 } \\\\\n &=& - \\frac{\\beta}{2} \\int_0^\\infty d \\omega \n \\int_{- r \\omega}^{r \\omega} d m\n ( \\omega^2 - \\frac{m^2}{r^2} ) \n \\frac{1}{e^{\\beta ( \\omega - m \\Omega_0 )} - 1 }, \n\\end{eqnarray}\nwhere we assume that the azimuthal quantum number $m$ is a continuous \nparameter. \nBy making the change of variable $ m = r \\omega u $ we obtain \nthe free energy \n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3} \\int d \\phi dz \\int_{r_1}^L \n\\frac{r}{( 1 - v^2)^2} dr, \\label{free1}\n\\end{equation}\nwhere $N$ is a constant and $v = r \\Omega_0$.\nNote that as $L$ goes to $1\/\\Omega_0$ this partition function diverges\nas $\\gamma^4$, where $\\gamma = ( 1 - v^2 )^{-1\/2}$.\n\nFrom the expression (\\ref{free1}) it is easy to obtain expressions \nfor the energy $E$, angular momentum $J$, and entropy $S$ of radiation\n\\begin{eqnarray}\nJ & = & \\langle m \\rangle_{av} = \\frac{1}{\\beta} \n \\frac{ \\partial}{\\partial \\Omega_0} (\\beta F) \n = 4 N \\frac{1}{\\beta^4} \\Omega_0\n \\int r^2 \\gamma^6 r dr d \\phi dz, \\\\\nE & = & \\langle \\omega \\rangle_{av} = \\Omega_0 \\cdot J - \n \\frac{\\partial}{\\partial \\beta } ( \\beta F) = N \\frac{1}{\\beta^4} \n \\int ( 3 + v^2 ) \\gamma^4 r dr d\\phi dz, \\\\\nS & = & \\beta^2 \\frac{ \\partial}{\\partial \\beta } F = \n4 N \\frac{1}{\\beta^3} \\int \\gamma^4 r dr d \\phi dz. \\label{entropy1}\n\\end{eqnarray}\nThese coincides with those in ref.\\cite{zurek}.\nSimilarly to the free energy $F$ these expressions $J, E,$ and $S$ \n diverge as $ L \\rightarrow 1\/\\Omega_0$.\nThe divergence is related to the rigid rotation.\nIn rigid rotating system the velocity of the comoving observer \ngrows as one move from the origin to infinity.\nSo beyond some point the velocity exceeds the velocity of the light. \nThis is unphysical. Thus a rotating system cannot have the size \ngreater than $1\/\\Omega_0$.\nTherefore to obtain a finite value for $J, E,$ and $S$, we must take\n$L < 1\/\\Omega_0$. In such a finite system $\\omega > m \\Omega_0$.\n \n\n\nNow let us consider above problem in co-moving coordinate that are\nrotating with angular velocity $\\Omega_0$. The line element in comoving\nframe is given by\n\\begin{equation}\nds^2 = - (1 - \\Omega_0^2 r^2 )dt^2 + 2 \\Omega_0 r d \\phi' dt \n+ dr^2 + dz^2, \\label{metric2}\n\\end{equation}\nwhere we have used $\\phi' = \\phi - \\Omega_0 t$.\nIn this coordinate the positive frequency field mode is written as\n$\\Phi_{q m} (x) = {\\bar f}_{qm} (r,z) e^{ - i \\omega' t + i m \\phi' }$.\n\nBecause in comoving frame the field has no rotation the free energy is \ngiven by\n\\begin{equation}\n\\beta F = \\int_0^\\infty d \\omega' g' ( \\omega') \\ln \n \\left( 1 - e^{- \\beta \\omega'} \\right), \\label{free3}\n\\end{equation}\nwhere $g' (\\omega')$ is the density of state for a fixed $\\omega'$. \nIn WKB approximation the Klein-Gordon equation $\\Box \\Phi = 0$ yields\nthe constraint\\cite{Mann} \n\\begin{equation}\ng^{ab} k_a k_b = 0 \\label{con2}\n\\end{equation}\nor\n\\begin{equation}\n- ( \\omega' - \\Omega_0 m)^2 + ( \\frac{1}{r^2} m^2 + k^2 + p^2 ) = 0,\n\\label{con3}\n\\end{equation}\nwhere $p = \\frac{\\partial S}{\\partial r}$.\nIn region where $\\Omega_0 r < 1$, for a fixed $\\omega'$, this expression\nrepresents the ellipsoid in momentum space. Therefore the total number \nof modes with energy less than $\\omega'$ is given by\n\\begin{eqnarray}\n\\Gamma' ( \\omega' ) &=& \\frac{1}{\\pi} \\sum_m d \\phi dz \\int dr\n\\int dk \\left(\n( \\omega' - m \\Omega_0)^2 - \\frac{m^2}{r^2 } - k^2 \\right)^{1\/2} \n\\label{star} \\\\\n&=& \\frac{4}{3} \\int d \\phi dz \\int_{r_1}^L dr \n \\frac{r}{(1 - \\Omega_0^2 r^2 )^2} ~\\omega^{'3}, \\label{pvol}\n\\end{eqnarray}\nwhich is the volume of the ellipsoid.\nThe expression (\\ref{star}) is just the same form as Eq.(\\ref{Pvol})\nwhen $\\omega \\rightarrow \\omega - m \\Omega_0$.\nThe phase volume (\\ref{pvol}) diverges as $L \\rightarrow 1\/\\Omega_0$.\nInserting the expression (\\ref{pvol}) into (\\ref{free3}) and integrating\nwe get\n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3} \\int d \\phi dz \\int_{r_1}^L dr\n \\frac{r}{( 1- \\Omega_0^2 r^2 )^2}. \\label{free4}\n\\end{equation}\nThis expression is the same with Eq(\\ref{free1}).\nFrom this we get the energy $E'$ and the entropy $S$:\n\\begin{eqnarray}\nE' &=& \\langle \\omega' \\rangle_{av} = \n - \\frac{\\partial}{\\partial \\beta } (\\beta F ) = 3 \n \\frac{N}{\\beta^4} A \\int_{r_1}^L dr \n \\frac{r}{(1 - \\Omega_0^2 r^2 )^2}, \\\\\nS &=& \\beta^2 \\frac{\\partial}{\\partial \\beta }( \\beta F) = \n 4 \\frac{N}{\\beta^3} A \\int_{r_1}^L dr \n \\frac{r}{(1 - \\Omega_0^2 r^2 )^2},\n\\end{eqnarray}\nwhere $A = \\int d \\phi dz$.\nIt is noted that the entropy $S$ is the same with Eq.(\\ref{entropy1}) \nand the energy $E'$ is satisfied with $E' = E - \\Omega_0 J$.\nThis fact show that the coordinate transformation to comoving frame\nonly change the energy and not change the entropy in WKB approximation.\nThus in the case of calculating the entropy or the free energy \nit is convenient to choose the comoving frame.\nIt is noted that in co-moving frame the divergence is related to the \ntime component $g_{tt}$ of the metric (\\ref{metric2}).\n\n\n\\section{A Thermal Bath in Rindler Spacetime with a Rotation}\n\nIn this section we will consider the thermal equilibrium \nstate of the scalar field with the mass $\\mu$ and an uniform \nrotation about $z-$axis in the Rindler spacetime. \nThe line element of the Rindler spacetime in cylindrical coordinates \nis given by \n\\begin{equation}\nds^2 = - \\xi^2 d \\eta^2 + d \\xi^2 \n + r^2 d \\phi^2 + d r^2. \\label{metric3} \n\\end{equation}\nIn this spacetime the event horizon is at $\\xi = 0$, and \n$\\xi = constant$ represent the trajectory of \nthe uniform acceleration\\cite{birrell}.\nThe importance of the Rindler space-time is that in the large\nblack hole mass limit the metric of the black space-time reduces\nto that of the Rindler space-time\\cite{uglum}.\n\nAs in Sec.2, the WKB approximation with \n$\\Phi (x) = e^{- i \\omega t + i m \\phi + i S(\\xi, r)}$\nyields\n\\begin{equation}\nK^2 ( \\xi, r,\\omega, m) = \n\\frac{\\omega^2}{\\xi^2} - \\frac{1}{r^2} m^2 - p_r^2 - \\mu^2,\n \\label{con4}\n\\end{equation}\nwhere $ K = \\partial_\\xi S$ and $p_r = \\partial_r S$.\nIn this section we will calculate the free energy by \nusing the slightly different method with that in section 2.\n\nIt is important to note that in WKB approximation the density \nof state $g(\\omega,m)$ is determined by the constraint (\\ref{con4}),\nand that the free energy is singular at $ \\omega = m \\Omega_0$. \nIn particular if $\\omega - m \\Omega_0 < 0 $ \nthe free energy becomes an imaginary number.\nHowever in the WKB approximation we can easily see \n$\\bar{\\omega} = \\omega - m \\Omega_0 > 0$ in the region such that\n$ \\xi -\\Omega_0 r > 0$. But in the region such that $ \\xi - \\Omega_0\nr < 0$ it is possible that $\\omega - m \\Omega_0 < 0$.\n( More details are in Sec.4.)\nTherefore to obtain the finite value for the free energy \nwe must require the system to be in the region such \nthat $ \\xi - \\Omega_0 r >0$. Then the free energy is written as\n\\begin{eqnarray}\n\\nonumber\n\\beta F &= & \\sum_m \\int_{m \\Omega_0}^\\infty d \\omega g( \\omega, m) \n\t \\ln \\left( 1 - e^{- \\beta (\\omega - m \\Omega_0)} \\right) \\\\\n \\nonumber\n\t &=& \\int_0^\\infty d \\omega \\sum_m g(\\omega + m \\Omega_0,m) \\ln \n\t \\left( 1 - e^{- \\beta \\omega} \\right) \\\\\n\t &=& - \\beta \\int_0^\\infty d \\omega \\frac{1}{e^{\\beta \\omega } \n\t -1 }\n\t \\int d m \\Gamma ( \\omega + m \\Omega_0,m),\n\\end{eqnarray}\nwhere we have integrated by parts and assumed that the quantum number\n$m$ is a continuous variable. \nThe total number of modes with energy less than $\\omega$ \nis obtained by integrating over the volume of phase space\n\\begin{eqnarray} \n\\nonumber\n\\Gamma(\\bar{\\omega}) &=& \\int d m \\Gamma (\\omega + m \\Omega_0,m) \\\\\n\\nonumber\n &=& \\int dm\n \\int d \\phi dr \\int_{r_1}^L d \\xi \\frac{1}{\\pi}\n \\int d p_r K(\\xi, r, \\omega + m \\Omega_0 ,m) \\\\\n &=& \\frac{1}{\\pi} \\int dm \\int d \\phi dr \\int_{r_1}^L d \\xi\n \\int d p_r \\left(\n \\frac{\\omega^2}{\\xi^2} + \\frac{2}{ \\xi^2}\n m \\Omega_0 \\omega + \n \\frac{m^2 \\Omega_0^2}{\\xi^2}\n - \\frac{1}{r^2} m^2 - p_r^2 - \\mu^2\n \\right)^{1\/2}. \\label{pvol2}\n\\end{eqnarray}\nThe integrations over $m$ and $p_r$ must be carried out over the \nphase space that satisfies $ K^2 ( \\omega + m \\Omega_0,m) \\ge 0$.\nAfter the integration we obtain \nthe number of states with energy less than $\\omega$, \nwhich is given by \n\\begin{equation}\n\\Gamma (\\omega) = \\frac{4}{3} \\int d^3 x \n\t\\frac{\\xi r}{\\sqrt{( \\xi^2 - \\Omega_0^2 r^2 )}}\n\t\\left( \\frac{\\omega^2}{ \\xi^2 - \\Omega_0^2 r^2 } \n\t - \\mu^2 \\right)^{3\/2}.\n\\end{equation}\nThus the free energy becomes\n\\begin{equation}\n\\beta F = - \\frac{4}{3} \\beta \\int d^3 x \n\\int_{\\mu \\sqrt{ \\xi^2 - \\Omega_0^2 r^2 } }^\\infty \nd \\omega \\frac{1}{e^{\\beta \\omega } - 1 } \n\t\\frac{\\xi r}{\\sqrt{( \\xi^2 - \\Omega_0^2 r^2 )}}\n\t\\left( \\frac{\\omega^2 }{ \\xi^2 - \\Omega_0^2 r^2 } \n\t - \\mu^2 \\right)^{3\/2}.\n\\end{equation}\nFor a massless scalar field ( $\\mu =0$ ) the free energy becomes\n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3} \\int d \\phi d r \\int_{\\xi_1}^L d \\xi\n\t\\frac{\\xi r}{( \\xi^2 - \\Omega^2 r^2 )^2}. \\label{Free}\n\\end{equation}\nFrom this we get the energy $E$, the angular momentum $J$, \nand the entropy $S$ of the field\n\\bea\nJ &=& \\bra m \\ket_{av} = 4 \\frac{N}{\\beta^4} \\Omega_0\n\\int \\frac{r^2}{( \\xi^2 - \\Omega_0^2 r^2 )^3} \\xi r d \\xi dr dz, \\\\\nE &=& \\bra E \\ket_{av} = \\frac{N}{\\beta^4} \n\\int \\frac{3 \\xi^2 + \\Omega_0^2 r^2 }{ ( \n \\xi^2 - \\Omega_0^2 r^2 )^3 } \\xi r d \\xi dr dz, \\\\\nS &=& 4 \\frac{N}{\\beta^3} \\int \\frac{1}{( \n \\xi^2 - \\Omega_0^2 r^2 )^2} \\xi r d \\xi dr dz. \\label{ent2}\n\\eea\nIt is noted that the thermodynamic quantities $F, E$, and $S$ \nare divergent\nas $\\xi \\rightarrow \\Omega_0 r$ rather than the event horizon.\nOnly in $\\Omega_0 = 0$ case the divergence occurs at the horizon\n$\\xi = 0$. Such a fact can be easily understand in the co-moving \nframe, of which line element is given by \n\\begin{eqnarray}\nds^2 &=& - \\xi^2 d \\eta^2 + r^2 ( d \\phi' + \\Omega_0 d \\eta )^2 + \n d \\xi^2 + d r^2 \\\\\n\\nonumber\n &=& - ( \\xi^2 - \\Omega_0^2 r^2 ) d \\eta^2 + 2 \\Omega_0 r^2 d \\eta \n d \\phi' + r^2 d {\\phi'}^2 + d \\xi^2 + d r^2, \n\\end{eqnarray}\nwhere we used $\\phi' = \\phi - \\Omega_0 \\eta $.\nIn this spacetime the event horizon is at $\\xi = 0$. In addition to the \nevent horizon there is a stationary limit surface \nat $\\xi = \\Omega_0 r$,\nwhere the Killing vector $\\partial_\\eta$ becomes null. That \nsurface is the elliptic hyper-surface\\cite{letaw}. In the \ninterval $ 0 < \\xi < \\Omega_0 r$, the Killing vector is spacelike.\nWe can also show that the entropy in the co-moving frame\nis the same form with Eq.(\\ref{ent2}).\n{\\it These facts imply that the divergence of the thermodynamic \nquantities is deeply related to the stationary\nlimit surface in the co-moving frame rather than the event horizon.}\n\n\n\n\\section{A Entropy of a Scalar Field in a Rotating Black Hole }\n\n\\subsection{General Formalism}\n\nLet us consider a scalar field with mass $\\mu$\nin thermal equilibrium at temperature $1\/\\beta$ in the \n rotating black hole background, \n of which line element is generally given by\n\\be\nds^2 = g_{tt}(r, \\theta) d t^2 \n + 2 g_{t \\phi}( r,\\theta) dt d \\phi \n + g_{\\phi \\phi}(r, \\theta) d\\phi^2 \n + g_{rr} (r,\\theta) d r^2 \n + g_{\\theta \\theta}(r, \\theta) d\\theta^2. \n \\label{Metric} \\\\\n\\ee \nThis metric has two Killing vector fields: the timelike \nKilling vector $\\xi^\\mu = (\\partial_t)^\\mu$ and the axial\nKilling vector $\\psi^\\mu =(\\partial_\\phi)^\\mu$.\nThe metrics, we concern, of the Kaluza-Klein, the Sen, and \nthe Kerr-Newman black holes are in the appendix.\nThe properties of those metrics are \n\\be\ng_{tt} g_{\\phi \\phi} - g^2_{t \\phi} = - \\Delta(r) \\sin^2 \\theta\n\\rightarrow 0 \n\\ee\nand\n\\be\n\\left( g_{tt} g_{\\phi \\phi} - g^2_{t \\phi}\\right) g_{rr} \n\\rightarrow finite\n\\ee\nas one approaches the horizon. \nAnother property is that\nthere are two important surfaces (the event horizon and the\nstationary limit surface), and the two surfaces does not coincide.\nOn the stationary limit surface the Killing vector $\\xi^\\mu$ vanishes,\nand the Killing vector $\\xi^\\mu + \\Omega_H \\psi^\\mu$ is null on\nthe horizon, where $\\Omega_H$ is the angular velocity of the\nhorizon.\n\nThe equation of motion of the field with mass $\\mu$ and \narbitrary coupled to the scalar curvature $R(x)$ is\n\\be\n\\left[ \\Del_\\mu \\Del^\\mu - \\xi R - \\mu^2 \\right] \n\\Psi = 0, \\label{equation}\n\\ee\nwhere $\\xi$ is an arbitrary constant.\n$\\xi = 1\/6$ and $\\mu =0$ case corresponds to the conformally\ncoupled one.\nWe assume that the scalar field is rotating with a constant \nazimuthal angular velocity $\\Omega_0$. \nThe associated conserved quantities are\nangular momentum $J$. \nThe free energy of the system is then given by \n\\be\n F = \\frac{1}{\\beta} \\sum_m \\int_0^\\infty d \\E g(\\E, m) \\ln \n \\left( 1 - e^{- \\beta( \\E - m \\Omega_0 )} \\right), \n\\ee\nwhere $g(\\E,m)$ is the density of state for a given $\\E$ and $m$. \n\n\nTo evaluate the free energy we will follow \nthe brick wall method of 't Hooft \\cite{tHooft}.\nFollowing the brick wall method we impose a small radial cut-off $h$\nsuch that \n\\begin{equation}\n\\Psi (x) = 0 ~~~~{\\rm for }~~~ r \\leq r_H + h,\n\\end{equation}\nwhere $r_H$ denotes the coordinate of the event horizon.\nTo remove the infra-red divergence we also introduce another \ncut-off $ L \\gg r_H$ such that \n\\be\n\\Psi (x) = 0~~~~ {\\rm for} ~~~r \\geq L.\n\\ee\nIt is noted that the brick wall is spherically symmetric.\nIn the WKB approximation with $\\Psi = \ne^{- i \\E t + i m \\phi + i S(r, \\theta)}$\nthe equation (\\ref{equation}) yields \nthe constraint \\cite{Mann}\n\\begin{equation}\n p_r^2 = \\frac{1}{g^{rr}} \\left[\n - g^{tt} \\E^2 + \n 2 g^{t \\phi} \\E m - g^{ \\phi \\phi } m^2 \n - g^{ \\theta \\theta } p_\\theta^2 \n - V(x) \\right], \\label{Con1}\n\\end{equation}\nwhere $ p_r = \\partial_r S$, $ p_\\theta = \\partial_\\theta S$, and\n$V(x) = \\xi R(x) + \\mu^2$.\nIn WKB approximation it is important to note that\nthe number of state for a given $\\E$ is determined by \n$p_\\theta, p_r$ and $m$.\nThe number of mode with energy less than $\\E$ and with a fixed\n$m$ is obtained by integrating over $p_\\theta$ in phase space. \n\\bea\n\\nonumber\n\\Gamma (\\E,m ) &=& \\frac{1}{\\pi} \\int d \\phi d \\theta \\int dr\n\\int d p_\\theta p_r ( \\E, m,x) \\\\\n&=& \\frac{1}{\\pi} \\int d \\phi d \\theta \\int dr\n\\int d p_\\theta \n \\left[ \\frac{1}{g^{rr}} \\left(\n - g^{tt} \\E^2 + \n 2 g^{t \\phi} \\E m - g^{ \\phi \\phi } m^2 \n - g^{ \\theta \\theta } p_\\theta^2 \n - V(x) \\right) \n \\right]^{\\frac{1}{2}}. \n\\eea\nThe integration over $p_\\theta$ must be carried over the phase space\nsuch that $p_r \\geq 0$.\n\nIn this point we need some remarks.\nIn a rotating system, in general, there is a superradiance effect,\nwhich occurs when $ 0 < \\E < m \\Omega_0$.\nFor this range of the frequency the free energy $F$ becomes a complex\nnumber. In case $\\E = m \\Omega_0$ the free energy is divergent.\nTherefore to obtain a real finite value for the free energy $F$,\nwe must require that $\\E > m \\Omega_0$. ( For $ 0 < \\E < m \\Omega_0$\nthe free energy diverges. See below.) This requirement say that\nwe must restrict the system to be in the region \nsuch that $g_{tt}^{'} \\equiv g_{tt} + 2 \\Omega_0 g_{t \\phi} \n+ \\Omega_0^2 g_{\\phi \\phi} < 0$. \nIn this region $ \\E - m \\Omega_0 >$,\nso the free energy is a finite real value.\nIt is easily showed as follows.\nLet us define $ E = \\E - m \\Omega_0$.\nThen it is written as \n\\bea\n\\nonumber\nE &=& \\left(\n \\frac{g^{t \\phi}}{ g^{tt} } - \\Omega_0 \n \\right) m\n + \\frac{1}{- g^{tt}} \n \\left[\n \\left( \n g^{t \\phi} m \n \\right)^2 \n + \\left( - g^{tt} \\right)\n \\left( V + g^{\\phi \\phi} m^2 + \n g^{rr} p_r^2 + g^{\\theta \\theta }p_\\theta^2 \n \\right)\n \\right]^{1\/2} \\\\\n &=& \\left( \n \\Omega - \\Omega_0 \n \\right) m + \\frac{ -\\D }{g_{\\phi \\phi}}\n \\left[\n \\frac{1}{-\\D} m^2 + \\frac{ g_{\\phi \\phi} }{- \\D }\n \\left( V + \\frac{p_r^2 }{g_{rr}}\n + \\frac{p_\\theta^2}{g_{\\theta \\theta} } \n \\right)\n \\right]^{1\/2}, \\label{Con2}\n\\eea\nwhere we used \n\\be\ng^{tt} = \\frac{g_{\\phi \\phi}}{ \\D},~~\ng^{t \\phi} = \\frac{ - g_{t \\phi}}{\\D},~~\ng^{\\phi \\phi} = \\frac{ g_{tt}}{\\D},\n\\ee\nand $ \\Omega = -\\frac{ g_{t \\phi}}{g_{\\phi \\phi}} $.\nHere $ - \\D = g_{t \\phi}^2 - g_{tt}g_{\\phi \\phi}$.\nFrom Eq.(\\ref{Con2}), for all $m, p_r$ and $p_\\theta$, \none can see the condition such that $ E >0$ is\n\\be\n \\frac{ \\sqrt{-\\D } }{ g_{\\phi \\phi}} \\pm \n\\left( \\Omega - \\Omega_0 \\right) > 0\n\\ee\nor\n\\be\ng_{tt}^{'} \\equiv g_{tt} + 2 \\Omega_0 g_{t \\phi} \n+ \\Omega_0^2 g_{\\phi \\phi} < 0.\n\\ee\n Therefore in the region such that $ - g_{tt}^{'} >0$ \n ( called region I) the free energy is a real, but\n in the region such that $- g_{tt}^{'} < 0$ (called region II)\n the free energy is complex.\nHowever in the region I the integration over the momentum \nphase space is convergent. \nBut in the region II the integration over the momentum \nphase is divergent.\n These facts become more apparent if we investigate the momentum \n phase space.\n In the region I \nthe possible points of $p_i$ satisfying $ \\E - \\Omega_0 p_\\phi = E$ \nfor a given $E$ are located on the following surface\n\\begin{equation}\n\\frac{p_r^2}{g_{rr}} + \\frac{p_\\theta^2}{g_{\\theta \\theta}} +\n \\frac{- {g'}_{tt}}{- \\cal D} \\left(\n p_\\phi + \\frac{g_{t \\phi } \n + \\Omega_0 g_{\\phi \\phi}}{{g'}_{tt}} \n E \\right)^2 \n = \\left( \\frac{ E^2}{- {g'}_{tt} } \n - V \\right), \\label{ellipsoid}\n\\end{equation}\nwhich is the ellipsoid, {\\it a compact surface}. \nHere $p_\\phi = m$.\nSo the density of state $g(E)$ for a given $E$ is finite and \nthe integrations \nover $p_i$ give a finite value.\nBut in the region II \nthe possible points of $p_i$ are located on the following surface\n\\begin{equation}\n\\frac{p_r^2}{g_{rr}} + \\frac{p_\\theta^2}{g_{\\theta \\theta}} -\n \\frac{ {g'}_{tt}}{- \\cal D} \\left( \n p_\\phi + \\frac{g_{t \\phi } + \n \\Omega_0 g_{\\phi \\phi}}{{g'}_{tt}} \n\t E \\right)^2 \n = - \\left( \n \\frac{ E^2 }{ {g'}_{tt}} \n + V \\right),\n\\end{equation}\nwhich is the hyperboloid, {\\it a non-compact surface}. So \n$g(E)$ diverges and the integration over $p_i$ diverges. \nIn case of $g^{'}_{tt} = 0$, the possible points are \ngiven by the surface\n\\begin{equation}\n\\frac{p_r^2}{g_{rr}} + \\frac{p_\\theta^2}{g_{\\theta \\theta}} =\n\\frac{p_\\phi - \\left( \\frac{ g_{\\phi \\phi} E^2 }{ \\cal D } + V \n \\right)\/ \\left( \\frac{ 2 g_{t \\phi} }{ \\cal D } E \n\t \\right) \n }{ \\frac{- {\\cal D} }{ 2 g_{t \\phi} E } \n },\n\\end{equation}\nwhich is elliptic paraboloid and also $non-compact$. Therefore \nthe value of the $p_i$ integration are divergent.\nActually the surface such that ${g'}_{tt} = 0$ is the velocity of the\nlight surface (VLS). Beyond VLS (in region II) the co-moving \nobserver must move more rapidly than the\nvelocity of light. \nThus we will assume that the system is in the region I.\n( For the possible region I see Sec. 4.2.)\nFor example, in the case of $\\Omega_0 = 0$ the points \nsatisfying ${g'}_{tt} =0$ are on the stationary limit \nsurface.\nThe region of the outside (inside) of the stationary \nlimit surface corresponds to the region I (II).\nIn the rotating system in Sec. 2 the region I is $ r < 1\/\\Omega_0$\nand $ r > 1\/\\Omega_0$ corresponds to the region II.\nIn the Rindler spacetime with a rotation\n$ \\xi > \\Omega_0 r$ corresponds to the region I, and $\\xi < \\Omega_0 r$\nto the region II.\n\n\n\n\n\nWith the assumption that the system is in the region I \nwe can obtain the free energy as follows\n\\begin{eqnarray}\n\\nonumber\n \\beta F &=& \\sum_m \\int_{m \\Omega_0}^\\infty d \\E g(\\E, m) \\ln \n \\left( 1 - e^{- \\beta( \\E - m \\Omega_0 )} \\right) \\\\\n\\nonumber\n &=& \\int_0^\\infty d\\E \\sum_m g(\\E + m \\Omega_0 , m) \\ln \n \\left( 1 - e^{- \\beta \\E } \\right) \\\\\n &=& - \\beta \\int_0^\\infty d\\E \\frac{1}{e^{\\beta \\E} - 1}\n \\int d m \\Gamma (\\E + m \\Omega_0, m),\n\\end{eqnarray}\nwhere we have integrated by parts and we assume that the quantum \nnumber $m$ is a continuous variable. \nThe integrations over $m$ and $p_\\theta$ yield \n\\begin{equation}\nF = - \\frac{4 }{3} \n\\int d \\phi d \\theta \\int_{r_H + h}^L dr\n\\int_{V(x) \\sqrt{- {g'}_{tt}}}^\\infty d\\E \n\\frac{1}{e^{\\beta \\E} - 1 }\n\\frac{ \\sqrt{g_4}}{\\sqrt{ - {g'}_{tt} }} \\left( \n\\frac{\\E^2}{ - {g'}_{tt} } - V(x) \\right)^{3\/2 }. \n\\label{freeenergy}\n\\end{equation} \nIn particular when $\\Omega_0 = 0$ and non-rotating case \n$g_{t \\phi} = 0$, \nthe free energy (\\ref{freeenergy}) \ncoincides with the expression in ref.\\cite{tHooft,barbon} and \nit is proportional to the volume of the optical space in the limit\n$V(x) = 0$ \\cite{optical}.\nIt is easy to see that the integrand diverges as \n$ r_H + h $ or $L$ approach the surface such that $g_{tt}^{'} = 0$. \nIn that case the contribution of the $V(x)$ can be negligible. \n\nFor a massless and minimally coupled scalar field case\n($\\mu = \\xi = 0$) the free energy reduces to \n\\begin{equation}\n\\beta F = - \\frac{N}{\\beta^3 } \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 } \n = - N \\int_0^\\beta d \\tau \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\sqrt{g_4} \n \\frac{1}{ \\beta_{local}^4}, \\label{freeenergy2}\n\\end{equation}\nwhere $\\beta_{local} = \\sqrt{ - g^{'}_{tt } } \\beta$ is \nthe reciprocal of the local Tolman temperature \\cite{Tolman}\nin the comoving frame.\nThis form is just the free energy of a gas of \nmassless particles at local\ntemperature $1\/\\beta_{local}$.\n\n\nFrom this expression (\\ref{freeenergy2}) \nit is easy to obtain expressions for \nthe total energy $U$, angular momentum $J$, and entropy $S$ of \na scalar field\n\\begin{eqnarray}\nJ &=& \\langle m \\rangle =\n - \\frac{1}{\\beta} \\frac{\\partial}{\\partial \\Omega_0}\n( \\beta F) = \\frac{4 N}{\\beta^4} \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 }\n\\frac{ g_{\\phi \\phi}}{( - {g'}_{tt})} \n\\left( \\Omega_0 - \\Omega \\right), \\\\\nU &=& \\langle \\E \\rangle = \\Omega_0 J + \\frac{\\partial}{\\partial \\beta} \n( \\beta F) = \\frac{N}{\\beta^4} \\int d \\theta d \\phi \n\\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 }\n\\left[ \n3 + 4 \\frac{ \\Omega_0 \\left( \\Omega_0 - \\Omega \\right)\n g_{\\phi \\phi } }{( - {g'}_{tt})} \\right], \\\\\nS &= & \\beta^2 \\frac{\\partial}{\\partial \\beta } F = \n \\beta ( U - F - \\Omega_0 J) = \n 4 \\frac{N}{\\beta^3 } \\int d \\theta d \\phi \n \\int_{r_H + h}^L dr \\frac{\\sqrt{g_4}}{ ( - g^{'}_{tt } )^2 },\n\\end{eqnarray}\nwhich are also divergent as one approach the surface such \nthat $ g_{tt}^{'} =0$.\n\n\n\\subsection{The region such that $ - g_{tt}^{'} > 0$.}\n\nIn this section we study where is the possible region I\nfor three black hole, the Kaluza-Klein,\nand the Sen, the Kerr-Newman black holes,\nfor $ \\Omega_0 = \\Omega_H, \\Omega_0\n< \\Omega_H$ and the extreme case with $\\Omega_0 = \\Omega_H$. \n\n\\subsubsection{The Kaluza-Klein black hole }\nA) $\\Omega_0 =\\Omega_H$ {\\it case}:\nIn $\\Omega_0 = \\Omega_H$ case the position of the light \nof velocity surface is exactly found. \nIn such a case $g^{'}_{tt}$ can be written as\n\\bea\ng^{'}_{tt} &=& g_{tt} + 2 \\Omega_H g_{t\\phi} + \n\t \\Omega_H^2 g_{\\phi \\phi} \\\\\n\\nonumber\n&=& \\frac{\\mu^2}{B \\Sigma} ( x - \\bar{r}_H ) \n \\left\\{\n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 }\n ( 1 - v^2) x^3 + \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } \n \\left( 2 - \\bar{r}_- (1 - v^2) \\right) x^2 \n \\right. \\\\\n\\nonumber\n & &~~ +\n \\left[ -1 + \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } \n\t \\left(\n 4 + y^2 ( 1- v^2) \\cos^2 \\theta - 2 \\bar{r}_- \n\t \\right) \n \\right] x \\\\\n\\nonumber\n& & ~~\\left.\n + \\left[\n \\bar{r}_- + \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } \n\t \\left(\n -4 \\bar{r}_H - \\bar{r}_- y^2 ( 1- v^2) \\cos^2 \\theta \n \\right) \n \\right] \n \\right\\} \\\\\n &\\equiv& \n \\frac{\\mu^2}{B \\Sigma} ( x - \\bar{r}_H ) \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 } ( 1- v^2)\n \\left( \n x^3 + a_1 x^2 + a_2 x + a_3 \n \\right)\n\\eea\nfor $\\theta \\neq 0$, where $x = \\frac{r}{\\mu },\ny = \\frac{a}{\\mu}, \\bar{r}_H = \\frac{r_H}{\\mu}$, and \n$ \\bar{r}_- = \\frac{ r_-}{\\mu}$.\nFrom this we can see that there are two VLS.\nOne is the horizon ($r = r_H$), and another \nlight of velocity surface (call outer VLS) is given by\\cite{Table}\n\\be\nr_{VLS} = 2 \\mu \\sqrt{-Q} \\cos \\left( \n \\frac{1}{3} \\Theta \\right)\n - \\frac{1}{3} a_1 \\mu,\n\\ee \nwhere\n\\be\n\\Theta = {\\rm arccos} \\left( \n\t \\frac{ P}{ \\sqrt{ - Q^2}} \\right)\n\\ee\nwith \n\\be\nQ = \\frac{ 3 a_2 - a_1^2}{9},~~~\nP = \\frac{9 a_1 a_2 - 27 a_3 - 2 a_1^3 }{54}.\n\\ee\nIn case of the slowly rotating black hole ($a$ is small)\nthe VLS is approximately given by\n\\be\nr_{VLS} \\sim 2 \\mu \\frac{r_H}{a\\sqrt{1 - v^2}\\sin\\theta} - \n\\frac{1}{3}\n\\left( \\frac{2 }{1 - v^2} - \\frac{r_-}{\\mu} \\right) \\mu,\n\\ee\nwhich is an open, roughly, cylindrical surface.\nAs $v \\rightarrow 1$ or $a \\rightarrow 0$ the VLS become more \ndistant, which came from the fact that as $v \\rightarrow 1$ or\n$ a \\rightarrow 0$ the coordinate angular velocity\n$\\frac{d \\phi}{d t} = - \\frac{g_{t \\phi}}{g_{\\phi \\phi}}$ becomes\nvanish.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure=fig1.eps,height=8cm, angle=0}}\n\\vspace{0.1cm}\n{\\footnotesize Figure 1: The position of the outer velocity of \nlight surface for the Kaluza-Klein black hole. }\n\\end{figure}\nFor $\\theta = 0$ it is always that $g_{tt}^{'} <0$ for $r >r_H$. \nAs $ a \\rightarrow \\mu$ the outer VLS approaches horizon. See\nFig.1.\n\n\nB) $ \\Omega_0 < \\Omega_H $ {\\it case}:\nIn this case $g^{'}_{tt} = 0 $ is a fourth order polynomial equation \nin $r$ for a given $\\theta$.\nThe region I corresponds to $ r_{in} < r < r_{VLS}$. \nAt $\\theta = \\pi\/2$\n$r_{in}$ is between the stationary limit surface and \nthe event horizon, \nand at $\\theta = 0$ $r_{in}$ contacts with the event horizon.\n Actually the inner VLS $r_{in}$ places between the stationary\n limit surface and the event horizon for all $\\theta$.\nThe particular point is that as $ \\Omega_0 \\rightarrow \\Omega_H$, \n$r_{in}$ approaches the horizon.\nHowever it does attach the horizon only when $\\Omega_0 = \\Omega_H$.\nWhile, the outer velocity of light surface locates at the very far \ndistance from the horizon, and it is a roughly cylindrical \nsurface as in case $\\Omega_0 = \\Omega_H$. \nFor the position of the inner VLS see Fig.2.\n\n\nC){ {\\it the extreme black hole case with }} $\\Omega_0 = \\Omega_H $: \nThe extreme black hole for the Kaluza-Klein black hole occurs when \n$ \\mu^2 = a^2$. In this case the inner horizon and outer horizon are \nat the same place.\nAt $\\theta = 1\/2 \\pi$, $g_{tt}^{'}$ is written as\n\\be\ng^{'}_{tt} = \\frac{\\mu^2}{B \\Sigma} \n( x - \\bar{r}_H )^2 x \\left( \n x + \\frac{2}{1 - v^2} \\right)\n\\frac{ 1 - v^2 }{4},\n\\ee\nwhich shows that the possible region such that $g^{'}_{tt} < 0 $ \ndoes not exist at $\\theta = 1\/2 \\pi$.\nTherefore in the extreme black hole case it is impossible to consider\nthe brick wall model of 't Hooft.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure=fig2.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 2: The position of $ r_{in}$ at $ \\theta = 0.5 \\pi$ for the \nKaluza-Klein black hole. $v = 0.5$.\n\\end{figure}\n\n\n\\subsubsection{The Sen black hole}\n\nA) $\\Omega_0 = \\Omega_H$ {\\it case}:\nIn $\\Omega_0 = \\Omega_H$ case \n$g^{'}_{tt}$ can be written as\n\\bea\ng^{'}_{tt} &=& g_{tt} + 2 \\Omega_H g_{t\\phi} + \n\t \\Omega_H^2 g_{\\phi \\phi} \\\\\n\\nonumber\n&=& \\frac{\\mu^2}{ \\Sigma} ( x - \\bar{r}_H ) \n \\left\\{\n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma}\n x^3 + \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n \\left( 2 \\cosh 2 \\gamma - \\bar{r}_- \\right) x^2 \n \\right. \\\\\n \\nonumber\n & &~~ +\n \\left[ -1 + \\frac{ y^2 \\sin^2 \\theta }{ \\bar{r}_H^2 } \n + \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n\t \\left(\n y^2 \\cos^2 \\theta - 2 \\bar{r}_- \\cosh 2 \\gamma\n\t \\right) \n \\right] x \\\\\n\\nonumber\n& & ~~\\left.\n + \\left[\n \\bar{r}_- + \\frac{ y^2 \\sin^2 \\theta }{ \\bar{r}_H } \n - \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n\t \\left(\n \\bar{r}_- y^2 \\cos^2 \\theta \n \\right) \n \\right] \n \\right\\} \\\\\n &\\equiv& \n \\frac{\\mu^2}{ \\Sigma} ( x - \\bar{r}_H ) \n \\frac{ y^2 \\sin^2 \\theta }{4 \\bar{r}_H^2 \\cosh^4 \\gamma} \n \\left( \n x^3 + a_1 x^2 + a_2 x + a_3 \n \\right)\n\\eea\nfor $\\theta \\neq 0$, where $x = \\frac{r}{\\mu },\ny = \\frac{a}{\\mu}, \\bar{r}_H = \\frac{r_H}{\\mu}$, and \n$ \\bar{r}_- = \\frac{ r_-}{\\mu}$.\nThen the exact position of the inner VLS and outer VLS are\nare given by\n\\be\nr_{in} = r_H,~~~ \nr_{VLS} = 2 \\mu \\sqrt{-Q} \\cos \\left( \n \\frac{1}{3} \\Theta \\right)\n - \\frac{1}{3} a_1 \\mu.\n\\ee \nThe position of the outer VLS for small $a$ is approximately given by\n\\be\nr_{VLS} \\sim \\frac{2 \\mu r_H \\cosh^2 \\gamma}{a\n \\sin \\theta }\n - \\frac{ 1}{3} \\left( 2 \\cosh(2\\gamma) - \\frac{r_-}{\\mu} \\right) \\mu,\n \\ee\nwhich is an open, roughly, cylindrical surface.\nAs $ a \\rightarrow 0 $ the VLS goes to the infinity, and it \ndisappears when $a = 0$.\nAs $ \\gamma$ or $a $ is increasing the VLS approaches the horizon.\nAt $ \\theta = \\frac{1}{2} \\pi$, similarly to the \nKaluza-Klein black hole, $g^{'}_{tt} < 0$ for $ r > r_H$.\nSee Fig.3.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig3.eps , height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 3: The position of the outer velocity of light surface for the \nSen black hole. $\\gamma = 5.0$.\n\\end{figure}\n\n\nB) $ \\Omega_0 < \\Omega_H $ {\\it case}:\nIn this case $g^{'}_{tt} = 0 $ is also a fourth order equation \nin $r$ for a given $\\theta$.\nSimilarly to the Kaluza-Klein black hole the region I is \n$ r_{in} < r < r_{VLS}$. \nAt $\\theta = 0 $ the inner VLS $r_{in}$ is at the horizon,\nand at $\\theta = \\pi\/2$\n$r_{in}$ locates at the between the stationary limit surface and \nthe event horizon. See Figure 4. \nAs $ \\Omega_0 \\rightarrow \\Omega_H$, $r_{in} $ approaches \nto the horizon. Only when $\\Omega_0 = \\Omega_H$ it coincides with the\nevent horizon.\nThe outer velocity of light surface, in case of small $a$, locates\nat the very far distance from the horizon, \nand it is a roughly cylindrical \nsurface.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure=fig4.eps, height= 8cm, angle=0}}\n\\vspace{0.1cm}\nFigure 4: The position of the inner velocity of light surface for \nthe Sen black hole. $\\gamma = 5.0, \\theta = 0.5 \\pi$. \n\\end{figure}\n\n\nC){ {\\it the extreme black hole case with }} $\\Omega_0 = \\Omega_H $: \nThe extreme black hole for the Kaluza-Klein black hole occurs when \n$ \\mu^2 = a^2$. \nIn this case the inner horizon \nand outer horizon are at the same place.\nAt $\\theta = \\frac{1}{2} \\pi $ $g_{tt}^{'}$ is written as\n\\be\ng^{'}_{tt} = \\frac{\\mu^2}{ \\Sigma} \n( x - \\bar{r}_H )^2 x \\left( \n x + 2 \\cosh 2 \\gamma \\right)\n\\ee\nwhich shows that the possible region such that $g^{'}_{tt} < 0 $ \ndoes not exist at $\\theta = 1\/2 \\pi$.\nTherefore in the extreme black hole case it is impossible to consider\nthe brick wall model of 't Hooft.\n\n \n\n\\subsubsection{The Kerr-Newman black hole}\n\n\nA) $\\Omega_0 = \\Omega_H$ {\\it case}:\nIn $\\Omega_0 = \\Omega_H$ case we can exactly find the \nposition of the light of velocity surface. \nIn such a case ${g'}_{tt}$ can be written as\n\\begin{eqnarray}\n{g'}_{tt} &=& g_{tt} + 2 \\Omega_H g_{t \\phi} + \\Omega_H^2 \n\t g_{\\phi \\phi} \\\\\n\\nonumber\n\t &=& \\frac{M^2}{\\Sigma} ( x - \\br_H) \\left\\{\n\t \\bar{\\Omega}_H^2 \\sin^2 \\theta ~x^3 + \\bar{r}_H\n\t \\bar{\\Omega}_H^2 \\sin^2 \\theta ~x^2 \\right. \\\\\n\\nonumber\n\t & & ~ + \\left[ -1 + \\bar{\\Omega}_H^2 \\sin^2 \\theta \\left(\n\t y^2 + y^2 \\cos^2 \\theta + \\bar{r}_H^2 \\right) \n\t \\right] x \\\\\n &+ & \\left. \\left[ \n 2 \\left( 1 - \\bar{\\Omega}_H y \\sin^2 \\theta \\right)^2 - \\br_H + \n \\br_H \\bar{\\Omega}_H^2 \\sin^2 \\theta \\left( \\br_H^2 + \n y^2 + y^2 \\cos^2 \\theta \\right)\n \\right] \\right\\} \\\\\n\t &\\equiv& \\frac{M^2}{\\Sigma} ( x - \\br_H ) \n\t \\bar{\\Omega}_H^2 \\sin^2 \\theta \\left(\n\t x^3 + a_1 x^2 + a_2 x + a_3 \\right)\n\\end{eqnarray}\nfor $\\theta \\neq 0$, where\n$x = r\/M, y = a\/M, z = e\/M, \\bar{\\Omega}_H = M \\Omega_H, \\br_H = r_H \/M $.\nThen the exact position of the outer light of velocity surface \nis given by \n\\be\n r_{VLS} = 2 M \\sqrt{\n - Q} \\cos \\left( \\frac{1}{3} \\Theta \\right) - \\frac{1}{3} a_1 M.\n \\label{sol}\n \\ee\nFor small $a$ Eq. (\\ref{sol}) is approximately given by \n \\be\n r_{VLS} \\sim \\frac{1}{\\Omega_H \\sin \\theta } - \\frac{r_H}{3 },\n \\ee\n which is an open, roughly, cylindrical surface.\n For $\\theta = 0$ it is always that ${g'}_{tt} < 0 $ for \n $r > r_H$. \n As $ a \\rightarrow 0$, $r_{VLS}$ goes to infinity, and\n as $ a \\rightarrow \\sqrt{M^2 + e^2 }$ it approaches \n the event horizon. See Fig.5.\n The inner VLS $r_{in}$ is the event horizon.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig5.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 5: The position of the outer light of velocity surface\nfor the Kerr-Newman black hole. $e = 0.0$. \n\\end{figure}\n\nB) $\\Omega_0 < \\Omega_H$ {\\it case}:\nIn this case, similarly to other black holes, \nthe inner VLS $r_{in}$ approaches to the horizon as \n$\\Omega_0 \\rightarrow \\Omega_H$. See Fig.6. \nThe inner VLS is a compact surface, which shrink to horizon as\n$\\Omega_0 \\rightarrow \\Omega_J$. See Fig.7.\nThe outer VLS is at far place, which disappears when $\\Omega_0 = 0$.\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig6.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 6: The position of the inner light of surface for the \nKerr-Newman black hole. $\\theta = 0.5 \\pi$. \n\\end{figure}\n\\begin{figure}[bh]\n\\vspace{0.5cm}\n\\centerline{\n\\epsfig{figure= fig7.eps, height=8 cm, angle=0}}\n\\vspace{0.1cm}\nFigure 6: The shape of the inner light of surface for the \nKerr-Newman black hole. $a = 0.8 M, e = 0 $. \n\\end{figure}\n\nC) {\\it the extreme black hole case with} $\\Omega_0 =\\Omega_H$: \n For extreme Kerr-Newman black hole case, which occurs when\n $M^2 = a^2 + e^2 $, \n $g_{tt}^{'}$ at $ \\theta = \\frac{1}{2} \\pi$ \n is written as \n \\be\ng_{tt}^{'}=\\frac{M^2}{\\Sigma} \\frac{y}{1 + y^2}\n( x - 1)^2 \n\\left(x + 1 - \\frac{1}{y} \\right)\n\\left(x + 1 + \\frac{1}{y} \\right)\n\\ee\nFrom this we obtain \n the position of VLS at $\\theta = \\frac{\\pi}{2} $ as\n\\bea\n r &=& M ~~~~~~{\\rm for ~} \\frac{1}{2} M~\\leq a \\leq M~ {\\rm and }~ a = 0, \\\\\n r &=& \\left( -1 + \\frac{M}{a} \\right) M ~~~~~{\\rm for ~} \n 0 < a < \\frac{1}{2} M. \n\\eea\nThe second case corresponds to the extreme black hole that\nis slowly rotating and has many charge. (In this case \n $ e > \\sqrt{3}\/2M \\approx 0.866 M $). \nIn particular in case of $e \\leq \\sqrt{3}\/2 M $ ( $a = M$ for $ e = 0$) \nthe horizon and the light of velocity surface \nare at the same position. \n Therefore in case of the extreme black hole with $a \\geq 1\/2 M $\n it is impossible to consider the brick wall model of 't Hooft. \n\n\n\n\n\\section{The Entropy in the Hartle-Hawking Vacuum}\n\nThe Hartle-Hawking vacuum state is one that\nthe angular velocity $\\Omega_0$ is equal to that of the \nevent horizon, and the temperature $\\beta$ is equal to the Hawking\ntemperature, where the Hawking temperature and the angular velocity of \nthe horizon are defined as\\cite{Wald}\n\\be\nT_H = \\frac{\\kappa}{2 \\pi},~~~ \\Omega_H = \\lim_{r \\rightarrow r_H}\n\\left( - \\frac{g_{t \\phi}}{g_{\\phi \\phi}} \\right).\n\\ee\nHere $\\kappa$ is the surface gravity of the horizon.\n\nFirst of all let us assume that $\\Omega_0 = \\Omega_H $.\nIn this case, as stated in Sec.4, \nthe possible region I is $r_H < r < L < r_{VLS}$.\nThe outer brick wall must locate inside the outer VLS.\nThis fact was already pointed out by Frolov and Thorne \\cite{Thorne} \nto remove the singular structure of the Hartle-Hawking vacuum and\nmodify it.\nNow recall that in general ${g'}_{tt}|_{r = r_H} =0$.\nThis came from that ${g'}_{tt}$ is the same form as \n$\\chi^\\mu \\chi_\\mu = (\\xi^\\mu + \\Omega_H \\psi^\\mu)(\n\\xi_\\mu + \\Omega_H \\psi_\\mu)$, and $\\chi^\\mu$ is null on the\nhorizon.\nSo it follows that ${g'}_{tt} = ( r - r_H) G(r, \\theta)$, where\n$G(r,\\theta)$ is a non-vanishing function at $r = r_H$ except \nthe extremal case.\n( We can not consider the extreme black hole case.) \n\nTherefore for the three black holes the leading behaviors of the free \nenergy $F$ for very small $h$ are then given by \n\\bea\n\\beta F &\\approx& - \\frac{N}{\\beta^3}\n\\int d \\theta d \\phi \\int_{r_H + h}^L dr \\frac{\n\\sqrt{g_4}}{ ( - g_{tt}^{'} )^2 } \\\\\n&=& - \\frac{N}{\\beta^3}\n\\int d \\theta d \\phi \\int_{r_H + h}^L dr \\frac{\nD(r)}{( r - r_H)^2 G^2(r, \\theta) },\n\\eea\nwhere $D(r,\\theta) = \\sqrt{g_4}$. Since $D(r,\\theta) $ and \n$G(r,\\theta)$ are non-vanishing functions at $r = r_H$ we can expand it\nabout $r= r_H$ as follows.\n\\bea\nD(r,\\theta) &=& D(r_H,\\theta) + D^{'} (r_H, \\theta) (r -r_H) +\nO((r- r_H )^2 ),\\\\\n\\frac{1}{ G^2(r, \\theta)} &=&\n\\frac{1}{ G^2(r_H, \\theta)} + \n\\left( \\frac{1}{G^2(r_H,\\theta)} \\right)^{'} \n+ O((r - r_H)^2 ),\n\\eea\nwhere $'$ denotes the partial derivative for $r$.\nSo the free energy is approximately given by\n\\bea\n\\nonumber\n\\beta F &\\approx & - \\frac{N}{\\beta^3}\n\\int d \\phi d \\theta \\int dr \n\\left\\{ \\frac{D(r_H, \\theta)}{G^2(r_H,\\theta)} \n\\frac{ 1}{(r - r_H)^2 } +\n\\left( \\frac{ D(r_H,\\theta) }{G^2(r_H,\\theta) } \\right)^{'} \n\\frac{1}{(r - r_H)}\n + O((r - r_H)^0) \n\\right\\} \\\\\n&=& \n- \\frac{2 \\pi N}{\\beta^3} \\left\\{\n\\frac{1}{h} \\int d \\theta \n \\frac{D(r_H, \\theta) }{G^2(r_H,\\theta) } \n- \\ln (h) \\int d \\theta \n\\left( \\frac{ D(r_H,\\theta) }{G^2(r_H,\\theta) } \\right)^{'} \n+ ... \\right\\},\n\\label{gen}\n\\eea\nwhich show that \ngenerally, in addition to the linear divergence term in $h$, \nthere is a logarithmic one in the case of rotating black hole.\n\nIf we written the free energy in terms of the proper distance \ncut-off $\\epsilon$, it become very simple form.\n \\bea\n \\nonumber\n\\beta F &\\approx& - \n\\frac{N}{ \\beta^3} \\int_{r = r_H} d \\phi d \\theta \n\\sqrt{g_{\\theta \\theta} g_{\\phi \\phi}} \n \\int_{r_H + h}^L dr \\sqrt{g_{rr}}\n\\left( \\frac{g_{\\phi \\phi}}{g^2_{t \\phi} -g_{tt}g_{\\phi \\phi}}\n\\right)^{3\/2} \\\\\n&\\approx& - \\frac{N}{ 2 ( \\kappa \\beta)^3 } \n \\frac{A_H}{\\epsilon^2},\n \\label{free2}\n\\eea\nwhere $A_H$ is the area of the event horizon,\n and $\\epsilon$ is the \nproper distance from the horizon to $r_H + h$.\n\\be\n\\epsilon = \\int_{r_H}^{r_H + h} dr \\sqrt{g_{rr}}.\n\\ee\nHowever the proper distance cut-off is dependent on the \ncoordinate $\\theta$, which is the general property of the \nrotating black hole.\n\nFrom the free energy $F$ we obtain the leading behaviors of \nthe entropy $S$ as\n\\bea\n\\nonumber\nS &=& \\beta^2 \\frac{\\partial}{\\partial \\beta} F \\\\\n &\\approx& \n \\frac{N}{\\beta^3} \\left( A ~ \\frac{1}{h} + B \\ln (h) + finite \n \\right),\n\\eea\nwhere $A$ and $B$ are in $c$-number in Eq.(\\ref{gen}),\n or\n\\be \nS \\approx \\frac{4 N}{ 2 ( \\kappa \\beta)^3 } \n\\frac{A_H}{\\epsilon^2}. \n\\label{Entropy} \n\\ee\nThe entropy $S$ is linearly and logarithmically divergent \nas $h \\rightarrow 0$. \nThe divergences arise because the density of state for a given $E$\n diverges as $h$ goes to zero.\n\nNow we take $T$ as the Hartle-Hawking temperature \n$T_H = \\frac{ \\kappa}{2 \\pi}$. \nThen the entropy becomes\n\\be\nS_H \\approx \\frac{N 8 \\pi^3 }{\\kappa^3} \\left(\nA~ \\frac{1}{h} + B \\ln (h) + finite \\right),\n\\ee\nor\n\\be\nS_H \\approx \\frac{N}{4 \\pi^3}\n\\frac{ A_H}{ \\epsilon^2}. \\label{result} \n\\ee\nThe entropy of a scalar field in Hartle-Hawking state \ndiverges quadratically \nin $\\epsilon^{-1}$ as the system approaches the horizon.\nOr it diverges in $h^{-1}$ and $ \\ln (h) $.\nIn case $ a= 0$ our result (\\ref{result}) agrees with the \nresult calculated by 't Hooft \\cite{tHooft} and \nwith one in ref.\\cite{ohta}.\nThese facts imply that the leading behaviors of entropy (\\ref{result})\nis general form.\n\n\n\\section{Summary and Conclusion}\n\nBy using the brick wall method we have calculated the entropies \nof the rotating systems with a rotation $\\Omega_0$ \nat thermal equilibrium with temperature $T$ in the \nrotating black holes. \nIn WKB approximation to get the real finite free energy and entropy \nthe system must be in the region I.\nAs the system approaches the VLS ( $r_{in}$ and $r_{VLS}$)\nthe thermodynamic quantities become divergent. From this fact \n{\\it we conclude that the divergence of the thermodynamic quantities \nincluding the entropy is related to the stationary limit\nsurface in the co-moving frame}. In spherical symmetric black hole\nthe stationary limit surface and the event horizon are coincide.\nOnly when $\\Omega_0 =\\Omega_H$ the system can be approach the horizon.\nThe entropy for this case is linearly and logarithmically divergent \nas the ultraviolet cut-off goes to zero.\nTo remove such a divergence, in addition to the renormalization of the \ngravitational constant, we need the renormalization of the \ncurvature square term\\cite{ohta}. But after the renormalization\nthe entropy does not proportional to the area of the event horizon.\nIf we use the proper distance cut-off \nthe entropy is proportional to the horizon area $A_H$. But \nthe cut-off depends on the coordinate $\\theta$.\n\nAnother particular point is that in the extremal black hole case\nwe can not consider the brick wall method of 't Hooft except for\nthe case $ 0 < a < 1\/2 M$ in Kerr-Newman black hole.\n\n\\section*{Appendix} \nFor the three rotating black holes the metrics, the surface \ngravities, , and the proper distances $\\epsilon$ \nare given as follows:\n\n1) the Kaluza-Klein black hole \\cite{Frolov} \n\\begin{eqnarray}\n\\nonumber\nds^2 &=& - \\frac{\\Delta - a^2 \\sin^2 \\theta }{B \\Sigma}dt^2\n -2 a \\sin^2 \\theta \\frac{1}{\\sqrt{1 - v^2 }} \n \\frac{Z}{B} dtd \\phi \\\\\n & & ~ + \\left[\n B \\left( r^2 + a^2 \\right) + a^2 \\sin^2 \\theta \n \\frac{Z}{B} \\right]\n \\sin^2 \\theta d \\phi^2 + \n \\frac{ B \\Sigma}{\\Delta} dr^2 + B \\Sigma d \\theta^2,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\Delta = r^2 - 2 \\mu r + a^2 ,~~~\n\\Sigma = r^2 + a^2 \\cos^2 \\theta,~~~\nZ = \\frac{2 \\mu r}{\\Sigma},~~~\nB = \\left( 1 + \\frac{v^2 Z}{1 - v^2 } \\right)^{\\frac{1}{2}}.\n\\end{equation}\nThe physical mass $M$, the charge $Q$, the angular momentum $J$,\nand the horizon\nare expressed by the parameters $v,\\mu,$ and $a$ as\n\\be\nM = \\mu \\left[\n 1 + \\frac{ v^2 }{2 ( 1 - v^2 )} \\right],~~~\n Q = \\frac{\\mu v}{ 1 - v^2 },~~~\n J = \\frac{\\mu a}{\\sqrt{1 -v^2}},\n r_H = \\mu + \\sqrt{ \\mu^2 - a^2}.\n\\ee\nThe surface gravity and proper distance are\n\\begin{eqnarray}\n\\kappa_{Kaluza-Klein} &=& \n \\frac{ \\sqrt{( 1 - v^2) ( \\mu^2 - a^2)}}{ r_H^2 + a^2},\\\\\n\\epsilon_{Kaluza-Klein} &=& \n 2 \\left( \\frac{B(r_H) \\Sigma (r_H) }{ 2 r_H - 2 \\mu} \\right)^{1\/2}\n \\sqrt{h}. \n\\end{eqnarray}\n\n\n2) the Sen black hole\\cite{sen}:\n\\bea\nds^2 &=& - \\frac{\\Delta - a^2 \\sin^2 \\theta}{\\Sigma}dt^2\n- \\frac{4 \\mu r a \\cosh^2 \\gamma \\sin^2 \\theta}{\\Sigma} dt d \\phi \\\\\n& &~+ \\frac{\\Sigma}{\\Delta} dr^2 + \\Sigma d\\theta^2\n+ \\frac{\\Lambda}{\\Sigma} \\sin^2 \\theta d \\phi^2,\n\\eea\nwhere\n\\bea\n\\Delta &=& r^2 - 2 \\mu r + a^2 ,~~~\n\\Sigma = r^2 + a^2 \\cos^2 \\theta + 2 \\mu r \\sinh^2 \\gamma, \\\\\n\\Lambda &= & \\left( r^2 + a^2 \\right) \\left(\n r^2 + a^2 \\cos^2 \\theta \\right) + 2 \\mu r a^2\n \\sin^2 \\theta \\\\\n & &~ + 4 \\mu r \\left( r^2 + a^2 \\right) \\sinh^2 \\gamma \n + 4 \\mu^2 r^2 \\sinh^4 \\gamma.\n\\eea\nThe mass $M$, the charge $Q$, the angular momentum $J$, and the\nhorizon are given \nby parameters $\\mu,\\beta$, and $a$ as\n\\be\nM = \\frac{\\mu}{2} \\left( 1 + \\cosh 2 \\gamma \\right),~~\nQ = \\frac{\\mu}{\\sqrt{2}} \\sinh 2 \\gamma,~~\nj = \\frac{ a \\mu }{2} \\left( 1 + \\cosh 2 \\gamma \\right),~~\nr_H = \\mu + \\sqrt{\\mu^2 - a^2}.\n\\ee\nThe surface gravity and proper distance are\n\\begin{eqnarray}\n\\kappa_{Sen} &=& \n \\frac{ \\sqrt{ ( 2 M^2 - e^2 )^2 - 4 J^2 }\n\t\t }{ 2 M \\left[\n\t 2 M^2 - e^2 + \\sqrt{ ( 2 M^2 - e^2 )^2 - 4 J^2 }\n\t\t\t \\right] }, \\\\\n\\epsilon_{Sen} &=& \n 2 \\left( \\frac{ r_H^2 + a^2 \\cos^2 \\theta + 2 \\mu r_H \\sinh^2 \n \\gamma }{ 2 r_H - 2 \\mu } \\right)^{1\/2} \\sqrt{h}.\n\\end{eqnarray}\n\n\n3) the charged Kerr black hole \\cite{kerr}\n\\begin{eqnarray}\n\\nonumber\nds^2 &= & - \\left( \n\t\t\\frac{ \\Delta - a^2 \\sin^2 \\theta }{\\Sigma}\n\t\t\\right) \n\t\tdt^2 - \\frac{2 a \\sin^2 \\theta ~( r^2 + a^2 - \\Delta)}{\n\t\t\\Sigma } dt d\\phi \\\\\n & &~ + \\left[ \\frac{(r^2 +a^2 )^2 - \\Delta a^2 \\sin^2 \\theta }{\n \\Sigma} \\right] \\sin^2 \\theta d \\phi^2 + \\frac{\\Sigma}{\\Delta} dr^2 +\n \\Sigma d \\theta^2, \n\\end{eqnarray} \nwhere \n\\begin{equation}\n\\Sigma = r^2 + a^2 \\cos^2 \\theta, ~~~~~ \n\\Delta = r^2 + a^2 + e^2 - 2 M r,\n\\end{equation}\nand $e,a,$ and $M$ are charge, angular momentum per unit mass, and\nmass of the spacetime respectively. \nThe event horizon is \n\\be\n r_H = M + \\sqrt{M^2 - a^2 - e^2 }.\n\\ee\nThe surface gravity and proper distance are\n\\begin{eqnarray}\n\\kappa_{Kerr}&=& \\frac{ \\sqrt{M^2 - a^2 -e^2}}{ 2 M \n\t\\left[ M + \\sqrt{M^2 - a^2 - e^2} \\right] - e^2}, \\\\\n\\epsilon_{Kerr} &= &\n 2 \\left( \\frac{ r_H^2 + a^2 \\cos^2 \\theta }{ 2 r_H - 2 M } \n\\right)^{1\/2} \\sqrt{h}. \n\\end{eqnarray}\n\n\n\\begin{flushleft}\n{\\bf Acknowledgment}\n\\end{flushleft}\n\nThis work is partially supported by Korea Science and Engineering \nFoundation.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\n\nAll graphs in this paper are finite and have neither loops nor parallel edges. We denote by $\\mathbb{N}$ the set of positive integers.\nA class $\\mathcal{C}$ of graphs is said to have the \\emph{Erd\\H{o}s-P\\'osa property} if there exists a function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$, called a \\emph{gap function}, such that\nfor every graph $G$ and a positive integer $k$, $G$ contains either\n\\begin{itemize}\n\\item $k+1$ pairwise vertex-disjoint subgraphs in $\\mathcal{C}$, or\n\\item a vertex set $T$ of $G$ such that $\\abs{T}\\le f(k)$ and $G- T$ has no subgraphs in $\\mathcal{C}$.\n\\end{itemize} \nErd\\H{o}s and P\\'osa~\\cite{ErdosP1965} showed that the class of all cycles has this property with a gap function $\\mathcal{O}(k\\log k)$.\nThis breakthrough result sparked an extensive research on finding min-max dualities of packing and covering for various graph families and combinatorial objects. \nErd\\H{o}s and P\\'osa also showed that the gap function cannot be improved to $o(k\\log k)$ using a probabilistic argument, and Simonovits~\\cite{Simonovits1967} provided a construction \nachieving the lower bound. \nThe result of Erd\\H{o}s and P\\'osa has been strengthened for cycles with additional constraints; for example, long cycles~\\cite{RobertsonS1986, BirmeleBR2007, FioriniH2014, MoussetNSW2016, BruhnJS2017}, directed cycles~\\cite{ReedRST1996, KakimuraK2012}, cycles with modularity constraints~\\cite{Thomassen1988, HuyneJW2017} or cycles intersecting a prescribed vertex set~\\cite{KakimuraKM2011, PontecorviW2012, BruhnJS2017, HuyneJW2017}. \nNot every variant of cycles has the Erd\\H{o}s-P\\'osa property; for example, Reed~\\cite{Reed1999} showed that the class of odd cycles does not satisfy the Erd\\H{o}s-P\\'osa property. \n\n\nWe generally say that a graph class $\\mathcal{C}$ has the \\emph{Erd\\H{o}s-P\\'osa property} under a graph containment relation $\\le_{\\star}$ \nif there exists a gap function $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ such that\nfor every graph $G$ and a positive integer $k$, $G$ contains either\n\\begin{itemize}\n\\item $k+1$ pairwise vertex-disjoint subsets $Z_1,\\ldots , Z_k$ such that each subgraph of $G$ induced by $Z_i$ contains a member of $\\mathcal{C}$ under $\\le_{\\star}$, or\n\\item a vertex set $T$ of $G$ such that $\\abs{T}\\le f(k)$ and $G- T$ contains no member of $\\mathcal{C}$ under $\\le_{\\star}$.\n\\end{itemize} \nHere, $\\le_{\\star}$ can be a graph containment relation such as subgraph, induced subgraph, minor, topological minor, induced minor, or induced subdivision. \nAn edge version and directed version of the Erd\\H{o}s-P\\'osa property can be similarly defined. \nIn this setting, the Erd\\H{o}s-P\\'osa properties of diverse undirected and directed graph families have been studied for graph containment relations such as minors~\\cite{RobertsonS1986}, immersions~\\cite{Liu2015, GianKRT2016}, and (directed) butterfly minors~\\cite{AmiriKKW2016}. It is known that the edge-version of the Erd\\H{o}s-P\\'osa property also holds for cycles~\\cite{diestel2012}. Raymond and Thilikos~\\cite{RaymondT16} provides an up-to-date overview on the Erd\\H{o}s-P\\'osa properties for a range of graph families.\n\nIn this paper, we study the Erd\\H{o}s-P\\'osa property for cycles of length at least $4$ under the induced subgraph relation.\nAn induced cycle of length at least $4$ in a graph $G$ is called a \\emph{hole} or a \\emph{chordless cycle}.\nConsidering the extensive study on the topic, it is somewhat surprising that \nwhether the Erd\\H{o}s-P\\'osa property holds for cycles of length at least $4$ under the induced subgraph relation has been left open till now. \nThis question was explicitly asked by Jansen and Pilipczuk~\\cite{JansenP17} in their study of the \\textsc{Chordal Vertex Deletion} problem,\nand was also asked by Raymond and Thilikos~\\cite{RaymondT16} in their survey. \nWe answer this question positively.\n\n\n\\begin{THM}\\label{thm:main}\nThere exist a constant $c$ and a polynomial-time algorithm \nwhich, given a graph $G$ and a positive integer $k$, finds either $k+1$ vertex-disjoint holes or a vertex set of size at most $ck^2\\log k$ hitting every hole of $G$.\n\\end{THM}\n\n\n\nOne might ask whether Theorem~\\ref{thm:main} can be extended to the class of cycles of length at least $\\ell$ for fixed $\\ell\\ge 5$.\nWe present a complementary result that for every fixed $\\ell\\ge 5$, the class of cycles of length at least $\\ell$ does not satisfy the Erd\\H{o}s-P\\'osa property under the induced subgraph relation.\n\n\\begin{THM}\\label{thm:main2}\nLet $\\ell\\ge 5$ be an integer. \nThen the class of cycles of length at least $\\ell$ does not have the Erd\\H{o}s-P\\'osa property under the induced subgraph relation.\n\\end{THM}\n\nTheorem~\\ref{thm:main} is closely related to the \\textsc{Chordal Vertex Deletion} problem.\nThe \\textsc{Chordal Vertex Deletion} problem asks whether, for a given graph $G$ and a positive integer $k$, there exists a vertex set $S$ of size at most $k$ such that $G-S$ has no holes; \n\tin other words, $G-S$ is a chordal graph.\n\tIn parameterized complexity, whether or not \\textsc{Chordal Vertex Deletion} admits a polynomial kernelization was one of major open problems since it was first mentioned by Marx~\\cite{Marx10}.\n\t A \\emph{polynomial kernelization} of a parameterized problem is a polynomial-time algorithm \n\t that takes an instance $(x,k)$ and outputs an instance $(x', k')$ such that \n\t (1) $(x,k)$ is a \\textsc{Yes}-instance if and only if $(x', k')$ is a \\textsc{Yes}-instance, and\n\t (2) $k'\\le k$, and $\\abs{x'}\\le g(k)$ for some polynomial function $g$.\n\nJansen and Pilipczuk~\\cite{JansenP17}, and independently Agrawal \\emph{et.~al.}~\\cite{AgrawalLMSZ17}, presented polynomial kernelizations for the \\textsc{Chordal Vertex Deletion} problem. In both works, an approximation algorithm for the optimization version of this problem emerges as an important subroutine. Jansen and Pilipczuk~\\cite{JansenP17} obtained an approximation algorithm of factor $\\mathcal{O}({\\sf opt}^2 \\log {\\sf opt} \\log n)$ using iterative decomposition of the input graph and linear programming. Agrawal \\emph{et.~al.}~\\cite{AgrawalLMSZ17} obtained an algorithm of factor $\\mathcal{O}({\\sf opt}\\log^2 n)$ based on divide-and-conquer. \nAs one might expect, the factor of an approximation algorithm for the \\textsc{Chordal Vertex Deletion} is intrinsically linked to the quality of the polynomial kernels \nattained in \\cite{JansenP17} and~\\cite{AgrawalLMSZ17}.\nWe point out that the polynomial-time algorithm of Theorem of~\\ref{thm:main} can be easily converted into an approximation algorithm of factor $O({\\sf opt}\\log {\\sf opt})$. \n\n\n\n\n\\begin{THM}\\label{thm:main3}\nThere is an approximation algorithm of factor $O({\\sf opt}\\log {\\sf opt})$ for {\\sc Chordal Vertex Deletion}. \n\\end{THM}\n\nIt should be noted that an $\\mathcal{O}(\\log^2 n)$-factor approximation algorithm \nwas presented recently by Agrawal \\emph{et.~al.}~\\cite{AgrawalLMSZ17b}, which outperforms \nthe approximation algorithm of Theorem~\\ref{thm:main3}. \n\nOur result has another application on packing and covering for weighted cycles.\nFor a graph $G$ and a non-negative weight function $w:V(G)\\rightarrow \\mathbb{N}\\cup \\{0\\}$, \nlet $\\pack(G, w)$ be the maximum number of cycles (repetition is allowed) such that each vertex $v$ used in at most $w(v)$ times, and\nlet $\\cover(G, w)$ be the minimum value $\\sum_{v\\in X} w(v)$ where $X$ hits all cycles in $G$.\nDing and Zang~\\cite{DingZ2002} characterized \\emph{cycle Mengerian graphs} $G$, which satisfy the property that for all non-negative weight function $w$, $\\pack(G,w)=\\cover(G,w)$.\nUp to our best knowledge, it was not previously known whether $\\cover(G,w)$ can be bounded by a function of $\\pack(G,w)$.\n\nAs a corollary of Theorem~\\ref{thm:main}, we show the following.\n\n\\begin{COR}\nFor a graph $G$ and a non-negative weight function $w:V(G)\\rightarrow \\mathbb{N}\\cup \\{0\\}$, \n$\\cover(G,w)\\le \\mathcal{O}(k^2\\log k)$, where $k=\\pack(G,w)$.\n\\end{COR}\n\n\n\n\nThe paper is organized as follows. Section~\\ref{sec:prelim} provides basic notations and previous results that are relevant to our result. \nIn Section~\\ref{sec:overview}, we explain how to reduce the proof of Theorem~\\ref{thm:main} to a proof under a specific premise, in which \nwe are given a shortest hole $C$ of $G$ \nsuch that $C$ has length more than $c\\cdot k\\log k$ for some constant $c$ and $G-V(C)$ is chordal.\nIn this setting, we introduce further technical notations and terminology. An outline of our proof will be also given in this section.\nWe present some structural properties of a shortest hole $C$ and its neighborhood in Section~\\ref{sec:lemmas}.\nIn Sections~\\ref{sec:hittingsunflower} and \\ref{sec:tulip}, we prove the Erd\\H{o}s-P\\'osa property for different types of holes intersecting $C$ step by step, and we conclude Theorem~\\ref{thm:main} at the end of Section~\\ref{sec:tulip}.\nSection~\\ref{sec:lowerbound} demonstrates that the class of cycles of length at least $\\ell$, for every fixed $\\ell\\ge 5$, does not have the Erd\\H{o}s-P\\'osa property under the induced subgraph relation. \nSection~\\ref{sec:applications} illustrates the implications of Theorem~\\ref{thm:main} to weighted cycles and to the \\textsc{Chordal Vertex Deletion} problem.\n\n\\section{Preliminaries}\\label{sec:prelim}\n\n\nFor a graph $G$, we denote by $V(G)$ and $E(G)$ the vertex set and the edge set of $G$, respectively.\nLet $G$ be a graph. \nFor a vertex set $S$ of $G$, let $G[S]$ denote the subgraph of $G$ induced by $S$, and \nlet $G-S$ denote the subgraph of $G$ obtained by removing all vertices in $S$.\nFor $v\\in V(G)$, we let $G-v:=G-\\{v\\}$.\nIf $uv\\in E(G)$, we say that $u$ is a \\emph{neighbor} of $v$. \nThe set of neighbors of a vertex $v$ is denoted by $N_G(v)$, and the \\emph{degree} of $v$ is defined as the size of $N_G(v)$.\nThe \\emph{open neighborhood} of a vertex set $A\\subseteq V(G)$ in $G$, denoted by $N_G(A)$, is the set of vertices in $V(G)\\setminus A$ having a neighbor in $A$. The set $N_G(A)\\cup A$ is called the \\emph{closed neighborhood} of $A$, and denoted by $N_G[A]$. \nFor convenience, we define these neighborhood operations for subgraphs as well; that is, for a subgraph $H$ of $G$, \nlet $N_G(H):=N_G(V(H))$ and $N_G[H]:=N_G[V(H)]$.\nWhen the underlying graph is clear from the context, we drop the subscript $G$. \nA vertex set $S$ of a graph is a \\emph{clique} if every pair of vertices in $S$ is adjacent, and \nit is an \\emph{independent set} if every pair of vertices in $S$ is non-adjacent.\nFor two subgraphs $H$ and $F$ of $G$, the \\emph{restriction} of $F$ on $H$ is defined as the graph $F[V(F)\\cap V(H)]$.\n\n\nA \\emph{walk} is a non-empty alternating sequence of vertices and edges of the form $(x_0,e_0,\\ldots ,e_{\\ell-1},x_{\\ell})$, beginning and ending with vertices, such that for every $0\\leq i\\leq \\ell-1$, $x_{i}$ and $x_{i+1}$ are endpoints of $e_i$. \n A \\emph{path} is a walk in which vertices are pairwise distinct. \nFor a path $P$ on vertices $x_0,\\ldots , x_{\\ell}$ with edges $x_ix_{i+1}$ for $i=0,1, \\ldots , \\ell-1$, we write $P=x_0x_1 \\cdots x_{\\ell}$.\nIt is also called an $(x_0, x_{\\ell})$-path.\nWe say $x_{i}$ is the $i$-th neighbor of $x_0$, and similarly, $x_{\\ell-i}$ is the $i$-th neighbor of $x_{\\ell}$ in $P$.\nA \\emph{cycle} is a walk $(x_0, e_0, \\ldots, e_{\\ell-1}, x_{\\ell})$ in which vertices are pairwise distinct except $x_0=x_{\\ell}$. \nFor a cycle $C$ on $x_0,x_1, \\ldots , x_{\\ell}$ with edges $x_ix_{i+1}$ for $i=0,1, \\ldots , \\ell-1$ and $x_{\\ell}x_0$, \nwe write $C=x_0x_1 \\cdots x_{\\ell}x_0$. If a cycle or a path $H$ is an induced subgraph of the given graph $G$,\nthen we say that $H$ is an induced cycle or an induced path in $G$, respectively. \n\nA subpath of a path $P$ starting at $x$ and ending at $y$ is denoted as $xPy$. \nIn the notation $xPy$, we may replace $x$ or $y$ with $\\mathring{x}$ or $\\mathring{y}$, to obtain a subpath starting from the neighbor of $x$ in $P$ closer to $y$ or ending at the neighbor of $y$ in $P$ closer to $x$, respectively.\nFor instance, $xP\\mathring{y}$ refers to the subpath of $P$ starting at $x$ and ending at the neighbor of $y$ in $P$ closer to $x$. \nGiven two walks $P=(v_0,e_0,\\ldots ,e_{p-1},v_{p})$ and $Q=(u_{0},f_{0},\\ldots ,f_{q-1},u_{q})$ such that $v_p=u_0$, the \\emph{concatenation} of $P$ and $Q$ is \ndefined as the walk $(v_0,e_0,\\ldots ,e_{p-1},v_{p}(=u_0),f_{0},\\ldots ,f_{q-1},u_{q})$, which we denote as $P\\odot Q$.\nNote that for two internally vertex-disjoint paths $P_1$ and $P_2$ from $v$ to $w$, \n$vP_1w\\odot wP_2v$ denotes the cycle passing through $P_1$ and $P_2$.\n\nGiven a graph $G$, the distance between two vertices $x$ and $y$ in $G$ is defined as the length of a shortest $(x,y)$-path and denoted as $\\dist_G(x,y)$. If $x=y$, then we define $\\dist_G(x,y)=0$, and $\\dist_G(x,y)=\\infty$ if there is no $(x,y)$-path in $G$. The distance between two vertex sets $X,Y\\subseteq V(G)$, written as $\\dist_G(X,Y)$, is the minimum $\\dist_G(x,y)$ over all $x\\in X$ and $y\\in Y$. If $X=\\{x\\}$, then we write $\\dist_G(X,Y)$ as $\\dist_G(x,Y)$. For a vertex subset $S$ of $G$, a vertex set $U$ is the \\emph{$r$-neighborhood} of $S$ in $G$ if it is the set of all vertices $w$ such that $\\dist_G(w, S)\\le r$. \nWe denote the $r$-neighborhood of $S$ in $G$ as $N^r_G[S]$. When the underlying graph $G$ is clear from the context, we omit the subscript $G$.\n\nGiven a cycle $C=x_0x_1 \\cdots x_{\\ell}x_0$, an edge $e$ of $G$ is a \\emph{chord} of $C$ if both endpoints of $e$ are contained in $V(C)$ \nbut $e$ is not an edge of $C$. A graph is \\emph{chordal} if it has no holes.\nA vertex set $T$ of a graph $G$ is called a \\emph{chordal deletion set} if $G-T$ is chordal.\n\nGiven a vertex set $S\\subseteq V(G)$, a path $P$ is called an \\emph{$S$-path} if $P$ connects two distinct vertices of $S$ and all internal vertices, possibly empty, are not in $S$. An $S$-path is called \\emph{proper} if it has at least one internal vertex. An $(A,B)$-path of a graph $G$ is a path $v_0v_1\\cdots v_{\\ell}$ such that $v_0\\in A$, $v_{\\ell}\\in B$ and all, possibly empty, internal vertices are in $V(G)\\setminus (A\\cup B)$. Observe that every path from $A$ to $B$ contains an $(A,B)$-path. If $A$ or $B$ is a singleton, then we omit the bracket from the set notation. A vertex set $S$ is an \\emph{$(A,B)$-separator} if $S$ disconnects all $(A,B)$-paths in $G$.\n\nWe recall the Menger's Theorem.\n\\begin{THM}[Menger's Theorem; See for instance \\cite{diestel2012}]\\label{thm:menger}\nLet $G$ be a graph and $A,B\\subseteq V(G)$. Then the size of a minimum $(A,B)$-separator in $G$ equals the maximum number of vertex-disjoint $(A,B)$-paths in $G$. \nFurthermore, one can output either one of them in polynomial time.\n\\end{THM}\nA bipartite graph is a graph $G$ with a vertex bipartition $(A,B)$ in which each of $G[A]$ and $G[B]$ is edgeless.\nA set $F$ of edges in a graph is a \\emph{matching} if no two edges in $F$ have a common endpoint.\nA vertex set $S$ of a graph $G$ is a \\emph{vertex cover} if $G-S$ has no edges.\nBy Theorem~\\ref{thm:menger}, \ngiven a bipartite graph with a bipartition $(A,B)$, \none can find a maximum matching or a minimum vertex cover in polynomial time.\n\n\n\nThe following result is useful to find many vertex-disjoint cycles in a graph of maximum degree $3$. We define $s_k$ for $k\\in \\mathbb{N}$ as\n\\begin{align*}\ns_k=\n\\begin{cases}\n4k(\\log k + \\log \\log k +4) \\quad &\\text{if } k\\geq 2\\\\\n2 &\\text{if } k=1.\n\\end{cases}\n\\end{align*}\n\\begin{THM}[Simonovitz~\\cite{Simonovits1967}]\\label{thm:simonovitz}\nLet $G$ be a graph all of whose vertices have degree $3$ and let $k$ be a positive integer. If $\\abs{V(G)}\\geq s_k$, then $G$ contains at least $k$ vertex-disjoint cycles. Furthermore, such $k$ cycles can be found in polynomial time.\n\\end{THM}\n\nLastly, we present lemmas which are useful for detecting a hole.\n\n\\begin{LEM}\\label{lem:twopaths}\nLet $H$ be a graph and $x,y\\in V(H)$ be two distinct vertices. Let $P$ and $Q$ be internally vertex-disjoint $(x,y)$-paths such that $Q$ contains an internal vertex $w$ having no neighbor in $V(P)\\setminus \\{x,y\\}$. If $Q$ is an induced path, then $H[V(P)\\cup V(Q)]$ has a hole containing $w$.\n\\end{LEM}\n\\begin{proof}\nLet $x_1$ and $x_2$ be the neighbors of $w$ in $Q$. As $xPy\\odot yQx$ is a cycle, $H[V(P)\\cup V(Q)]-w$ is connected.\nLet $R$ be a shortest $(x_1, x_2)$-path in $H[V(P)\\cup V(Q)]-w$. As the only neighbors of $w$ contained in $(V(P)\\cup V(Q)) \\setminus \\{w\\}$ are $x_1$ and $x_2$, \n$w$ has no neighbors in the internal vertices of $R$. Note that $R$ has length at least two since $x_1,x_2\\in V(Q)$ and $Q$ is an induced path.\nTherefore, $x_1Rx_2\\odot x_2wx_1$ is a hole containing $w$, as required.\n\\end{proof}\n\nA special case of Lemma~\\ref{lem:twopaths} \nis when there is a vertex $w$ in a cycle $C$ such that $w$ has no neighbors in the internal vertices of $C-w$ and the \nneighbors of $w$ on $C$ are non-adjacent. \nIn this case, $C$ has a hole containing $w$ by Lemma~\\ref{lem:twopaths}.\n\nOne can test in polynomial time whether a graph contains a hole or not.\n\\begin{LEM}\\label{lem:detectinghole}\nGiven a graph $G$, one can test in polynomial time whether it has a hole or not.\nFurthermore, one can find in polynomial time a shortest hole of $G$, if one exists.\n\\end{LEM}\n\\begin{proof}\nWe guess three vertices $v,w,z$ where $vw, wz\\in E(G)$ and $vz\\notin E(G)$, and \ntest whether there is a path from $v$ to $z$ in $G-(N_G[w]\\setminus \\{v,z\\})$.\nIf there is such a path, then we choose a shortest path $P$ from $v$ to $z$. As $w$ has no neighbors in the set of internal vertices of $P$, \n$V(P)\\cup \\{w\\}$ induces a hole. Clearly if $G$ has a hole, then we can find one by the above procedure.\n\nTo find a shortest one, for every such a guessed tuple $(v,w,z)$, we keep the length of the obtained hole.\nThen it is sufficient to output a hole with minimum length among all obtained holes.\n\\end{proof}\n\n\n\n\\section{Terminology and a proof overview.}\\label{sec:overview} \n\nThe proof of Theorem~\\ref{thm:main} begins by finding a sequence of shortest holes. Let $G$ be the input graph and let $G_1=G$. \nFor each $i=1,2,\\ldots$, we iteratively find a shortest hole $C_i$ in $G_i$ and set $G_{i+1}:=G_i - V(C_i)$. If the procedure fails to find a hole at $j$-th iteration, then $G_j$ is a chordal graph. \nThis iterative procedure leads us to the following theorem, which is the core component of our result.\n\nFor $k\\in \\mathbb{N}$, we define\n$\\mu_k=76s_{k+1}+3217k+1985$.\n\n\\begin{THM}\\label{thm:core}\nLet $G$ be a graph, $k$ be a positive integer and $C$ be a shortest hole of $G$\nsuch that $C$ has length strictly larger than $\\mu_k$ and $G-V(C)$ is chordal. Given such $G$, $k$, and $C$, one can find in polynomial time either $k+1$ vertex-disjoint holes or a vertex set $X\\subseteq V(G)$ of size at most $\\mu_{k}$ hitting every hole of $G$.\n\\end{THM}\n\n\nIt is easy to derive our main result from Theorem~\\ref{thm:core}.\n\n\\smallskip\n\n\\begin{proofof}{Theorem~\\ref{thm:main}}\nWe construct sequences $G_1,\\ldots, G_{\\ell+1}$ and $C_1,\\ldots , C_{\\ell}$ \nsuch that \n\\begin{itemize}\n\\item $G_1=G$, \n\\item for each $i\\in \\{1, 2, \\ldots, \\ell\\}$, $C_i$ is a shortest hole of $G_i$, and \n\\item for each $i\\in \\{1, 2, \\ldots, \\ell\\}$, $G_{i+1}=G_i-V(C_i)$.\n\\item $G_{\\ell+1}$ is chordal.\n\\end{itemize}\nSuch a sequence can be constructed in polynomial time repeatedly applying Lemma~\\ref{lem:detectinghole} to find a shortest hole.\nIf $\\ell \\geq k+1$, then we have found a packing of $k+1$ holes. \nHence, we assume that $\\ell \\leq k$. \n\nWe prove the following claim for $j=\\ell+1$ down to $j=1$. \n\\begin{quote}\nOne can find in polynomial time either $k+1$ vertex-disjoint holes, or \na chordal deletion set $T_{j}$ of $G_{j}$ of size at most $(\\ell+1-j)\\mu_k$. \n\\end{quote}\n\n\nThe claim trivially holds for $j=\\ell+1$ with $T_{\\ell+1}=\\emptyset$ because $G_{\\ell+1}$ is chordal.\nLet us assume that for some $j\\leq \\ell$, we obtained a chordal deletion set $T_{j+1}$ of $G_{j+1}$ of size at most $(\\ell-j)\\mu_k$.\nThen in $G_{j}-T_{j+1}$, $C_{j}$ is a shortest hole, and $\\left( G_{j}-T_{j+1} \\right)-V(C_{j})$ is chordal.\nIf $C_{j}$ has length at most $\\mu_k$, then we set $T_{j}:=T_{j+1}\\cup V(C_{j})$. Clearly, $\\abs{T_j}\\leq (\\ell-j+1)\\mu_k$.\nOtherwise, by applying Theorem~\\ref{thm:core} to $G_{j}-T_{j+1}$ and $C_{j}$, \none can find in polynomial time either $k+1$ vertex-disjoint holes or a chordal deletion set $X$ of size at most $\\mu_k$ of $G_{j}-T_{j+1}$. \nIn the former case, we output $k+1$ vertex-disjoint holes, and we are done. If we obtain a chordal deletion set $X$, then we set $T_{j}:=T_{j+1}\\cup X$. Observe that the set $T_{j}$ is a chordal deletion set of $G_{j}$ and $\\abs{T_{j}}\\le (\\ell-j+1)\\mu_k$ as claimed. \n\nFrom the claim with $j=1$, we conclude that \nin polynomial time, one can find either $k+1$ vertex-disjoint holes, \nor a chordal deletion set of $G_1=G$ of size at most $\\ell\\mu_k\\le k\\mu_k=\\mathcal{O}(k^2\\log k)$. \n\\end{proofof}\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (12,-.5) node [w] {};\n \t\\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \\foreach \\y in {2,3,4}{\n\t\\draw (3,2)--(a\\y);\n }\n \\foreach \\y in {4,5}{\n\t\\draw (4.5,2)--(a\\y);\n }\n \\draw(3,2)--(4.5,2);\n\n \\foreach \\y in {7,8,9}{\n\t\\draw (8,2)--(a\\y);\n }\n \\draw (3,2) node [w] {};\n \\draw (8,2) node [w] {};\n \\draw (4.5,2) node [w] {};\n\n\\draw[dashed, rounded corners] (4, 2.3)--(5, 2.3)--(4, -.5)--(2.5, 2.3)--(4, 2.3);\n\n\\draw[rounded corners] (-3,3)--(-3,2)--(0,2)--(0,4)--(-3,4)--(-3,3);\n \\foreach \\y in {-2,-1.5,-1, -.5,0}{\n\t\\draw (-1.5,2.5)--(\\y, 1.5);\n }\n \\draw (-1.5, 2.5) node [w] {};\n\n\n \\node at (-1.5, 3) {$D$};\n \\node at (-2, -1) {$C$};\n \\node at (4, 2.7) {$Z_v$};\n \\node at (4, -.8) {$v$};\n\n \\end{tikzpicture} \\caption{The set of vertices adjacent to all vertices of $C$ is denoted by $D$, and for each $v\\in V(C)$, $Z_v$ \n denotes the set $\\{v\\}\\cup (N(v)\\setminus V(C)\\setminus D)$. Using the fact that $C$ is chosen as a shortest hole and it is long, we will prove in Lemma~\\ref{lem:consecutive} that each vertex in $N(C)\\setminus D$ has at most $3$ neighbors on $C$ and they are consecutive in $C$. }\\label{fig:setting}\n\\end{figure}\n\n\\medskip\n\nThe rest of this section and Sections~\\ref{sec:lemmas}-\\ref{sec:tulip} are devoted to establish Theorem~\\ref{thm:core}. \nThroughout these sections, we shall consider the input tuple $(G,k, C)$ of Theorem~\\ref{thm:core} as fixed. \n\nLet us introduce the notations that are frequently used (see Figure~\\ref{fig:setting}).\nA vertex $v\\in N(C)$ is \\emph{$C$-dominating} if $v$ is adjacent to every vertex on $C$.\nWe reserve $D$ to denote the set of all $C$-dominating vertices.\nFor each vertex $v$ in $C$, we denote by $Z_v:=\\{v\\}\\cup (N(v)\\setminus V(C)\\setminus D )$, and \nfor a subset $S$ of $V(C)$, we denote by $Z_S:=\\bigcup_{v\\in S}Z_v$.\nWe also define\n\\begin{itemize}\n\\item $G_{deldom}:=G-D$ and $G_{nbd}:=G[N[C]\\setminus D]$. \n\\end{itemize}\nFor a subpath $Q$ of $C$, the subgraph of $G$ induced by $Z_{V(Q)}$ is called a \\emph{$Q$-tunnel}. \nBy definition of $Z_{V(Q)}$, $Q$-tunnel is an induced subgraph of $G_{nbd}$. \nWhen $q, q'$ are endpoints of $Q$, we say that $Z_{q}$ and $Z_{q'}$ are \\emph{entrances} of the $Q$-tunnel.\n\nWe distinguish between two types of holes, namely \\emph{sunflowers} and \\emph{tulips}. \nA hole $H$ is said to be a \\emph{sunflower} if $V(H)\\subseteq N[C]$, that is, its entire vertex set is placed within the closed neighborhood of $C$.\nA hole that is not a sunflower is called a \\emph{tulip}. Every tulip contains at least one vertex not contained in $N[C]$.\nAlso, we classify holes depending on whether one contains a $C$-dominating vertex or not.\nA hole is \\emph{$D$-traversing} it contains a $C$-dominating vertex (which is a vertex of $D$), and \\emph{$D$-avoiding} otherwise.\n\nIn the remainder of this section, we present a proof outline of Theorem~\\ref{thm:core}. \nHere are three basic observations, necessary to give the ideas of our proofs.\n\\begin{itemize}\n\\item (Lemma~\\ref{lem:consecutive})\nFor every vertex $v$ of $N(C)$, either it has at most $3$ consecutive neighbors in $C$ or it is $C$-dominating.\n\\item (Lemma~\\ref{lem:farnonadj})\nLet $x,y$ be two vertices in $C$ such that $\\dist_C(x,y)\\geq 4$. Then there is no edge between $Z_x$ and $Z_y$. \n\\item (Lemma~\\ref{lem:dominating})\n$D$ is a clique.\n\\end{itemize}\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (12,-.5) node [w] {};\n \t\\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \n \n \\draw[dashed](8,2) [in=30,out=150] to (3,2);\n\n \\foreach \\y in {2,3,4}{\n\t\\draw (3,2)--(a\\y);\n }\n \n \\foreach \\y in {7,8,9}{\n\t\\draw (8,2)--(a\\y);\n }\n \\draw (3,2) node [w] {};\n \\draw (8,2) node [w] {};\n \n \n\\draw[rounded corners] (-3,3)--(-3,2)--(0,2)--(0,4)--(-3,4)--(-3,3);\n \\foreach \\y in {-2,-1.5,-1, -.5,0}{\n\t\\draw (-1.5,2.5)--(\\y, 1.5);\n }\n \\draw (-1.5, 2.5) node [w] {};\n\n\n \\node at (-1.5, 3) {$D$};\n \\node at (-2, -1) {$C$};\n \\node at (2, -.8) {$x$};\n\t\\node at (9, -.8) {$y$};\n \\node at (2.6, 2.4) {$v$};\n \\node at (8.4, 2.4) {$w$};\n \n \\end{tikzpicture} \\caption{Illustration of Lemma~\\ref{lem:farnonadj}: if $\\dist_C(x,y)\\ge 4$, then there are no edges between $Z_x$ and $Z_y$.\n For instance, suppose $v$ and $w$ are adjacent. \n Note that $v$ and $w$ have at most $3$ consecutive neighbors in $C$.\n If the distance from $N(v)\\cap V(C)$ to $N(w)\\cap V(C)$ in $C$ is at least $1$, then we can find a hole shorter than $C$ using the shortest path from $N(v)\\cap V(C)$ to $N(w)\\cap V(C)$ in $C$.\n Otherwise, we have $\\dist_C(x,y)=4$ and $\\abs{N(v)\\cap V(C)}=\\abs{N(w)\\cap V(C)}=3$ and thus, the longer path between $N(v)\\cap V(C)$ to $N(w)\\cap V(C)$ in $C$ creates a hole shorter than $C$.}\\label{fig:distance}\n\\end{figure}\n\n\n\\subsection{$D$-avoiding sunflowers.} \n\nThe set of $D$-avoiding sunflowers is categorized into two subgroups, \\emph{petals} and \\emph{full sunflowers}. Roughly speaking, petals are \\emph{seen} by a small number of consecutive vertices on $C$, while a full one is \\emph{seen} by every vertex of $C$. For the precise definition, we introduce the notion of support. \n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\t \\draw(3,3) [in=120,out=-120] to (3,0);\n\t \\draw(3.4,3) [in=120,out=-120] to (3,0);\n\t \\draw(4.1,3) [in=120,out=-140] to (3,0);\n\t \\draw(4.5,3) [in=120,out=-140] to (3,0);\n\t \\draw(4.5,2) -- (3,0);\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (12,-.5) node [w] {};\n \t\\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \\foreach \\y in {2,3,4}{\n\t\\draw (3,2)--(a\\y);\n }\n \\foreach \\y in {4,5}{\n\t\\draw (4.5,2)--(a\\y);\n }\n \\draw (3,2)--(3,3);\n \\draw(4.5,3)--(4.5,2);\n\t\\draw(3,3)--(3.4,3);\\draw[dotted](3.4,3)--(4.1,3);\\draw(4.1,3)--(4.5,3);\n\t \n \\foreach \\y in {7,8,9}{\n\t\\draw (8,2)--(a\\y);\n }\n \\draw (3,2) node [w] {};\n \\draw (3,3) node [w] {};\n \\draw (8,2) node [w] {};\n \\draw (4.5,2) node [w] {};\n \\draw (4.5,3) node [w] {};\n\n\t \\draw (3.4,3) node [w] {};\n \\draw (4.1,3) node [w] {};\n \n\\draw[dashed, rounded corners] (4, 3.3)--(5,3.3)--(5, 2.3)--(4, -.5)--(2.5, 2.3)--(2.5,3.3)--(4, 3.3);\n\n\\draw[rounded corners] (-3,3)--(-3,2)--(0,2)--(0,4)--(-3,4)--(-3,3);\n \\foreach \\y in {-2,-1.5,-1, -.5,0}{\n\t\\draw (-1.5,2.5)--(\\y, 1.5);\n }\n \\draw (-1.5, 2.5) node [w] {};\n\n\n \\node at (-1.5, 3) {$D$};\n \\node at (-2, -1) {$C$};\n \\node at (4, 3.7) {$H$};\n\n \\node at (2, -.8) {$v_1$};\n \\node at (3, -.8) {$v_2$};\n \\node at (4, -.8) {$v_3$};\n \\node at (5, -.8) {$v_4$};\n\n \\end{tikzpicture} \\caption{The cycle $H$ is a petal having the support $\\{v_1, v_2, v_3, v_4\\}$. A petal can be arbitrarily long.}\\label{fig:petal}\n\\end{figure}\n\nFor a subgraph $H$ of $G$, the \\emph{support} of $H$, denoted by $\\operatorname{\\textsf{sp}} (H)$, is the set of all vertices $v\\in V(C)$ such that $(Z_v\\cup D)\\cap V(H)\\neq\\emptyset$. \nObserve that if $H$ contains a vertex of $D$, then trivially $\\operatorname{\\textsf{sp}}(H)=V(C)$.\nFor a $D$-avoiding sunflower $H$, we say that \n\\begin{itemize}\n\\item it is a \\emph{petal} if $\\abs{\\operatorname{\\textsf{sp}}(H)}\\leq 7$, and \n\\item it is \\emph{full} if $\\operatorname{\\textsf{sp}}(H)=V(C)$.\n\\end{itemize}\nSee Figure~\\ref{fig:petal} for an illustration of a petal.\n\\medskip\n\n\\noindent {[Subsection~\\ref{subsec:petal}.]} We first obtain a small hitting set of petals, unless $G$ has $k+1$ vertex-disjoint holes.\nFor this, we greedily pack petals and mark their supports on $C$.\nClearly, if there are $k+1$ petals whose supports are pairwise disjoint, then we can find $k+1$ vertex-disjoint holes. \nThus, we can assume that there are at most $k$ petals whose supports are pairwise disjoint. \nWe take the union of all those supports and call it $T_1$. By construction, for every petal $H$, $\\operatorname{\\textsf{sp}}(H)\\cap T_1\\neq\\emptyset$.\nThen we take the $6$-neighborhood of $T_1$ in $C$ and call it $T_{petal}$.\nIt turns out that \n\\begin{itemize}\n\\item[($\\ast$)] for every petal $H$, $\\operatorname{\\textsf{sp}} (H)$ is fully contained in $T_{petal}$, \n\\end{itemize} \nand in particular, $V(H)\\cap T_{petal}\\neq \\emptyset$. The size of $T_{petal}$ is at most $19k$.\n\\medskip\n\n\\noindent {[Subsection~\\ref{subsec:allisfull}.]} \nSomewhat surprisingly, we show that every $D$-avoiding sunflower that does not intersect $T_{petal}$ is a full sunflower. \nIt is possible that there is a sunflower with support of size at least $8$ and less than $\\abs{V(C)}$.\nWe argue that if such a sunflower $H$ exists, then there is a vertex $v\\in V(C)\\cap V(H)$ and a petal whose support contains $v$. \nBut the property $(\\ast)$ of $T_{petal}$ implies that $T_{petal}$ contains $v$, which implies that such a sunflower should be hit by $T_{petal}$.\nTherefore, it is sufficient to hit full sunflowers for hitting all remaining $D$-avoiding sunflowers.\n\\medskip\n\n\\noindent {[Subsection~\\ref{subsec:sunflowerwithout}.]} \nWe obtain a small hitting set of full sunflowers, when $G$ has no $k+1$ vertex-disjoint holes.\nFor this, we consider two vertex sets $Z_v$ and $Z_w$ for some $v$ and $w$ on $C$, \nand apply Menger's theorem for two directions, say `north' and `south' hemispheres around $C$, between $Z_v$ and $Z_w$ in the graph $G_{nbd}$.\nWe want to argue that if there are many paths in both directions, then we can find many vertex-disjoint holes. \nHowever, it is not clear how to link two families of paths.\n\nTo handle this issue, we find two families of paths whose supports cross on constant number of vertices. \nSince $C$ is much larger than the obtained hitting set $T_{petal}$ for petals, we can find $25$ consecutive vertices that contain no vertices in $T_{petal}$.\nLet $v_{-2}, v_{-1}, v_0, \\ldots, v_{22}$ be such a set of consecutive vertices.\nLet $\\mathcal{P}$ be the family of vertex-disjoint paths from $Z_{v_0}$ to $Z_{v_{20}}$ whose supports are contained in $\\{v_{-2}, v_{-1}, v_0, v_1, \\ldots, v_{20}, v_{21}, v_{22}\\}$, and let $\\mathcal{Q}$ be the family of vertex-disjoint paths from $Z_{v_5}$ to $Z_{v_{16}}$ whose supports do not contain $v_8$ and $v_{13}$. \nNon-existence of petals with support intersecting $\\{v_{-2}, v_{-1}, \\ldots, v_5\\}$ implies that \npaths in $\\mathcal{P}$ and $\\mathcal{Q}$ are `well-linked' at $Z_{v_0}$ except for few paths, and a symmetric argument holds at $Z_{v_{20}}$. \nThis allows us to link any pair of paths from $\\mathcal{P}$ and $\\mathcal{Q}$. \nIf one of $\\mathcal{P}$ and $\\mathcal{Q}$ is small, then we can output a hitting \nset of full sunflowers using Menger's theorem. The size of the obtained set $T_{full}$ will be at most $3k+14$.\n\n\\subsection{$D$-traversing sunflowers}\n[Subsection~\\ref{subsec:sunflowerwith}.] \nIt is easy to see that every $D$-traversing hole $H$ contains exactly one vertex of $D$ (since $D$ is a clique), and every vertex of $V(C)\\cap V(H)$ is a neighbor of the vertex in $D\\cap V(H)$. Let $v\\in V(C)\\cap V(H)$ and $d\\in D\\cap V(H)$ be such an adjacent pair.\nWe argue that $H$ contains a subpath $Q$ in $N[C]\\setminus D$ that starts from $v$ and is contained in $Z_{\\{v, v_2, v_3\\}}$ for some three consecutive vertices $v, v_2, v_3$ of $C$, such that\n\\begin{itemize}\n\\item $G[V(Q)\\cup \\{v, v_2, v_3, d\\}]$ contains a $D$-traversing sunflower containing $d$ and $v$.\n\\end{itemize}\n In other words, even if $H-d$ has large support, we can find another $D$-traversing sunflower $H'$ \n containing $d$ and $v$ where $H'-d$ has support on small number of vertices. \n The fact that $H$ and $H'$ share $v$ is important as we will take one of $d$ and $v$ as a hitting set for such $H'$, and this will hit $H$ as well.\n\nTo this end, we create an auxiliary bipartite graph in which one part is $D$ and the other part consists of sets of $3$ consecutive vertices $v_1, v_2, v_3$ of $C$, and we add an edge \nbetween $d\\in D$ and $\\{v_1, v_2, v_3\\}$ if $G[Z_{\\{v_1, v_2, v_3\\}}\\cup \\{d\\}]$ contains a $D$-traversing hole.\nWe argue that if this bipartite graph has a large matching, then we can find many vertex-disjoint holes, \nand otherwise, we have a small vertex cover. The union of all vertices involved in the vertex cover suffices \nto cover all $D$-traversing sunflowers. The hitting set $T_{trav:sunf}$ will have size at most $15k+9$.\n\n\n\\subsection{$D$-avoiding tulips.} \nWe follow the approach of constructing a subgraph of maximum degree $3$ used in proving the Erd\\H{o}s-P\\'osa property for various types of cycles: roughly speaking, if there is a cycle after removing the vertices of degree $3$ in the subgraph constructed so far, we augment the construction by adding some path or cycle.\nSimonovitz~\\cite{Simonovits1967} first proposed this idea and proved that if the number of degree $3$ vertices is $s_{k+1}$, then there are $k+1$ vertex-disjoint cycles. \nIf a maximal construction has less than $s_{k+1}$ vertices of degree 3, then we can hit all cycles of the input graph by taking all vertices of degree $3$ and a few more vertices.\n\\medskip \n\n\\noindent [Subsection~\\ref{subsec:tuliphive}.] The major obstacle for employing Simonovitz' approach is that for our purpose, we need to guarantee that every cycle of such a construction gives a hole.\nFor this reason, we will carefully add a path so that every cycle in a construction contains some hole.\nWe arrive at a notion of an $F$-extension, which is a path to be added iteratively with $C$ at the beginning.\nBy adding $F$-extensions recursively, we will construct a subgraph such that all vertices have degree $2$ or $3$ and it contains $C$.\nFor a subgraph $F$ of $G_{deldom}$ such that all vertices have degree $2$ or $3$ and it contains $C$< \nan $F$-extension is a proper $V(F)$-path $P$ in $G_{deldom}$, but has additional properties that \n\\begin{enumerate}[(i)]\n\\item both endpoints of $P$ are vertices of degree $2$ in $F$,\n\\item one of its endpoints should be in $C$, and \n\\item $P$ has at least one endpoint in $C$ whose second neighbor on $P$ has no neighbors in $F$, \nand the path obtained from $P$ by removing its endpoints is induced.\n\\end{enumerate}\nAn almost $F$-extension is a similar object, but its endpoints on $F$ is the same. Note that an almost $F$-extension is a cycle and is not an $F$-extension.\nWe depict an (almost) $F$-extension in Figure~\\ref{fig:wextension}. \nWhen we recursively choose an $F$-extension to add, we apply two priority rules:\n\\begin{itemize}\n\\item We choose a shortest $F$-extension among all possible $F$-extensions. \n\\item We choose an $F$-extension $Q$ with maximum $\\abs{V(Q)\\cap V(C)}$ among all shortest $F$-extensions. \n\\end{itemize}\nFollowing these rules, we recursively add $F$-extensions until there are no $F$-extensions. \n\nLet $W$ be the resulting graph.\nThe properties (ii), (iii) together with Lemma~\\ref{lem:twopaths} guarantee that the subgraph induced by the vertex set of every cycle of $W$ contains a hole. Therefore, in case when $W$ contains $s_{k+1}$ vertices of degree $3$, we can find $k+1$ vertex-disjoint holes by Theorem~\\ref{thm:simonovitz}. \nWe may assume that it contains less than $s_{k+1}$ vertices of degree $3$. \nLet $T_{branch}$ be the set of degree $3$ vertices in $W$.\nWe also separately argue that we can hit all of almost $W$-extensions by at most $5k+4$ vertices.\nLet $T_{almost}$ be the hitting set for almost $W$-extensions.\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw[rounded corners] (6,0)--(11,0)--(12,-1)--(11,-2)--(-2,-2)--(-3,-1)--(-2,0)--(6,0);\n \n \\draw[rounded corners] (7,0)--(7,1)--(9,1)--(9,0);\n \\draw[rounded corners] (8,1)--(8,2)--(10,2)--(10,0);\n \\draw[rounded corners] (6,0)--(6,3)--(9,3)--(9,2);\n\n \\foreach \\y in {3.5,4.5}{\n\t\\draw[dashed] (3,2)--(\\y, 1.5);\n }\n \n \\draw (-.1,0.3)--(-.1,-0.3);\n \\draw (.1,0.3)--(.1,-0.3);\n \\draw (4-.1,0.3)--(4-.1,-0.3);\n \\draw (4.1,0.3)--(4.1,-0.3);\n \n \\draw (2,0)--(2,1)--(3,2)--(4,2.5);\n \\draw (4,2.5)--(6,2.5);\n \\draw (2,0) node [w] {};\n \\draw (2,1) node [w] {};\n \\draw (3,2) node [w] {};\n \\draw (4,2.5) node [w] {};\n \\draw (6,2.5) node [w] {};\n \n \\node at (7, 2) {$W$};\n \\node at (-2, 2) {$U$};\n\n \\node at (2, -.5) {$v$};\n \\node at (2.7, 2.3) {$w$};\n \n\t\\draw[rounded corners] (-1,0)--(-1.5, 1)--(-1.5, 4)--(-.5,4)--(-.5,1)--(-1,0);\n \\node at (3.2, 3) {$P$};\n\n\n \\node at (-2, -1) {$C$};\n\n \\end{tikzpicture} \\caption{A brief description of the construction $W$. \n Each extension contains at least one endpoint in $C$ whose second neighbor in the extension \n has no neighbor in $W$ hitherto constructed.\n For instance, $P$ is a $W$-extension, and $v$ is the vertex in $V(C)\\cap V(P)$, and its second neighbor $w$ in $P$ has no neighbors in $W$.\n The subgraph $U$ depicts an almost $W$-extension.}\\label{fig:wextension}\n\\end{figure}\n\n\n\nNow, let $T_{ext}$ be the union of \n\\[T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost}\\] and \n\\[N^{20}_C[(T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})\\cap V(C)].\\] \nNote that \n\\[ \\abs{T_{ext}}\\le 41(19k+(3k+14)+(15k+9)+(s_{k+1}-1)+(5k+4))\\le 41(s_{k+1}+42k+26). \\]\n\nFurthermore, $C-(T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})$ contains at most $s_{k+1}+42k+26$ connected components, and thus $C-T_{ext}$ does as well.\n\\medskip\n\n\\noindent [Subsection~\\ref{subsec:cfragment}] We discuss the patterns of the remaining tulips in $G_{deldom}-T_{ext}$.\nSince we will add all components of $C-T_{ext}$ having at most $35$ vertices to the deletion set for $D$-avoiding tulips, \nwe focus on components of $C-T_{ext}$ with at least $36$ vertices.\nLet $H$ be a $D$-avoiding tulip in $G_{deldom}-T_{ext}$.\nLet $Q=q_1q_2 \\cdots q_m$ be a connected component of $C-T_{ext}$, and we consider the $Q$-tunnel $R$.\n\nWe argue that there is no edge $vw$ in $H$ such that \n$v$ is in the $Q$-tunnel, and $w$ is not in $N[C]$. \nSee Figure~\\ref{fig:qtunnel} for an illustration.\nSuppose there is such a pair, and let $q\\in V(Q)$ be a neighbor of $v$. We mainly prove that since $q$ is sufficiently far from degree $3$ vertices of $W$ in $C$, $w$ has no neighbors in $W$ \n(this is why we take the $20$-neighborhood of $V(C)\\cap (T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})$ in $C$).\nThis is formulated in Lemma~\\ref{lem:distance2}. \nNote that $qvw$ is a path where $q\\in V(C)$ and $w$ has no neighbors in $W$, and furthermore, $H$ contains a vertex in $V(C)\\setminus T_{ext}$ which is a vertex of degree $2$ in $W$.\nTherefore, if we traverse in $H$ following the direction from $v$ to $w$, we meet some vertex having a neighbor which is a vertex of degree $2$ in $W$. This gives a $W$-extension or an almost $W$-extension. But it is a contradiction as there is no $W$-extension, and $T_{ext}$ hits all of almost $W$-extensions.\nSo, there are no such edges $vw$.\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw[rounded corners] (6,0)--(11,0)--(12,-1)--(11,-2)--(-2,-2)--(-3,-1)--(-2,0)--(6,0);\n \n \\draw[rounded corners] (7,0)--(7,1)--(9,1)--(9,0);\n \\draw[rounded corners] (8,1)--(8,2)--(10,2)--(10,0);\n \\draw[rounded corners] (6,0)--(6,3)--(9,3)--(9,2);\n \\draw[rounded corners] (6,2.5)--(-1,2.5)--(-1,0);\n\n\t\\draw[fill=black] (-2,-.3)--(-2,.3)--(1,.3)--(1,-.3)--(-2,-.3);\n\t\\draw[fill=black] (4,-.3)--(4,.3)--(7,.3)--(7,-.3)--(4,-.3);\n\t\n\t\\draw[red, dotted, very thick, rounded corners] (1,0)--(1,1)--(4,1)--(4,0);\n\t\n\t\\draw[rounded corners] (0, .5)--(2,.5)--(2.3, -.2)--(2.6,.5)--(3.5,.5)--(3.5,1.5)--(4, 2);\n\t \\draw (3.45,.5) node [w] {};\n \\draw (3.5,1.5) node [w] {};\n \\node at (3.45, 0) {$v$};\n \\node at (3, 1.5) {$w$};\n\n \\node at (7, 2) {$W$};\n\n \\node at (-2, -1) {$C$};\n \\node at (2.5, -1) {$Q$};\n \\node at (5.5, -1) {$T_{ext}$};\n\n \\end{tikzpicture} \n \\caption{A description of the set $T_{ext}$ and a component $Q$ of $C-T_{ext}$. \n When there is an edge $vw$ where $v\\in Z_{V(Q)}\\setminus V(Q)$ and $w\\notin N[C]$, \n we will prove in Lemma~\\ref{lem:tunnellemma1} that $w$ has no neighbors in $W$.\n In particular, if there is a $D$-avoiding tulip containing such an edge, then we can find a $W$-extension or an almost $W$-extension starting with $qvw$ for some $q\\in V(Q)$.\n }\\label{fig:qtunnel}\n\\end{figure}\n\nThis argument leads to an observation that \nif $H$ contains some vertex $q_i$ with $6\\le i\\le m-5$, then the restriction of $H$ on $R$ should be a path from $Z_{q_1}$ to $Z_{q_m}$. Let $P$ be the restriction of $H$ on $R$. \nWe additionally remove $\\{q_j:1\\le j\\le 15, m-14\\le j\\le m\\}$ and assume $H$ is not removed.\nThen there are two ways that $P$ can be placed inside the $Q$-tunnel $R$: either\nthe endpoints of $P$ are in the same connected component of $R-(T_{ext}\\cup V(Q))$ or not.\nIn the former case, we could reroute this path so that this part does not contain a vertex of $Q$.\nSo, we could obtain a $D$-avoiding tulip containing less vertices of $C$.\nHowever, since $G-V(C)$ is chordal, \nthere should be some subpath $Q'$ of $C-T_{ext}$ such that \nthe restriction of $H$ on the $Q'$-tunnel is of the second type.\nWe will show that such a path can be hit by removing $5$ more vertices in $Q'$.\nThis will give a vertex set $T_{avoid:tulip}$ of size at most $35(s_{k+1}+42k+26)$ hitting all the remaining $D$-avoiding tulips.\n\n\\subsection{$D$-traversing tulips}\n[Subsection~\\ref{subsec:Dtulip}.]\nThis case can be handled similarly as the case of $D$-traversing sunflowers. \nIt turns out that $T_{ext}$ hits every $D$-traversing tulip that contains precisely two vertices of $C$. \nUsing a matching argument between $D$ and the set of three consecutive vertices of $C$, we show that an additional set $T_{trav:tulip}$ of size at most $25k+9$ \nplus $T_{avoid:tulip}$ \nhits all the remaining $D$-traversing tulips unless $G$ contains $k+1$ vertex-disjoint holes.\n\nIn total, we can output in polynomial time either $k+1$ vertex-disjoint holes in $G$, or a vertex set of size at most \n\\begin{align*}\n&\\abs{T_{ext}\\cup T_{avoid:tulip}\\cup T_{trav:tulip}} \\\\\n&\\le 41(s_{k+1}+42k+26)+ 35(s_{k+1}+42k+26)+25k+9 \\\\\n\t\t\t\t\t\t\t\t\t\t&\\le 76(s_{k+1}+42k+26)+25k+9= 76s_{k+1}+3217k+1985\n\t\t\t\t\t\t\t\t\t\t\\end{align*}\n\t\t\t\t\t\t\t\t\t\t hitting all holes.\n\n\n\\section{Structural properties of $G$}\\label{sec:lemmas}\n\nIn this section, we present structural properties of a graph $G$ with a shortest hole $C$.\nIn Subsection~\\ref{subsec:distance}, we derive a relationship between the distance between $Z_v$ and $Z_w$ in $G_{deldom}$ for two vertices $v, w\\in V(C)$ and the distance between $v$ and $w$ in $C$.\nBriefly, we show that the distance between $Z_v$ and $Z_w$ in $G_{deldom}$ is at least some constant times the distance between $v$ and $w$ in $C$. \nWe also prove that every connected subgraph in $G_{nbd}$ has a connected support.\nIn Subsection~\\ref{subsec:dominating}, we obtain some basic properties of $C$-dominating vertices.\nRecall that we assume that the length of $C$ exceeds $\\mu_k$.\n\n\\subsection{Distance lemmas}\\label{subsec:distance}\n\nThe following lemma classifies vertices in $N(C)$ with respect to the number of neighbors in $C$.\n\n\\begin{LEM}\\label{lem:consecutive}\nFor every vertex $v$ of $N(C)$, either it has at most $3$ neighbors in $C$ that are consecutive in $C$ or it is $C$-dominating. \n\\end{LEM}\n\\begin{proof}\nLet us write $N_i:=\\{v\\in N(C):\\abs{N(v)\\cap V(C)}=i\\}$ for $i\\geq 1$.\nWe first show that $N(C)=N_1\\uplus N_2\\uplus N_3\\uplus D$.\nLet $v\\in N(C)\\setminus D$.\n\nWe claim that $v$ has no two neighbors $w_1$ and $w_2$ in $C$ \nsuch that \n\\begin{itemize}\n\\item[($\\ast$)] there is a $(w_1, w_2)$-subpath $Q$ of $C$ where $Q$ has length at least $2$ and at most $\\abs{V(C)}-3$ and $v$ has no neighbor in the internal vertices of $Q$.\n\\end{itemize}\nIf there is such a path $Q$, then $w_1vw_2\\odot w_2Qw_1$ is a hole of length at most $\\abs{V(C)}-1<\\abs{V(C)}$, which contradicts the assumption that $C$ is a shortest hole. So the claim holds.\n\nThis implies that $v$ has no neighbors $z_1$ and $z_2$ with $\\dist_C(z_1, z_2)\\ge 3$. Indeed, if such neighbors exist, then let $Q$ be a $(z_1,z_2)$-subpath of $C$ containing at least one internal vertex non-adjacent to $v$. Since $v\\notin D$, such $Q$ exists. Note that the length of $Q$ is at least 3 and at most $\\abs{V(C)}-3$. Then there exist two neighbors $w_1$ and $w_2$ of $v$ in $V(Q)$ satisfying $(\\ast)$, a contradiction. \nTherefore, the neighbors of $v$ in $C$ are contained in three consecutive vertices of $C$. \nThis implies that $v\\in N_1\\uplus N_2\\uplus N_3$.\n\nFurthermore, if $v\\in N(C)\\setminus D$ has exactly two neighbors with distance $2$ in $C$, then $G$ contains a hole of length $4$, a contradiction. \nTherefore, such a vertex has at most $3$ neighbors in $C$ that are consecutive in $C$, as required.\n\\end{proof}\n\nThe next lemma is illustrated in Figure~\\ref{fig:distance}.\n\\begin{LEM}\\label{lem:farnonadj}\nLet $x$ and $y$ be two vertices in $C$ such that $\\dist_C(x,y)\\geq 4$. Then there is no edge between $Z_x$ and $Z_y$. \n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:consecutive}, the neighbors of any vertex in $N(C)\\setminus D$ lie within distance at most two, and thus $x$ has no neighbors in $Z_y$ and $y$ has no neighbors in $Z_x$. \nSuppose $v\\in Z_x\\setminus \\{x\\}$ and $w\\in Z_y\\setminus \\{y\\}$ are adjacent. \nLet $P$ and $Q$ be $(x,y)$-subpaths of $C$ such that $P$'s length is not greater than $Q$'s. \n\nWe may assume that $Q$ does not contain a common neighbor of $v$ and $w$. \nIf the length of $Q$ is at least five, then $N(v)\\cap V(Q)$ is included in $N_C^2(x)\\cap V(Q)$ and likewise we have \n$N(w)\\cap V(Q)\\subseteq N_C^2(y)\\cap V(Q)$. Thus, $Q$ does not contains a common neighbor of $v$ and $w$ by Lemma~\\ref{lem:consecutive}. Otherwise, both $P$ and $Q$ have length four and at least one of the two paths satisfy the assumption.\n\nSince the length of $Q$ is at most $\\abs{V(C)}-4$, a shortest $(v,w)$-path $Q'$ in $G[\\{v,w\\}\\cup V(Q)]-vw$ has length at most $\\abs{V(C)}-2$. \nMoreover, $Q'$ has length at least three due to the above assumption. \nIt remains to observe that the closed walk $vQ'w\\odot wv$ is a hole strictly shorter than $C$, a contradiction.\n\\end{proof}\n\nWe prove a generalization of Lemma~\\ref{lem:farnonadj}.\n\\begin{LEM}\\label{lem:generalfarnonadj}\nLet $m$ be a positive integer, and let $P$ be a $V(C)$-path in $G_{deldom}$ with endpoints $x$ and $y$.\nIf $P$ has length at most $m+2$, then $\\dist_C(x,y)\\leq 4m-1$.\n\\end{LEM}\n\\begin{proof}\nWe prove by induction on $m$. Lemma~\\ref{lem:farnonadj} settles the case when $m=1$.\nLet us assume $m\\ge 2$.\nLet $P=p_1p_2 \\cdots p_n$ be a $V(C)$-path of length at most $m+2$ from $p_1=x$ and $p_n=y$ such that \nall of $p_2, \\ldots, p_{n-1}$ are contained in $V(G_{deldom})\\setminus V(C)$, and suppose that $\\dist_C(x,y)\\geq 4m$.\nFor $\\dist_C(x,y)\\ge 4m\\ge 4$, Lemma~\\ref{lem:farnonadj} implies that $p_2$ is not adjacent to $p_{n-1}$.\nTherefore, $P$ contains at least $5$ vertices.\nWe distinguish cases depending on whether $\\{p_3, \\ldots, p_{n-2}\\}$ contains a vertex in $N(C)$ or not.\n\n\\medskip\n\\noindent {\\bf Case 1.} $\\{p_3, \\ldots, p_{n-2}\\}$ contains a vertex in $N(C)$. \\\\\nWe choose an integer $i\\in \\{3, \\ldots, n-2\\}$ such that $p_i\\in N(C)$, and\nchoose a neighbor $z$ of $p_i$ in $C$.\nSince there is a $V(C)$-path from $p_1$ to $z$ of length $i$, by induction hypothesis, $\\dist_C(p_1, z)< 4(i-2)$.\nBy the same reason, we have $\\dist_C(z, p_n)<4(n-i+1-2)=4(n-i-1)$.\nTherefore, we have \n\\[\\dist_C(p_1, p_n)\\le \\dist_C(p_1, z)+\\dist_C(z, p_n)<4(n-3)\\le 4m,\\]\na contradiction.\n\n\\medskip\n\\noindent {\\bf Case 2.} $\\{p_3, \\ldots, p_{n-2}\\}$ contains no vertices in $N(C)$.\n\\\\\nLet $Q$ be a shortest path from $N(p_2)\\cap V(C)$ to $N(p_{n-1})\\cap V(C)$ in $C$, and let $q, q'$ be its endpoints.\nObserve that $p_2Pp_{n-1}$ and $p_2q\\odot qQq' \\odot q'p_{n-1}$ are two paths from $p_2$ to $q'$ where there are no edges between their internal vertices.\nTherefore, $G[V(Q)\\cup (V(P)\\setminus \\{p_1, p_n\\})]$ is a hole.\n\nSince $\\dist_C(x,y)\\le \\frac{\\abs{V(C)}}{2}$, we have $m\\le \\frac{\\abs{V(C)}}{8}$. Therefore, the hole\n$G[V(Q)\\cup (V(P)\\setminus \\{p_1, p_n\\})]$ has length at most \n\\[ \\abs{V(Q)}+\\abs{V(P)}\\le \\frac{\\abs{V(C)}}{2}+ \\frac{\\abs{V(C)}}{8} +1 < \\abs{V(C)}.\\]\nThis contradicts the assumption that $C$ is a shortest hole of $G$.\n\n\\medskip\nThis concludes the proof.\n\\end{proof}\n\nNext, we show that every connected subgraph in $G_{nbd}$ has a connected support.\nThe following observation is useful.\n\n\\begin{LEM}\\label{lem:threeconsecutive}\nLet $a,b$ be vertices of $C$ with $\\dist_C(a,b)\\in \\{2,3\\}$ and \nlet $S$ be the set of internal vertices of the shortest $(a,b)$-path of $C$.\nThen there is no edge between $Z_a\\setminus Z_S$ and $Z_b\\setminus Z_S$.\n\\end{LEM}\n\\begin{proof}\nSuppose there is an edge between $x\\in Z_a\\setminus Z_S$ and $y\\in Z_b\\setminus Z_S$.\nIf $x$ is adjacent to $b$, then $x\\neq a$, and by Lemma~\\ref{lem:consecutive}, $x$ has a neighbor in $S$, contradicting the assumption that $x\\notin Z_S$.\nTherefore, $x$ is not adjacent to $b$.\nFor the same reason, $y$ is not adjacent to $a$.\nTherefore, the distance between $N(x)\\cap V(C)$ and $N(y)\\cap V(C)$ in $C$ is $2$ or $3$, \nand the vertex set of the shortest path from $N(x)\\cap V(C)$ to $N(y)\\cap V(C)$ in $C$ with $\\{x,y\\}$ induces a hole of length $5$ or $6$. \nThis contradicts with the assumption that $C$ is a shortest hole in $G$ and it has length greater than $6$.\n\\end{proof}\n\n\\begin{LEM}\\label{lem:connectedsupport}\nLet $H$ be a connected subgraph in $G_{nbd}$.\nThen $C[\\operatorname{\\textsf{sp}}(H)]$ is connected.\n\\end{LEM}\n\\begin{proof}\nSuppose $\\operatorname{\\textsf{sp}}(H)$ is not connected.\nThen $H$ contains an edge $xy$ such that $\\operatorname{\\textsf{sp}}(G[\\{x\\}])$ and $\\operatorname{\\textsf{sp}}(G[\\{y\\}])$ are contained in distinct components of $C[\\operatorname{\\textsf{sp}}(H)]$.\nNotice that $\\operatorname{\\textsf{sp}}(G[\\{x\\}])=N(x)\\cap V(C)$ and $\\operatorname{\\textsf{sp}}(G[\\{y\\}])=N(y)\\cap V(C)$, and it follows that $x,y\\notin V(C)$ by Lemma~\\ref{lem:consecutive}. \nWe choose $a\\in \\operatorname{\\textsf{sp}}(G[\\{x\\}])$ and $b\\in \\operatorname{\\textsf{sp}}(G[\\{y\\}])$ with minimum $\\dist_C(a,b)$.\nBy Lemma~\\ref{lem:farnonadj}, we have $\\dist_C(a,b)\\le 3$, and since \n$\\operatorname{\\textsf{sp}}(G[\\{x\\}])$ and $\\operatorname{\\textsf{sp}}(G[\\{y\\}])$ are contained in distinct components of $C[\\operatorname{\\textsf{sp}}(H)]$, \nwe have $\\dist_C(a,b)\\ge 2$.\nLet $S$ be the set of internal vertices of the shortest $(a,b)$-path in $C$.\nBy the choice, $x\\in Z_a\\setminus Z_S$ and $y\\in Z_b\\setminus Z_S$.\nThen by Lemma~\\ref{lem:threeconsecutive}, \nthere is no edge between $Z_a\\setminus Z_S$ and $Z_b\\setminus Z_S$.\nThis contradicts the assumption that $x$ is adjacent to $y$.\n\\end{proof}\n\nThe following lemma provides a structure of a $(Z_x, Z_y)$-path in $G_{nbd}$ for two vertices $x,y\\in V(C)$.\n\n\\begin{LEM}\\label{lem:pathsupport}\nLet $x,y$ be two distinct vertices in $C$ and $P_1$, $P_2$ be two $(x,y)$-paths in $C$.\nLet $Q$ be a $(Z_x, Z_y)$-path in $G_{nbd}$ such that $\\operatorname{\\textsf{sp}}(Q)\\neq V(C)$.\nThen either $Q$ is contained in $Z_{V(P_1)}$ or $Z_{V(P_2)}$.\n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:connectedsupport}, \n$\\operatorname{\\textsf{sp}}(Q)$ is connected, and thus $\\operatorname{\\textsf{sp}}(Q)$ contains either $V(P_1)$ or $V(P_2)$.\nWithout loss of generality, we assume that $\\operatorname{\\textsf{sp}}(Q)$ contains $V(P_1)$.\nBy the definition of a $(Z_x, Z_y)$-path, $Q$ contains no vertex of $Z_{\\{x,y\\}}$ as an internal vertex.\nLet $s$ and $t$ be the two endpoints of $Q$ contained in $Z_x$ and $Z_y$, respectively.\n\n\nWe claim that $Q$ contains no vertex of $V(G_{nbd})\\setminus Z_{V(P_1)}$, which immediately implies the statement.\nSuppose for contradiction that $Q$ contains a vertex $v\\in V(G_{nbd})\\setminus Z_{V(P_1)}$. Clearly, we have $v\\neq s$ and $v\\neq t$.\nLet $u$ be a vertex in $C$ such that $v\\in Z_u$. Observe that $u\\neq x$ and $u\\neq y$, as\n$Q$ contains no vertex of $Z_{\\{x,y\\}}$ as an internal vertex.\nLet $Q_s$ and $Q_t$ be the $(s,v)$- and $(t,v)$-subpath of $Q$, respectively.\n\nBy Lemma~\\ref{lem:connectedsupport}, $\\operatorname{\\textsf{sp}}(Q_s)$ contains an $(x,u)$-subpath of $C$.\nAssume $\\operatorname{\\textsf{sp}}(Q_s)$ contains the $(x,u)$-subpath of $C$ containing $y$.\nThis means that $Q_s$ contains a vertex of $Z_y$, other than $t$, contradicting the fact that $Q$ contains no vertex of $Z_{\\{x,y\\}}$ as an internal vertex.\nTherefore, \n$\\operatorname{\\textsf{sp}}(Q_s)$ contains the vertex set of the $(x,u)$-subpath of $C$ avoiding $y$. \nSimilarly, $\\operatorname{\\textsf{sp}}(Q_t)$ contains the vertex set of the $(y,u)$-subpath of $C$ avoiding $x$. \nNow, observe that $\\operatorname{\\textsf{sp}}(Q)=\\operatorname{\\textsf{sp}}(Q_s)\\cup \\operatorname{\\textsf{sp}}(Q_t) \\supseteq V(P_2)$ and also by assumption, we have $V(P_1)\\subseteq \\operatorname{\\textsf{sp}}(Q)$. \nConsequently, we have $\\operatorname{\\textsf{sp}}(Q)=V(C)$, a contradiction. This completes the proof.\n\\end{proof}\n\n\n\n\n\nThe following lemma is useful to find a hole with a small support.\n\\begin{LEM}\\label{lem:overlay}\nLet $P$ and $Q$ be two vertex-disjoint induced paths of $G_{nbd}$ such that\n\\begin{itemize}\n\\item there are no edges between $V(P)$ and $V(Q)$, and \n\\item $\\operatorname{\\textsf{sp}}(P)\\neq V(C)$ and $\\operatorname{\\textsf{sp}}(Q)\\neq V(C)$.\n\\end{itemize}\nIf $\\abs{\\operatorname{\\textsf{sp}}(P)\\cap \\operatorname{\\textsf{sp}}(Q)}\\geq 3$ and $x,y,z\\in \\operatorname{\\textsf{sp}}(P)\\cap \\operatorname{\\textsf{sp}}(Q)$ are three consecutive vertices on $C$, then $Z_{\\{x,y,z\\}}$ contains a hole. \n\\end{LEM}\n\\begin{proof}\nSince $x,y,z\\in \\operatorname{\\textsf{sp}}(P)\\cap \\operatorname{\\textsf{sp}}(Q)$ and there is no edge between $V(P)$ and $V(Q)$, $P$ and $Q$ contains no vertex of $\\{x,y,z\\}$.\nLet $P'$ be a shortest $(Z_x,Z_z)$-subpath of $P$, and \nlet $Q'$ be a shortest $(Z_x,Z_z)$-subpath of $Q$.\nAs $\\operatorname{\\textsf{sp}}(P)\\neq V(C)$ and $\\operatorname{\\textsf{sp}}(Q)\\neq V(C)$, \n$V(P')$ and $V(Q')$ are contained in $Z_{\\{x,y,z\\}}$ by Lemma~\\ref{lem:pathsupport}.\nBy the preconditions, $P'$ and $Q'$ are vertex-disjoint and there are no edges between $P'$ and $Q'$. \nThus, by Lemma~\\ref{lem:twopaths}, $G[V(P')\\cup V(Q')\\cup \\{x,z\\}]$ contains a hole, which is in $Z_{\\{x,y,z\\}}$.\n\\end{proof}\n\n\\subsection{$C$-dominating vertices}\\label{subsec:dominating}\n\nWe recall that $D$ is the set of $C$-dominating vertices.\nWe observe that $D$ is a clique because $G$ does not contain a hole of length 4.\n\n\\begin{LEM}\\label{lem:dominating}\nThe set $D$ is a clique. Furthermore, every hole contains at most one vertex of $D$.\n\\end{LEM}\n\\begin{proof}\nNote that $G$ contains no hole of length $4$. This implies that any two vertices of $D$ are adjacent, which proves the first statement. To see the second statement, suppose that $H$ is a hole containing two distinct vertices $u,v$ of $D$ and let $x\\in V(H)\\cap V(C)$ (there are no holes in $G-V(C)$). Then $\\{x,u,v\\}$ forms a triangle, contradicting the assumption that $H$ is a hole. \n\\end{proof}\n\n\\begin{LEM}\\label{lem:neighborofdominating}\nIf $H$ is a $D$-traversing hole, then it contains at most two vertices of $C$. Furthermore, every vertex of $V(H)\\cap V(C)$ is adjacent to the unique $C$-dominating vertex on $H$. \n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:dominating}, $H$ contains exactly one vertex of $D$, say $v$. \nIf there is a vertex $x\\in V(C)\\cap V(H)$, then $x$ is adjacent to $v$ as $v$ is $C$-dominating. Therefore, any vertex of $V(C)\\cap V(H)$ is adjacent to $v$ on $H$. Since $H$ is a cycle, $H$ contains at most two vertices of $C$.\n\\end{proof}\n\n\n\n\n\n\\section{Hitting all sunflowers}\\label{sec:hittingsunflower}\n\nIn this section, we obtain a hitting set for sunflowers, unless $G$ contains $k+1$ vertex-disjoint hols.\nLike in the previous section, we assume that $(G,k,C)$ is given as an input such that $C$ is a shortest hole of $G$ of length strictly greater than $\\mu_k$, \nand $G-V(C)$ is chordal.\n\n\n\\subsection{Hitting all petals.} \\label{subsec:petal}\n\n\\begin{LEM}\\label{lem:petalcover}\nThere is a polynomial-time algorithm which finds either $k+1$ vertex-disjoint holes in $G$ or a vertex set $T_{petal}\\subseteq V(C)$ of at most $19k$ vertices such that \n\\begin{itemize}\n\\item for every petal $H$, we have $\\operatorname{\\textsf{sp}}(H)\\subseteq T_{petal}$.\n\\end{itemize} \n\\end{LEM}\n\\begin{proof}\nSet $X:=\\emptyset$, $\\mathcal{C}=\\emptyset$, and $counter:=0$ at the beginning. We recursively do the following until the counter reaches $k+1$. For every set of nine consecutive vertices $v_0, v_1, v_2, \\ldots, v_7, v_8$ of $C$ with $\\{v_1, v_2, \\ldots, v_7\\}\\cap X=\\emptyset$, we test if $G[Z_{\\{v_1, v_2, \\ldots, v_7\\}}\\setminus Z_{\\{v_0, v_8\\}}]$ contains a hole $H$, and if so, add vertices in $\\{v_1, v_2, \\ldots, v_7\\}$ to $X$, and add $H$ to $\\mathcal{C}$ and increase the counter by 1. If the counter reaches $k+1$, then we stop. \nIf the counter does not reach $k+1$, then\nwe have $\\abs{X}\\leq 7k$. In this case, we set $T_{petal}$ as the $6$-neighborhood of $X$ in $C$.\n\nBy construction, any hole $H \\in \\mathcal{C}$ has a support that is fully contained in the considered set $\\{v_1, v_2, \\ldots, v_7\\}$. \nObserve that we choose this set to be disjoint from $X$ constructed thus far. \nTherefore, holes in $\\mathcal{C}$ are pairwise vertex-disjoint; otherwise, their supports have a common vertex. \nThis implies that if the counter reaches $k+1$, then we can output $k+1$ vertex-disjoint holes.\n\nAssume the counter does not reach $k+1$. In this case, we claim that for every petal $H$, $\\operatorname{\\textsf{sp}}(H)\\subseteq T_{petal}$.\nLet $H$ be a petal. By the definition of a petal, there is a set of $7$ consecutive vertices $w_1, w_2, \\ldots, w_7$ in \n$C$ such that $\\operatorname{\\textsf{sp}}(H)\\subseteq \\{w_1, w_2, \\ldots, w_7\\}$. If the set $\\{w_1, w_2, \\ldots, w_7\\}$ is disjoint from $X$, then the above procedure must have considered this set and added it to $X$, a contradiction. \nTherefore, $\\{w_1, w_2, \\ldots, w_7\\}\\cap X\\neq \\emptyset$. Then during the step of adding 6-neighborhood of $X$ to $T_{petal}$, \n$\\{w_1, w_2, \\ldots, w_7\\}$ is added to $T_{petal}$, and thus we have $\\operatorname{\\textsf{sp}}(H)\\subseteq \\{w_1, w_2, \\ldots, w_7\\} \\subseteq T_{petal}$ as claimed.\n\\end{proof}\n\nIn what follows, we reserve $T_{petal}$ to denote a vertex subset of $V(C)$ that contains the support of every petal. \n\n\\subsection{Polarization of $D$-avoiding sunflowers.}\\label{subsec:allisfull}\n\n\nWe show that every $D$-avoiding sunflower in $G-T_{petal}$ is full. \nThis will imply that, in order to hit every $D$-avoiding sunflower it is sufficient to find a hitting set for full sunflowers.\n\n\n\n\\begin{LEM}\\label{lem:sppdichotomy}\nEvery $D$-avoiding sunflower $H$ in $G-T_{petal}$ is full, that is, $\\operatorname{\\textsf{sp}}(H)=V(C)$. \n\\end{LEM}\n\\begin{proof}\nSuppose $H$ is a $D$-avoiding sunflower in $G-T_{petal}$ such that $8\\le \\abs{\\operatorname{\\textsf{sp}}(H)}< \\abs{V(C)}$. \nBy Lemma~\\ref{lem:connectedsupport}, $\\operatorname{\\textsf{sp}}(H)$ is a subpath of $C$. \nLet $\\operatorname{\\textsf{sp}}(H)=v_1v_2 \\cdots v_{\\ell}$. Choose $x\\in V(H)\\cap Z_{v_1}$ and $y\\in V(H)\\cap Z_{v_{\\ell}}$, and let $P$ and $Q$ be the two $(x,y)$-paths on $H$. \nAs each of $C[\\operatorname{\\textsf{sp}}(P)]$ and $C[\\operatorname{\\textsf{sp}}(Q)]$ is connected by Lemma~\\ref{lem:connectedsupport}, \nwe have $\\operatorname{\\textsf{sp}}(P)=\\operatorname{\\textsf{sp}}(Q)= \\operatorname{\\textsf{sp}}(H)$.\n\n\n\nRecall that $V(H)\\cap V(C)\\neq \\emptyset$ and $V(H)\\cap V(C)$ must be contained in $\\operatorname{\\textsf{sp}}(H)$. We argue that any $v_i\\in \\operatorname{\\textsf{sp}}(H)$ with $i\\in \\{4, 5, \\ldots, \\ell-3\\}$ does not lie on $H$. Suppose $v_i\\in V(H)\\cap V(C)$ for some $4\\leq i\\leq \\ell-3$. Notice that both $x$ and $y$ are distinct from $v_i$. Therefore, $v_i$ belongs to exactly one of $P$ and $Q$. Without loss of generality, we assume $v_i\\in V(P)$. Since $v_i\\in \\operatorname{\\textsf{sp}}(Q)$, $Z_{v_i}\\cap V(Q)\\neq \\emptyset$ and thus we can choose a vertex $v'_i$ from the set $Z_{v_i}\\cap V(Q)$. Lemma~\\ref{lem:consecutive} and $v'_i\\notin D$ imply that $v'_i$ is not adjacent to $v_1$ or $v_{\\ell}$, and thus $v'_i\\notin Z_{v_1}$ and $v'_i\\notin Z_{v_{\\ell}}$. This means that $v'_i$ is distinct from $x$ and $y$, especially $v'_i$ is an internal vertex of $Q$. However, $v_iv'_i\\in E(G)$ is a chord of $H$, a contradiction. \n\nTherefore, $v_i \\notin V(H)$ for every $4\\leq i\\leq \\ell-3$. At least one of $\\{v_1,v_2,v_3\\}$ and $\\{v_{\\ell-2},v_{\\ell-1},v_{\\ell}\\}$ intersects with $V(H)$, and we assume that $\\{v_1,v_2,v_3\\}$ intersects with $V(H)$ without loss of generality (a symmetric argument works in the other case). From each of $P$ and $Q$, choose the first vertex (starting from $x$) that lies in $Z_{v_4}$ and call them $p\\in V(P)$ and $q\\in V(Q)$ respectively; the existence of such vertices follows from $v_4\\in \\operatorname{\\textsf{sp}}(P)=\\operatorname{\\textsf{sp}}(Q)$. Let $H'$ be the cycle $pPx\\odot xQq\\odot qv_4p$.\n\nSince $p$, $q$ are the first vertices contained in $Z_{v_4}$, \n$v_4$ has no neighbors in $(V(xPp)\\cup V(xQq))\\setminus \\{p, q\\}$, and thus $H'$ is a hole.\nBy Lemma~\\ref{lem:consecutive}, we have $\\operatorname{\\textsf{sp}}(H')\\subseteq \\{v_1, v_2, \\ldots, v_6\\}$, i.e. $H'$ is a petal. \nHowever, $\\{v_1,v_2,v_3\\}\\cap V(H)\\neq \\emptyset$ and $V(H)\\cap T_{petal}=\\emptyset$ implies $\\{v_1,v_2,v_3\\}\\setminus T_{petal}\\neq \\emptyset$. This contradicts \nthe assumption $T_{petal}$ contains the support of every petal. This completes the proof. \n\\end{proof}\n\n\n\\subsection{Hitting all $D$-avoiding sunflowers.}\\label{subsec:sunflowerwithout}\nIn this subsection we focus on full sunflowers.\n\n\\begin{PROP}\\label{prop:hitsunflower}\nThere is a polynomial-time algorithm which finds either $k+1$ vertex-disjoint holes in $G$ or a vertex set $T_{full}\\subseteq V(G)\\setminus T_{petal}$ of at most $3k+14$ vertices such that $T_{petal}\\cup T_{full}$ hits all full sunflowers. \n\\end{PROP}\n\nOur strategy is to find two collections of many vertex-disjoint paths so that we can link the paths to obtain many vertex-disjoint holes.\nThe following lemma explains how to do this.\nNote that since $C$ has length greater than $\\mu_k$, $V(C)\\setminus T_{petal}$ contains $25$ vertices that are consecutive in $C$.\n\n\\begin{LEM}\\label{lem:connectingpaths}\nLet $v_{-2}v_1v_0v_1 \\cdots v_{20}v_{21}v_{22}$ be a subpath of $C$ that does not intersect $T_{petal}$, \nand $X$ be the $(v_0, v_{20})$-subpath of $C$ containing $v_1$ \nand $Y$ be the $(v_5, v_{15})$-subpath of $C$ containing $v_4$.\nLet $\\mathcal{P}$ be a collection of vertex-disjoint $(Z_{v_0},Z_{v_{20}})$-paths in $G[Z_{V(X)}]$, and \nlet $\\mathcal{Q}$ be a collection of vertex-disjoint $(Z_{v_5},Z_{v_{15}})$-paths in $G[Z_{V(Y)}]$. \nGiven such $\\mathcal{P}$ and $\\mathcal{Q}$, if $\\abs{\\mathcal{P}}\\ge k+13$ and $\\abs{\\mathcal{Q}}\\ge 3k+15$, then one can output $k+1$ vertex-disjoint holes in polynomial time.\n\\end{LEM}\n\\begin{proof}\nWe begin with the observation that there is no petal whose support contains a vertex of $\\{v_{-2}, v_{-1}, \\ldots, v_{22}\\}$. \nThis is because $\\{v_{-2}, v_{-1}, \\ldots, v_{22}\\}\\cap T_{petal}=\\emptyset$ by assumption and \n$T_{petal}$ contains the support of every petal of $G$.\nWe may assume that every path in $\\mathcal{P}$ is induced, \nand similarly every path in $\\mathcal{Q}$ is induced.\n\nWe take a subset $\\mathcal{P}_1$ of $\\mathcal{P}$ with $\\abs{\\mathcal{P}_1}=k+1$ that consists of paths containing no vertices of $\\{v_0, v_1, \\ldots, v_5\\}\\cup \\{v_{15}, v_{16}, \\ldots, v_{20}\\}$. Such a collection $\\mathcal{P}_1$ exists because the paths of $\\mathcal{P}$ are vertex-disjoint and at most 12 of them \nintersect with $\\{v_0, v_1, \\ldots, v_5\\}\\cup \\{v_{15}, v_{16}, \\ldots, v_{20}\\}$. \nSimilarly we take a subset $\\mathcal{Q}_1$ of $\\mathcal{Q}$ with $\\abs{\\mathcal{Q}_1}=3k+3$ \nthat consists of paths containing no vertices of $\\{v_0, v_1, \\ldots, v_5\\} \\cup \\{v_{15}, v_{16}, \\ldots, v_{20}\\}$. \n\nFor each $P\\in \\mathcal{P}_1$, let $\\ell(P)$ and $r(P)$ be the endpoints of $P$ contained in $Z_{v_0}$ and $Z_{v_{20}}$, respectively.\nFor each $Q\\in \\mathcal{Q}_1$, let $a(Q)$ be the vertex of $Z_{v_0}\\cap V(Q)$ that is closest to $Z_{v_5}\\cap V(Q)$ in $Q$, and \nlet $b(Q)$ be the vertex in $Z_{v_{20}}$ that is closest to $Z_{v_{15}}\\cap V(Q)$ in $Q$.\nBy definition, no internal vertex of the subpath of $Q$ from $a(Q)$ to the vertex in $V(Q)\\cap Z_{v_5}$ contains a neighbor of $v_0$, and \nsimilarly no internal vertex of the subpath of $Q$ from $b(Q)$ to the vertex in $V(Q)\\cap Z_{v_{15}}$ contains a neighbor of $v_{20}$. See Figure~\\ref{fig:pathsystems} for an illustration.\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(5,0);\\draw[thick, dotted](5,0)--(6,0);\n \\draw (6,0)--(13,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(13, 0)--(14,-0.5);\n \\draw (-3,-.5) node [w] {};\n \\draw (14,-.5) node [w] {};\n \t\\draw[dashed](15, -1)--(14,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n \n \\foreach \\y in {-2,...,13}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n\n \\foreach \\y in {-1, 4, 7, 12}{\n\\draw[dashed, rounded corners] (\\y, 3.3)--(\\y+.5, 3.3)--(\\y, -.5)--(\\y-.5, 3.3)--(\\y, 3.3);\n}\n\n \\node at (-1, 4) {$Z_{v_0}$};\n \\node at (4, 4) {$Z_{v_5}$};\n \\node at (7, 4) {$Z_{v_{15}}$};\n \\node at (12, 4) {$Z_{v_{20}}$};\n \\node at (-1, -.8) {$v_0$};\n \\node at (4, -.8) {$v_5$};\n \\node at (7, -.8) {$v_{15}$};\n \\node at (12, -.8) {$v_{20}$};\n\n \\node at (-2, -1) {$C$};\n\n \\node at (1.5, 2.8) {$P$};\n \\node at (1.5, 1.6) {$Q$};\n\n \\node at (-2, 2.4) {$\\ell(P)$};\n \\node at (13, 2.4) {$r(P)$};\n\n\t \\node at (-1.8, 0.8) {$a(Q)$};\n \\node at (12.8, 0.8) {$b(Q)$};\n\n\t \\draw (-1,2.4)--(12,2.4);\n \\draw[rounded corners] (4,1.2)--(-2,1.2)--(-3,0.7);\\draw[dashed] (-3, 0.7)--(-4,0.2);\n \\draw[rounded corners] (7,1.2)--(13,1.2)--(14,0.7);\\draw[dashed] (14, 0.7)--(15,0.2);\n \\draw (-1,2.4) node [w] () {};\n \\draw (4,1.2) node [w] () {};\n \\draw (7,1.2) node [w] () {};\n \\draw (12,2.4) node [w] () {};\n \\draw (-1,1.2) node [w] () {};\n \\draw (12,1.2) node [w] () {};\n\n \\end{tikzpicture} \\caption{Paths $P\\in \\mathcal{P}$ and $Q\\in \\mathcal{Q}$ in Lemma~\\ref{lem:connectingpaths}. The vertices $\\ell(P)$ and $a(Q)$ are completely adjacent, otherwise, we can find a hole $Z_{\\{v_0, v_1, v_2\\}}$, which is a petal. For the same reason, $r(P)$ is completely adjacent to $b(Q)$.}\\label{fig:pathsystems}\n\\end{figure}\n\n\n\n\n\\begin{CLAIM}\\label{claim:ef}\nLet $w\\in \\{\\ell(P):P\\in \\mathcal{P}_1\\}$ and $z\\in \\{a(Q):Q\\in \\mathcal{Q}_1\\}$. If $w\\neq z$, then $wz\\in E(G)$.\nLet $w\\in \\{r(P):P\\in \\mathcal{P}_1\\}$ and $z\\in \\{b(Q):Q\\in \\mathcal{Q}_1\\}$. If $w\\neq z$, then $wz\\in E(G)$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose $w\\in \\{\\ell(P):P\\in \\mathcal{P}_1\\}$ and $z\\in \\{a(Q):Q\\in \\mathcal{Q}_1\\}$ such that $w\\neq z$ and they are not adjacent. \nLet $P_w\\in \\mathcal{P}_1$ and $Q_z\\in \\mathcal{Q}_1$ such that \n$\\ell(P_w)=w$ and $a(Q_z)=z$. Let $Q_z'$ be the subpath of $Q_z$ from $z$ to the vertex in $Z_{v_5}$.\nNote that $v_0$ has no neighbors in $V(P_w)\\setminus \\{w\\}$ and $V(Q_z')\\setminus \\{z\\}$.\nIn case when $P_w$ and $Q_z'$ meet somewhere in $\\bigcup_{i\\in \\{1, 2\\}}Z_{v_i}$, \nwe obtain a hole contained in $Z_{\\{v_0, v_1, v_2\\}}$ by Lemma~\\ref{lem:twopaths}.\nWhen $P_w$ and $Q_z'$ do not meet in $\\bigcup_{i\\in \\{1, 2\\}}Z_{v_i}$, \nthere is a hole contained in $Z_{\\{v_0, v_1, v_2\\}}$ by Lemma~\\ref{lem:twopaths} since $v_2$ has a neighbor in both $P_w$ and $Q_z'$. \nIn both cases, there is a petal with support contained in $\\{v_{i}: -2\\le i\\le 4\\}$, a contradiction. \nWe conclude that $wz\\in E(G)$.\nThe proof of the latter statement is symmetric.\n\\end{proofofclaim}\n\nFor every $P\\in \\mathcal{P}_1$, $\\ell(P)$ is the unique vertex of $Z_{v_0}\\cap V(P)$. Therefore, for fixed $P\\in \\mathcal{P}_1$,\nthere is at most one path $Q\\in \\mathcal{Q}_1$ such that $V(Q)\\cap V(P)\\cap Z_{v_0}\\neq \\emptyset$. \nSimilarly, there is at most one path of $\\mathcal{Q}_1$ intersecting with $P$ at a vertex of $Z_{v_{20}}$. \nWe construct a new collection $\\mathcal{Q}_2$ so that\n\\begin{quote}\nfor every $Q\\in \\mathcal{Q}_1$, $\\mathcal{Q}_2$ contains the subpath $a(Q)Qb(Q)$ if and only if \n$Q$ does not intersect with any $P\\in \\mathcal{P}_1$ at a vertex of $Z_{v_0}\\cup Z_{v_{20}}$.\n\\end{quote}\nObserve that $\\mathcal{Q}_2$ contains at least $k+1$ paths because each path of $\\mathcal{P}_1$ can make \nat most two paths of $\\mathcal{Q}_1$ drop out. For our purpose, taking precisely $k+1$ paths is sufficient. \nLet $\\mathcal{P}_1=\\{P_1,\\ldots , P_{k+1}\\}$ and $\\mathcal{Q}_2=\\{Q_1, Q_2, \\ldots, Q_{k+1}\\}$.\nFor each $i\\in \\{1, 2, \\ldots, k+1\\}$, we create a cycle $C_i$ from the disjoint union of $P_i\\in \\mathcal{P}_1$ and $Q_i\\in \\mathcal{Q}_2$ \nby adding two edges $a(Q_i)\\ell(P_i)$ and $b(Q_i)r(P_i)$. \nSuch edges exist by Claim~\\ref{claim:ef}.\n\nWe observe that each $C_i$ contains a hole. To see this, take $v\\in Z_{v_{10}}\\cap V(P_i)$. \nAs $Q_i\\in \\mathcal{Q}_2$ is a path of $G[Z_{V(Y)}]$, Lemma~\\ref{lem:consecutive} implies that $v$ is not adjacent to any vertex of $Q_i$. \nNote that $v$ is an internal vertex of \nthe induced path $P_i$. Therefore, $G[V(C_i)]$ contains a hole by Lemma~\\ref{lem:twopaths}. \n\nLastly, we verify that two holes contained in distinct cycles of $\\{C_i:1\\le i\\le k+1\\}$ are vertex-disjoint. To prove this, it is sufficient to show that \nfor two integers $a,b\\in \\{1, 2, \\ldots, k+1\\}$, no internal vertex of $P_a\\in \\mathcal{P}_2$ is an internal vertex of $Q_b\\in \\mathcal{Q}_2$. \nSuppose the contrary, that is, $w$ is an internal vertex of $P_a\\in \\mathcal{P}_2$ and $Q_b\\in \\mathcal{Q}_2$ simultaneously for some $a,b$.\nSince $Q_b$ is a path of $G[Z_{Y}]$ and $w\\notin Z_{v_0}\\cup Z_{v_{20}}$ is an internal vertex of $P_a$, we have $w\\in Z_{\\{v_1, v_2, v_3, v_4,v_5\\}}\\cup Z_{\\{v_{15}, v_{16}, v_{17}, v_{18},v_{19}\\}}$. \nWithout loss of generality, we assume $w\\in Z_{\\{v_1, v_2, v_3, v_4,v_5\\}}$. \nIn fact, $w$ cannot be in $Z_{v_5}$ since otherwise, the path $Q'_b\\in \\mathcal{Q}_1$ having $Q_b$ as a proper subpath contains a vertex of $Z_{v_5}$ as an internal vertex; \nviolating the definition of $(Z_{v_5},Z_{v_{15}})$-path. \nNow observe that $Q_b$ contains, as a subpath, a $Z_{v_0}$-path $Q'$ having all internal vertices in $Z_{\\{v_1, v_2, v_3, v_4\\}}\\setminus Z_{\\{v_0, v_5\\}}$. \nLet $x,y$ be the endpoints of $Q'$. Since $v_0$ is adjacent to $x,y$ but is not adjacent to any internal vertices of $Q'$,\n$G[\\{v_0\\}\\cup V(Q')]$ contains a hole by Lemma~\\ref{lem:twopaths}, a contradiction. A symmetric argument holds for the case \n$w\\in Z_{\\{v_{15}, v_{16}, v_{17}, v_{18},v_{19}\\}}$. Therefore, $\\{C_i:1\\le i\\le k+1\\}$ \nis a vertex-disjoint holes of $G$, which completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:hitsunflower}]\nNote that $\\abs{T_{petal}}\\le 19k$ and $\\abs{V(C)}>\\mu_k> 25\\abs{T_{petal}}$ by assumption. Thus, there are $25$ consecutive vertices on $C$ having no vertices in $T_{petal}$.\nWe choose a subpath $v_{-2}v_{-1}v_0v_1 \\cdots v_{20}v_{21}v_{22}$ of $C$ that contains no vertices in $T_{petal}$.\n Let $P_1$ be the $(v_0, v_{20})$-subpath of $C$ containing $v_1$ \n and $P_2$ be the $(v_5, v_{15})$-subpath of $C$ containing $v_4$.\n\nWe apply Menger's Theorem for $(Z_{v_0},Z_{v_{20}})$-paths in $G[Z_{V(P_1)}]$,\nand then for $(Z_{v_5},Z_{v_{15}})$-paths in $G[Z_{V(P_2)}]$. \nWe have one of the following.\n\\begin{itemize}\n\\item The first application of Menger's Theorem outputs a vertex set $X$ with $\\abs{X}\\leq k+12$ hitting all $(Z_{v_0},Z_{v_{20}})$-paths in $G[Z_{V(P_1)}]$.\n\\item The second application of Menger's Theorem outputs a vertex set $X$ with $\\abs{X}\\leq 3k+14$ hitting all $(Z_{v_5},Z_{v_{15}})$-paths in $G[Z_{V(P_2)}]$.\n\\item The first algorithm outputs at least $k+13$ vertex-disjoint paths, and the second algorithm outputs at least $3k+15$ vertex-disjoint paths. \n\\end{itemize} \n\nIn the third case, by Lemma~\\ref{lem:connectingpaths}, we can construct $k+1$ vertex-disjoint holes in polynomial time.\n\nSuppose we obtained a vertex set $X$ in the first case. We claim that $T_{petal}\\cup X$ hits all full sunflowers. Suppose that there is a full sunflower $H$ avoiding every vertex of $T_{petal}\\cup X$. By definition, $\\operatorname{\\textsf{sp}} (H)=V(C)$. In particular, $H$ contains at least one vertex of $Z_{v_{10}}$, say $w$. \nLet $F$ be the connected component of the restriction of $H$ on $G[Z_{V(P_1)}]$ containing $w$. \nClearly $F$ is a path. \nWe argue that its endpoints are contained in $Z_{\\{v_0, v_{20}\\}}$ because of Lemma~\\ref{lem:connectedsupport}.\n\nSuppose that the endpoints of $F$ are contained in distinct sets of $Z_{v_0}$ and $Z_{v_{20}}$, respectively. \nLet $F'$ be a subpath of $F$ that is a $(Z_{v_0}, Z_{v_{20}})$-path. Note that $F'$ is a $(Z_{v_0}, Z_{v_{20}})$-path of $G[Z_{V(P_1)}]$ \nbecause $F$ is a path of $G[Z_{V(P_1)}]$.\nBut it contradicts with the fact that $X$ hits all such paths. \n\nSuppose that both endpoints of $F$ are contained in one of $Z_{v_0}$ or $Z_{v_{20}}$, say $Z_{v_0}$.\nLet $F_1$ and $F_2$ be the two subpaths of $F$ from $w$ to its endpoints.\nThen by Lemma~\\ref{lem:connectedsupport}, both $\\operatorname{\\textsf{sp}}(F_1)$ and $\\operatorname{\\textsf{sp}}(F_2)$ contain the $(v_0, v_{10})$-subpath of $P_1=v_0v_1 \\cdots v_{20}$. \nThis implies that $\\operatorname{\\textsf{sp}}(F_1-w)\\cap \\operatorname{\\textsf{sp}}(F_2-w)$ contains $\\{v_0, v_1, v_2\\}$.\nSince there are no edges between $F_1-w$ and $F_2-w$, Lemma~\\ref{lem:overlay} implies that \nthere is a hole contained in $Z_{\\{v_0, v_1, v_2\\}}$.\nThis is a contradiction because we assumed $\\{v_{-2}, v_{-1}, \\ldots, v_{22}\\}\\cap T_{petal}=\\emptyset$ while \n$T_{petal}$ contains the support of every petal of $G$.\nTherefore, $T_{petal}\\cup X$ hits every full sunflower. The case when both endpoints of $F$ are contained on $Z_{v_{20}}$ \nfollows from a symmetric argument.\n\nThe second case when we obtain the vertex set $X$ with $\\abs{X}\\leq 3k+14$ can be handled similarly. Hence, in the first or second case, we can output a required vertex set $T_{full}$ of size at most $3k+14$ hitting every full sunflower in polynomial time.\n\\end{proof}\n\n\n\\subsection{Hitting all $D$-traversing sunflowers.}\\label{subsec:sunflowerwith}\n\nOur proof builds on the observation that any $D$-traversing sunflower entails another $D$-traversing sunflower $H'$ where the support of the path $H'- D$ is `small'. Then we exploit the min-max duality of vertex cover and matching on bipartite graphs\nin order to cover such $D$-traversing sunflowers with small support.\n\nThe following lemma describes how to obtain such a sunflower $H'$. \nWe depict the setting of Lemma~\\ref{lem:smallsunflower} in Figure~\\ref{fig:traversingsunflower}.\n\n\n\\begin{LEM}\\label{lem:smallsunflower}\nLet $v_1v_2 \\cdots v_5$ be a subpath of $C$ such that $\\{v_1,\\ldots , v_5\\}\\cap T_{petal}=\\emptyset$ and let $P=p_1p_2 \\cdots p_m$ be a path in $G[D\\cup Z_{\\{v_1, v_2, \\ldots, v_5\\}}]$ such that\n\\begin{enumerate}[(i)]\n\\item $p_1$ is a $C$-dominating vertex and $p_2=v_3$, \n\\item $p_m\\in Z_{\\{v_1, v_5\\}}\\setminus \\{v_1, v_5\\}$, \n\\item all internal vertices of $P$ are in $Z_{\\{v_2, v_3, v_4\\}}\\setminus Z_{\\{v_1, v_5\\}}$, and \n\\item $E(G[V(P)])\\setminus E(P)\\subseteq \\{p_1p_m\\}$; that is, $G[V(P)]$ is either an induced path $p_1p_2 \\cdots p_m$ or an induced cycle $p_1p_2 \\cdots p_mp_1$, \n\\item if $G[V(P)]$ is an induced cycle, then $m\\ge 4$.\n\\end{enumerate}\nThen there exists a $D$-traversing sunflower $H$ containing $p_1$ and $p_2$ such that $V(H)\\setminus \\{p_1\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}} \\cap (V(P)\\cup \\{v_1\\})$\n or $ V(H)\\setminus \\{p_1\\} \\subseteq Z_{\\{v_3, v_4, v_5\\}}\\cap (V(P)\\cup \\{v_5\\})$. \n\\end{LEM}\n\\begin{proof}\nWe claim the following:\n\\begin{quote}\nIf $p_m\\in Z_{v_1}$, then $P-p_1$ is contained in $Z_{\\{v_1, v_2, v_3\\}}$. Likewise, \nif $p_m\\in Z_{v_5}$, then $P-p_1$ is contained in $Z_{\\{v_3, v_4, v_5\\}}$.\n\\end{quote} \n\nWe only prove the first statement; the proof of the second statement will be symmetric.\nLet us assume $p_m\\in Z_{v_1}$. We observe that $P$ contains no vertex in $Z_{v_5}$ because \nall internal vertices of $P$ are in $Z_{\\{v_2, v_3, v_4\\}}\\setminus Z_{\\{v_1, v_5\\}}$.\n\nWe first show that $P$ contains no vertex in $Z_{v_4}\\setminus Z_{v_3}$.\nSuppose the contrary and let $w\\in V(P)\\cap (Z_{v_4}\\setminus Z_{v_3})$. \nSince both $C[\\operatorname{\\textsf{sp}}(p_2Pw)]$ and $C[\\operatorname{\\textsf{sp}}(wPp_m)]$ are connected by Lemma~\\ref{lem:connectedsupport}, \n$P$ contains a $Z_{v_3}$-subpath $P'$ whose internal vertices are all contained in $Z_{v_4}\\setminus Z_{v_3}$.\nThen $v_3$ is not adjacent to any internal vertex of $P'$ by Lemma~\\ref{lem:twopaths}, and thus $G[V(P')\\cup \\{v_3\\}]$ contains a hole, which is a petal.\nThis contradicts the fact that $v_3\\notin T_{petal}$, because by the construction of $T_{petal}$ in Lemma~\\ref{lem:petalcover}, $T_{petal}$ fully contains the support of every petal.\nHence, $P$ contains no vertex of $Z_{v_4}\\setminus Z_{v_3}$ and we have $V(P)\\setminus \\{p_1\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}}$.\n\n\nWe claim that there is a $D$-traversing sunflower as claimed. \nIf $p_1p_m\\in E(G)$, then $G[V(P)]$ is a hole as claimed by (iv) and due to the previous claim.\nHence, we may assume $p_1$ is not adjacent to $p_m$.\nObserve that $v_1p_1p_2$ is an induced path.\nAlso, $G[(V(P)\\setminus \\{p_1\\})\\cup \\{v_1\\}]$ is a path from $v_1$ to $v_3$, and it does not contain $v_2$. Indeed, \nif the path $G[(V(P)\\setminus \\{p_1\\})\\cup \\{v_1\\}]$ contains $v_2$, then $\\{p_1,p_2=v_3,v_2\\}$ forms a triangle, a contradiction to (iv). \nNow, $p_1$ has no neighbors in $V(P)\\setminus \\{p_1, p_2\\}$ as we assumed that $p_1$ is not adjacent to $p_m$.\nTherefore, we can apply Lemma~\\ref{lem:twopaths} with the induced path $v_1p_1p_2$ with $p_1$ as an internal vertex and \nthe path $p_2Pp_m\\odot p_mv_1$. It follows that there is a hole in $G[V(P)\\cup \\{v_1\\}]$ containing $p_1$, $p_2$ \nsuch that $V(H)\\setminus \\{p_1\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}}$. The statement follows immediately.\n\\end{proof}\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw (-2,0)--(11,0);\n\t\\draw(-2, 0)--(-3,-0.5);\n\t\\draw(11, 0)--(12,-0.5);\n \\draw[dashed](13, -1)--(12,-0.5);\n\t\\draw[dashed](-4,-1)--(-3,-0.5);\n\t\n\t\n \n \\draw(5,3.5) [in=80,out=150] to (.5,0);\n \\draw(.5,0)--(.5,1.5);\n \\node at (0, 1.5) {$p_m$};\n \n \t\\draw (5, 3.5)--(4.5,0)--(4, 2)--(3,2.7)--(2, 2)--(1.5, 0.5)--(.5, 1.5);\n \\draw (4,2) node [w] {};\n \\draw (3,2.7) node [w] {};\n \\draw (2,2) node [w] {};\n \\draw (1.5,.5) node [w] {};\n \\draw (.5,1.5) node [w] {};\n\n \\node at (3, 1.5) {$P$};\n \n \\foreach \\y in {-1.5,0.5, ..., 11}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n \\node at (0.5, -.8) {$v_1$};\n \\node at (2.5, -.8) {$v_2$};\n \\node at (4.5, -.8) {$v_3=p_2$};\n \\node at (6.5, -.8) {$v_4$};\n \\node at (8.5, -.8) {$v_5$};\n\n \\foreach \\y in {0.5, 8.5}{\n\\draw[dashed, rounded corners] (\\y, 3.3)--(\\y+.5, 3.3)--(\\y, -.5)--(\\y-.5, 3.3)--(\\y, 3.3);\n}\n\n \\node at (0.5, 4) {$Z_{v_1}$};\n \\node at (8.5, 4) {$Z_{v_5}$};\n\n \n\\draw[rounded corners] (-3-.5+7,4)--(-3-.5+7,3)--(0-.5+7,3)--(0-.5+7,5)--(-3-.5+7,5)--(-3-.5+7,4);\n \\draw (-2+7, 3.5) node [w] {};\n \\node at (5.5, 3.5) {$p_1$};\n\n\n \\node at (-1.5-.5+7, 4.5) {$D$};\n \n \\end{tikzpicture} \\caption{Obtaining another sunflower in Lemma~\\ref{lem:smallsunflower}.\n As $v_1p_1p_2$ is an induced path and $p_1$ has no neighbors in the set of internal vertices of $p_2Pp_m\\odot p_mv_1$, $G[V(P)\\cup \\{v_1\\}]$ contains a hole.}\\label{fig:traversingsunflower}\n\\end{figure}\n\n\\begin{LEM}\\label{lem:small}\nLet $H$ be a $D$-traversing sunflower in $G-T_{petal}$ containing a $C$-dominating vertex $d$. Then there exist three consecutive vertices $x,y,z$ on $C$ and a $D$-traversing sunflower $H'$ containing $d$ such that $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$ and $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{x,y,z\\}}$. \n\\end{LEM}\n\\begin{proof}\nBy Lemma~\\ref{lem:neighborofdominating}, $H$ contains at most $2$ vertices of $C$, and every vertex in $V(H)\\cap V(C)$ is neighboring the \n(unique) vertex of $V(H)\\cap D$ on $H$.\nLet $P$ be the connected component of $H-(V(C)\\cup \\{d\\})$.\nNote that $P$ contains no vertices of $D$.\nLet $a\\in V(H)\\cap V(C)$.\nIf the support of $P$ is contained in $N_C[a]$, then we are done as $\\abs{N_C[a]}\\le 3$ and we can take $H'=H$ and $\\{x,y, z\\}=N_C[a]$.\nWe may assume that the support of $P$ contains a vertex in $C$ whose distance to $a$ in $C$ is $2$ by Lemma~\\ref{lem:connectedsupport}.\nLet $v_1, v_2, v_3, v_4, v_5$ be the consecutive vertices of $C$ where $a=v_3$.\nThis assumption implies that $P$ contains a vertex in either $Z_{v_1}$ or $Z_{v_5}$.\n\nLet $w$ be the vertex of $Z_{\\{v_1, v_5\\}}\\cap V(H)$ that is closest to $N_H(a)\\setminus \\{d\\}$.\nLet $Q$ be the $(d,w)$-subpath of $H$ containing $a$.\nWe verify the preconditions of Lemma~\\ref{lem:smallsunflower} with $(p_1, p_2, P)=(d, a, Q)$.\nThe first condition is clear.\nNote that $H$ contains neither $v_2$ nor $v_4$; otherwise, $dav_2$ or $dav_4$ is a triangle in $H$, contradicting the assumption that $H$ is a hole.\nThus $Q$ contains neither $v_2$ nor $v_4$.\nIf $w$ is $v_1$ or $v_5$, then the neighbor of $w$ in $Q$ is also in $Z_{\\{v_1, v_5\\}}$, contradicting the choice of $w$.\nThus, $w\\in Z_{\\{v_1, v_5\\}}\\setminus \\{v_1, v_5\\}$.\nClearly, all internal vertices of $Q$ are in $Z_{\\{v_2, v_3, v_4\\}}\\setminus Z_{\\{v_1, v_5\\}}$; otherwise by Lemma~\\ref{lem:connectedsupport}, \n$Q$ must contain an internal vertex from $Z_{\\{v_1, v_5\\}}$, contradicting the choice of $w$.\nThe last two conditions are satisfied because $H$ is a hole.\nThen $(d,a,Q)$ meets the preconditions of Lemma~\\ref{lem:smallsunflower}. \n\nTherefore, there exists a $D$-traversing sunflower $H'$ containing $d$ and $a$ such that $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{v_1, v_2, v_3\\}}$ or $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{v_3, v_4, v_5\\}}$.\nAs $a=v_3\\in V(H)$, we have $V(H)\\cap \\{v_1, v_2, v_3\\}\\neq\\emptyset$ or $V(H)\\cap \\{v_3, v_4, v_5\\}\\neq\\emptyset$, respectively.\n\\end{proof}\n\nBased on Lemma~\\ref{lem:small}, we prove the following.\n\n\n\n\n\\begin{PROP}\\label{prop:skew}\nThere is a polynomial-time algorithm which finds either $k+1$ vertex-disjoint holes in $G$ or a vertex set $T_{trav:sunf}\\subseteq (D\\cup V(C))\\setminus T_{petal}$ of size at most $15k+9$ such that $T_{petal}\\cup T_{trav:sunf}$ hits all $D$-traversing sunflowers.\n\\end{PROP}\n\\begin{proof}\nLet $C=v_0v_1 \\cdots v_{m-1}v_0$. All additions are taken modulo $m$. \nWe create an auxiliary bipartite graph $\\mathcal{G}_i=(D\\uplus \\mathcal{A}_i, \\mathcal{E}_i)$ for each $0\\leq i \\leq 4$, such that\n\\begin{itemize}\n\\item $\\mathcal{A}_i=\\{\\{v_{5j+i},v_{5j+i+1},v_{5j+i+2}\\}: j=0, 1, \\ldots ,\\lfloor \\frac{m}{5} \\rfloor -1\\}$,\n\\item there is an edge between $d\\in D$ and $\\{x,y,z\\}\\in \\mathcal{A}_i$ if and only if \nthere is a hole $H$ containing $d$ such that $V(H)\\setminus \\{d\\}\\subseteq Z_{\\{x,y,z\\}}$ (thus, $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$).\n\\end{itemize}\nClearly, the auxiliary graph $\\mathcal{G}_i$ can be constructed in polynomial time using Lemma~\\ref{lem:detectinghole}. \n\nNow, we apply Theorem~\\ref{thm:menger} to each $\\mathcal{G}_i$ and outputs either a matching of size $k+1$ or a vertex cover of size at most $k$.\n\nSuppose that there exists $i\\in \\{0,1,\\ldots, 4\\}$ such that $\\mathcal{G}_i$ contains a matching $M$ of size at least $k+1$. \nWe argue that there are $k+1$ vertex-disjoint holes in this case. Let $e=(d,\\{x,y,z\\})$ and $e'=(d',\\{x',y',z'\\})$ be two distinct edges of $M$. \nBy construction, \nthere exist two holes $H$ and $H'$ such that \n\\begin{itemize}\n\\item $H$ contains $d$ and $V(H)\\setminus \\{d\\} \\subseteq Z_{\\{x,y,z\\}}$,\n\\item $H'$ contains $d'$ and $V(H')\\setminus \\{d'\\} \\subseteq Z_{\\{x',y',z'\\}}$.\n\\end{itemize}\nRecall that any vertex of $N(C)\\setminus D$ has at most three neighbors on $C$, which are consecutive by Lemma~\\ref{lem:consecutive}. On the other hand, the distance between $\\{x,y,z\\}$ and $\\{x',y',z'\\}$ on $C$ is at least three by the construction of the family $\\mathcal{A}_i$. Therefore the two sets $Z_{\\{x,y,z\\}}$ and $Z_{\\{x',y',z'\\}}$ are disjoint, \nwhich implies \n$H$ and $H'$ are vertex-disjoint. We conclude that one can output $k+1$ vertex-disjoint holes when there is a matching $M$ of size $k+1$ in one of $\\mathcal{G}_i$'s.\n\nConsider the case when for every $0\\le i\\leq 4$, $\\mathcal{G}_i$ admits a vertex cover $S_i$ of size at most $k$. For $S_i$, let $S^*_i$ be the vertex set \n\\[ (S_i\\cap D) \\cup \\bigcup_{\\{x,y,z\\}\\in S_i\\cap \\mathcal{A}_i} \\{x,y,z\\} \\]\nand let $T_{trav:sunf}:=\\left( \\bigcup_{i=0}^4 S^*_i \\right) \\cup \\{v_{5\\lfloor \\frac{m}{5} \\rfloor+i}:-2\\le i\\le 6\\}$. Notice that $\\abs{T_{trav:sunf}}\\leq 15k+9$. \n\n\\begin{CLAIM}\nThe vertex set $T_{petal}\\cup T_{trav:sunf}$ hits all $D$-traversing sunflowers. \n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose $G-(T_{petal}\\cup T_{trav:sunf})$ contains a $D$-traversing sunflower $H$ having a vertex $d\\in D$. \nBy Lemma~\\ref{lem:small}, \nthere exist $x,y,z$ that are consecutive vertices on $C$ and a $D$-traversing sunflower $H'$ containing $d$ such that $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$ and $V(H')\\setminus \\{d\\}\\subseteq Z_{\\{x,y,z\\}}$. Clearly, we have either \n\\begin{itemize}\n\\item $d$ is adjacent to $\\{x,y,z\\}$ in one of the bipartite graphs $\\mathcal{G}_i$, or\n\\item $\\{x,y,z\\}\\subseteq \\{v_{5\\lfloor \\frac{m}{5} \\rfloor+i}:-2\\le i\\le 6\\}$.\n\\end{itemize}\nIn the first case, $S^*_i$ contains $\\{d\\}$ or $\\{x,y,z\\}$, as $S_i$ is a vertex cover of $\\mathcal{G}_i$. Since $V(H)\\cap \\{x,y,z\\}\\neq \\emptyset$ and $H$ contains $d$, $S^*_i$ contains a vertex of $H$, which contradicts the assumption that $H$ is a $D$-traversing sunflower in $G-(T_{petal}\\cup T_{trav:sunf})$. In the second case, \n$T_{trav:sunf}$ contains $\\{x,y,z\\}$, which again contradicts that $H$ is a $D$-traversing sunflower in $G-(T_{petal}\\cup T_{trav:sunf})$.\n\\end{proofofclaim}\n\nThis completes the proof. \n\\end{proof}\n\n\n\n\\section{Hitting all tulips}\\label{sec:tulip}\n\n\n\nIn this section, \nwe show that one can find in polynomial time either $k+1$ vertex-disjoint holes or a vertex set hitting all tulips.\nAgain, we assume that $(G,k,C)$ is given as an input such that $C$ is a shortest hole of $G$ of length \nstrictly greater than $\\mu_k$ and $G-V(C)$ is chordal.\n\nThe first few sections will focus on $D$-avoiding tulips. In Subsection~\\ref{subsec:Dtulip}, we settle the case of $D$-traversing tulips.\nSubsection~\\ref{subsec:final} will establish the main theorem for holes in general, Theorem~\\ref{thm:core}.\nFor $D$-avoiding tulips, it is sufficient to consider the graph $G_{deldom}=G-D$. \n\n\\subsection{Constructing a nested structure of partial tulips}\\label{subsec:tuliphive}\n\nWe recursively construct a subgraph of $G_{deldom}$ in which all vertices have degree $2$ or $3$ and it contains $C$. \nA subgraph of $G$ is called a \\emph{$(2,3)$-subgraph} if its all vertices have degree $2$ or $3$.\nFor a $(2,3)$-subgraph $F$, a vertex $v$ of degree $3$ in $F$ is called a \\emph{branching point} in $F$,\n and other vertices are called \\emph{non-branching points}. \n\nGiven a $(2,3)$-subgraph $F$ of $G_{deldom}$ containing $C$, an $(x,y)$-path $P$ of $G_{deldom}$ is a \\emph{$F$-extension} if it satisfies the following.\n\\begin{enumerate}[(i)]\n\\item $x$ and $y$ are distinct non-branching points of $F$.\n\\item $\\{x,y\\}\\cap V(C)\\neq \\emptyset$.\n\\item $P$ is a proper $V(F)$-path and $P-V(F)$ is an induced path of $G_{deldom}$.\n\\item There exists a vertex $v\\in V(P)$ such that $\\dist_P(v,\\{x,y\\}\\cap V(C))=2$ and $v\\notin N[F]$. \n\\end{enumerate} \nNote that by condition (iv), the length of an $F$-extension is at least $4$.\n\nA cycle $H$ of $G_{deldom}$ is an \\emph{almost $F$-extension} \nif it satisfies the following. \n\\begin{enumerate}[(i)]\n\\item[(i')] $\\abs{V(H)\\cap V(C)}=1$ and the vertex in $V(H)\\cap V(C)$ is a non-branching point of $F$ in $C$. \n\\item[(ii')] $H-V(C)$ is an induced path of $G_{deldom}$ and contains no vertex of $F$.\n\\item[(iii')] There exists a vertex $v\\in V(H)$ such that $\\dist_H(v,V(H)\\cap V(C))=2$ and $v\\notin N[F]$. \n\\end{enumerate} \nWe call the vertex in $V(H)\\cap V(C)$ the \\emph{root} of the almost $F$-extension $H$. \n\nIt is not difficult to see that given a $(2,3)$-subgraph $F$ containing $C$, \nthere is a polynomial-time algorithm to find a shortest $F$-extension $P$ or correctly decides that there is no $F$-extension. \nFor this, we exhaustively guess five vertices $x,y,x',y',v$ such that \n\\begin{itemize}\n\\item $x$ and $y$ are non-branching points of $F$ such that $x\\in V(C)$,\n\\item $x'$ and $y'$ are neighbors of $x$ and $y$ in $V(G_{deldom})\\setminus V(F)$, respectively, \n\\item $v$ is a neighbor of $x'$ in $V(G)\\setminus N[F]$.\n\\end{itemize}\nSince we are looking for a $(x,y)$-path $P$ where $\\mathring{x}P\\mathring{y}$ is induced, \nwe check whether there is a path from $v$ to $y'$ in $G_{deldom}-((V(F)\\cup N[x'])\\setminus \\{v\\})$. \nIf there is such a path, then we find a shortest one $Q$. Then $xx'v\\odot vQy'\\odot y'y$ is an $F$-extension. \nAmong all possible choices of five vertices $x,y,x',y',v$, we find a shortest $F$-extension using these five vertices.\nClearly if there is an $F$-extension, then we can find a tuple of such five vertices that outputs a shortest $F$-extension in the above procedure.\n\nThroughout this section, we heavily rely on the structure of a maximal subgraph obtained by adding a sequence of $F$-extensions exhaustively. We additionally impose a tie breaking rule for the choice of $F$-extensions.\n\n\n\\begin{description}\n\\item [Initialize] $W_1=C$, $B_1=\\emptyset$, and $i=1$.\n\\item [At step $i$] We perform the following.\n\\begin{enumerate}\n\\item Find a shortest $W_i$-extension $P_{i}$ such that\n\\begin{itemize}\n\\item[] (\\textbf{Tie break}) $\\abs{V(P_i)\\cap V(C)}$ is maximum.\n\\end{itemize}\nIf no $W_i$-extension exists, then terminate. Let $x_i$, $y_i$ be the endpoints of $P_{i}$ otherwise. \n\\item Set $W_{i+1}:= (V(W_i)\\cup V(P_{i}), E(W_{i})\\cup E(P_{i}))$. \n\\item Set $B_{i+1}:=B_{i} \\cup \\{x_i,y_i\\}$ and increase $i$ by one.\n\\end{enumerate}\n\\end{description}\n\nNotice that every vertex of $W_i$ has degree 2 or 3. Let $W_1, W_2, \\ldots , W_{\\ell}$ be \nthe sequence of subgraphs constructed exhaustively until there is no $W_{\\ell}$-extension. Let $W=W_{\\ell}$ and $T_{branch}=B_{\\ell}$. \nThroughout this section, we fix those sequences $W_1, W_2, \\ldots , W_{\\ell}=W$ and $P_1, P_2, \\ldots, P_{\\ell-1}$, and $B_1, B_2, \\ldots, B_{\\ell}=T_{branch}$. \nClearly, the construction of $W$ requires at most $n$ iterations, and thus we can construct these sequences in polynomial time.\n\n\nThe first observation is that if $T_{branch}$ has size at least $s_{k+1}$, then $G[V(W)]$ contains $k+1$ vertex-disjoint holes. \nIn fact, the construction of $W$ is calibrated so that every cycle of $W$ contains a hole of $G$. \nFor this, the condition (iv) of $W$-extension is crucial. Due to the next lemma, we may assume that $\\abs{T_{branch}}< s_{k+1}$.\n\n\\begin{LEM}\\label{lem:manybranching}\nIf $W$ has at least $s_{k+1}$ branching points, then there are $k+1$ vertex-disjoint holes and they can be detected in polynomial time. \n\\end{LEM}\n\\begin{proof}\nBy Theorem~\\ref{thm:simonovitz}, $\\abs{T_{branch}}\\geq s_{k+1}$ implies that $W$ has at least $k+1$ cycles, and such a collection of cycles can be found in polynomial time.\n We shall prove that for each cycle $H$ of $W$, there is a hole in the subgraph of $G$ induced by $V(H)$. Clearly, this immediately establishes the statement.\n We fix a cycle $H$ of $W$. We may assume that $H\\neq C$.\n Recall that for each $i$, $P_i$ is a $W_i$-extension added to $W_i$.\n\nLet $i$ be the minimum integer such that $E(H)\\subseteq E(W_{i+1})$. We claim that $P_{i}$ is entirely contained in $H$ as a subgraph. \nNotice that for any $W_{j}$-extension $P_j$, every branching point $v\\in T_{branch}$ which is an internal vertex of $P_j$ has been added at iteration $j'>j$. \nTherefore, if $P_{i}$ is not entirely contained in $H$ as a subgraph, then for some $i'>i$, there exists a subpath of $P_{i'}$ \nsuch that $E(P_{i'})\\cap E(H)\\neq \\emptyset$, contradicting the choice of $i$. \n\nLet $x,y$ be the endpoint of $P_i$. \nLet $v$ be an internal vertex of $P_{i}$ that is not contained in $N[W_i]$.\nSuch a vertex exists by the condition (iv) of the definition of a $W$-extension. \nLet $Q:=H-v$. Since $\\mathring{x}P_i\\mathring{y}$ is induced, the neighbors of $v$ in $P_i$ are not adjacent, and $v$ has no neighbors in the set of internal vertices of $Q$. \nTherefore, by Lemma~\\ref{lem:twopaths}, $G[V(H)]$ contains a hole, as claimed.\n\\end{proof}\n\nIn the next step, we exhaustively find almost $W$-extensions and cover them if there are no $k+1$ vertex-disjoint holes. We show that if there are two almost $W$-extensions with roots $x_1$ and $x_2$ and $\\dist_C(x_1, x_2)\\ge 5$, then these two almost $W$-extensions do not intersect. This is because if they meet, then we can obtain a $W$-extension, contradicting the maximality of $W$. Using this, we can deduce that if there are $5k+5$ almost $W$-extensions with distinct roots, then there are $k+1$ vertex-disjoint holes. \n\n\n\n\n\\begin{PROP}\\label{prop:almostpacking}\nThere is a polynomial-time algorithm that\nfinds either $k+1$ vertex-disjoint holes or a vertex set $T_{almost}\\subseteq V(C)\\setminus T_{branch}$ of size at most $5k+4$ such that $T_{almost}\\cup T_{branch}$ hits all almost $W$-extensions. \n\\end{PROP}\n\\begin{proof}\nLet $C=v_0v_1 \\cdots v_{m-1}v_0$. All additions are taken modulo $m$. \nWe greedily construct a collection of almost $W$-extensions $\\mathcal{Y}=\\{Y_1, Y_2, \\ldots , Y_t\\}$ (not necessarily vertex-disjoint) with distinct roots $v_{a_1}, v_{a_2}, \\ldots, v_{a_t}\\in V(C)\\setminus T_{branch}$, and stop if $t$ reaches $5k+5$. \nTo construct such a collection, we do the following for each vertex $v\\in V(C)\\setminus T_{branch}$:\n\\begin{enumerate}\n\\item Choose three vertices $w_1, w_2, w_3$ such that $w_1, w_3\\in Z_v\\setminus \\{v\\}$, $w_2\\notin N[W]$, and $w_1$ is adjacent to $w_2$ but not adjacent to $w_3$.\n\\item Test whether there is a path from $w_2$ to $w_3$ in $G_{deldom}-((V(W)\\cup N[w_1])\\setminus \\{w_2\\})$.\nIf there is such a path $P$, then we add the cycle $H=w_1w_2\\odot w_2Pw_3\\odot w_3vw_1$ \nto $\\mathcal{Y}$.\n\\end{enumerate}\nIt is not difficult to verify that there is an almost $W$-extension with root $v$ if and only if \nthe algorithm outputs such a cycle $H$.\n\n\nWe claim that if $v_{a_p}$ and $v_{a_q}$ have distance at least $5$ in $C$, \nthen $Y_p$ and $Y_q$ do not meet.\n\n\\begin{CLAIM}\\label{claim:distancealmost}\nLet $p,q\\in \\{1, 2, \\ldots, t\\}$. \nIf $\\dist_C(v_{a_p}, v_{a_q})\\ge 5$, \nthen $V(Y_p)\\cap V(Y_q)=\\emptyset$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose for contradiction that $Y_p$ and $Y_q$ meet at a vertex $z$.\nLet $Y_p=p_1p_2 \\cdots p_rv_{a_p}p_1$ and $Y_q=q_1q_2 \\cdots q_sv_{a_q}q_1$.\nFor convenience let $p_0:=v_{a_p}$ and $q_0:=v_{a_q}$.\nBy the condition (iii') of an almost $W$-extension, \nwe may assume that $p_2, q_2\\notin N[W]$.\nLet $t_1$ be the minimum integer such that $p_{t_1}$ has a neighbor in $Y_q$.\nWe choose a neighbor $q_{t_2}$ of $p_{t_1}$ in $Y_q$ with minimum $t_2$.\nLet $R:=p_0Y_pp_{t_1}\\odot p_{t_1}q_{t_2}\\odot q_{t_2}Y_qq_0$.\nIt is not difficult to see that $R$ is an induced path.\n\nSince $\\dist_C(p_0, q_0)\\ge 5$, the length of $R$ is at least $4$ by Lemma~\\ref{lem:generalfarnonadj}. Therefore, \n$R$ contains either $p_2$ or $q_2$.\nIt implies that $R$ is a $W$-extension, contradicting the maximality of $W$.\n\\end{proofofclaim}\n\nSuppose $t\\geq 5k+5$. There exists $M=\\{b_1, b_2, \\ldots, b_{k+1}\\}\\subseteq \\{a_1, a_2, \\ldots, a_{t}\\}\\setminus \\{v_i: m-4 \\le i\\le m-1\\}$ \nsuch that for all $b_i, b_j\\in M$, $b_i\\equiv b_j\\pmod 5$. \nAs we exclude the vertices of $\\{v_i: m-4 \\le i\\le m-1\\}$, for every $b_i, b_j\\in M$, $\\dist_C(v_{b_i}, v_{b_j})\\ge 5$.\nBy Claim~\\ref{claim:distancealmost}, \nfor $i, j\\in \\{1, 2, \\ldots, k+1\\}$, corresponding almost $W$-extensions $Y_{b_{i}}$ and $Y_{b_{j}}$ are vertex-disjoint.\nThus, we can output $k+1$ vertex-disjoint holes in polynomial time. \nOtherwise, $\\abs{\\mathcal{Y}}\\le 5k+4$, and thus the set of all roots $\\{v_{a_1}, v_{a_2}, \\ldots, v_{a_t}\\} \\subseteq V(C)\\setminus T_{branch}$ \nof $\\mathcal{Y}$ \ncontain at most $5k+4$ vertices. Clearly, the set of all roots \nhits every element of $\\mathcal{Y}$, that is, every almost $W$-extension.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{$Q$-tunnels}\\label{subsec:cfragment}\n\nWe define \n\\begin{align*}\nT_{ext}:=&T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost}\\cup\\\\\n &N_C^{20}[(T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost})\\cap V(C)].\n\\end{align*}\nNote that \n\\[ \\abs{T_{ext}}\\le 41(19k+(3k+14)+(15k+9)+(s_{k+1}-1)+(5k+4))\\le 41(s_{k+1}+42k+26). \\]\nSince $\\abs{T_{petal}\\cup T_{full}\\cup T_{trav:sunf} \\cup T_{branch}\\cup T_{almost}}\\le s_{k+1}+42k+26$, $C-(T_{petal}\\cup T_{full}\\cup T_{trav:sunf} \\cup T_{branch}\\cup T_{almost})$ contains at most $s_{k+1}+42k+26$ connected components, and so does $C-T_{ext}$.\nLet $\\mathcal{Q}$ be the set of connected components of $C-T_{ext}$ and we call each element of $\\mathcal{Q}$ a \\emph{$C$-fragment}.\n\n \n We want to show that for every $D$-avoiding tulip $H$ not hit by $T_{ext}$ and for every $Q\\in \\mathcal{Q}$, \n if $H$ contains a vertex of $Q$ far from the endpoints of $Q$, then $H$ must traverse the $Q$-tunnel from one entrance to the other entrance. \n To argue this, we show that $H$ contains no edge $vw$ where $v\\in Z_{V(Q)}$ and $w\\notin N[C]$.\n We first need to show that in such a case, we have $w\\notin N[W]$. \n The next lemma states a useful distance property of $W$.\n See Figure~\\ref{fig:distancelemma} for an illustration.\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\draw[rounded corners] (6,0)--(11,0)--(12,-1)--(11,-2)--(-2,-2)--(-3,-1)--(-2,0)--(6,0);\n \n \\draw[rounded corners] (7,0)--(7,1)--(9,1)--(9,0);\n \\draw[rounded corners] (8,1)--(8,2)--(10,2)--(10,0);\n \\draw[rounded corners] (6,0)--(6,3)--(9,3)--(9,2);\n \n \\draw (-.1,0.3)--(-.1,-0.3);\n \\draw (.1,0.3)--(.1,-0.3);\n \\draw (4-.1,0.3)--(4-.1,-0.3);\n \\draw (4.1,0.3)--(4.1,-0.3);\n \n \\draw (2,0)--(2,1)--(3,2)--(6,2.5);\n \\draw (2,0) node [w] {};\n \\draw (2,1) node [w] {};\n \\draw (3,2) node [w] {};\n \\draw (6,2.5) node [w] {};\n \n \\node at (7, 2) {$P_i$};\n\n \\node at (2, -.5) {$v_0$};\n \\node at (1.5, 1) {$v_1$};\n \\node at (2.7, 2.3) {$v_2$};\n \\node at (5.5, 2.8) {$v_3$};\n \n\n\n \\node at (-2, -1) {$C$};\n\n \\end{tikzpicture} \\caption{The setting in Lemma~\\ref{lem:distance2} where $v_0$ is a vertex of $C-T_{ext}$ and there is a path of length $3$ from $v_0$ to $V(W)\\setminus V(C)$ whose internal vertices are in $V(G_{deldom})\\setminus V(W)$. By Lemma~\\ref{lem:distance2}, $v_2$ should have a neighbor in $C$. }\\label{fig:distancelemma}\n\\end{figure}\n\n\n\\begin{LEM}\\label{lem:distance2}\nLet $Q\\in \\mathcal{Q}$ be a $C$-fragment, and \n$v\\in V(Q)$ and $u\\in V(W)$ with $v\\neq u$. \nThen every $V(W)$-path $R=v_0v_1 \\cdots v_{s}$ from $v_0=v$ to $v_s=u$ satisfies one of the following.\n\\begin{enumerate}[(1)]\n\\item $u$ is a vertex of $C$ such that $\\dist_C(u, V(Q))\\le 4$, \n\\item $R$ has length $3$ and $v_2$ is adjacent to a vertex of $C$, \n\\item $R$ has length at least 4.\n\\end{enumerate}\n\\end{LEM}\n\\begin{proof}\nRecall that sequences $W_1, W_2, \\ldots , W_{\\ell}=W$ and $P_1, P_2, \\ldots, P_{\\ell-1}$, and $B_1, B_2, \\ldots, B_{\\ell}=T_{branch}$ \nrespectively denote the sequences of subgraphs, $W_i$-extensions and branching points during the construction of $W$.\n\nSuppose $u\\in V(C)$. If a $V(W)$-path $R$ between $u$ and $v$ has length at most $3$, \nthen there is an edge between $Z_u$ and $Z_v$ or we have $Z_u\\cap Z_v\\neq \\emptyset$.\nThen Lemma~\\ref{lem:farnonadj} implies that $\\dist_C(u,v)\\le 3$, and $R$ satisfies (1). \nTherefore, we may assume $u\\notin V(C)$. In particular, the following claim for every $i\\in \\{1, \\ldots, \\ell-1\\}$ establishes the statement immediately. \nWe prove by induction on $i$: \n\\begin{itemize}\n\\item[$(\\ast)$] if $u$ is an internal vertex of $P_i$, then every $V(W)$-path $R$ between $v$ and $u$ satisfies (2) or (3).\n\\end{itemize}\n\n\nLet $P_{i}=u_0u_1 \\cdots u_p$. \nBy definition of a $W_{i}$-extension, we may assume $u_0\\in V(C)$ and $u_2\\notin N[W_{i}]$. \nSuppose there exists a $V(W)$-path from $v$ to an internal vertex of $P_i$ violating (3). \nSuch a path has length at most $3$.\nLet $s\\in \\{1,2,3\\}$ be the minimum integer such that \nthere is a $V(W)$-path of length $s$ between $v$ and an internal vertex of $P_i$. \nWe choose the minimum integer $j\\in \\{1, 2, \\ldots, p-1\\}$ such that \nthere is a $(v,u_j)$-path $R$ of length $s$.\nLet $R:=v_0v_1 \\cdots v_s$ with $v_0=v$ and $v_s=u_j$, and \n $R_1=u_0P_iu_j\\odot R$. \n\nWe verify that $R_1$ is a $W_i$-extension. \n\\begin{CLAIM}\\label{claim:r1}\n$R_1$ is a $W_i$-extension containing $u_2$. \n\\end{CLAIM}\n\\begin{proofofclaim}\nBy the choice of $s$ and $u_j$, every vertex in $\\mathring{v_0}R\\mathring{u_j}$ has no neighbors in $\\mathring{u_0}P_i\\mathring{u_j}$. Therefore, $\\mathring{u_0}R_1\\mathring{v_0}$ is an induced path. Also, $v_0$ is a non-branching point of $W_i$. \nHence, $R_1$ satisfies the conditions (i)-(iii) of $W_i$-extension. \nFor (iv), it is sufficient to show that $j\\ge 2$.\nSuppose $j=1$. \nThen $R_1$ has length at most $4$, and by Lemma~\\ref{lem:generalfarnonadj} with $m=2$, we have $\\dist_C(v, u_0)\\le 7$.\nSince $u_0\\in T_{branch}\\cap V(C)$, this contradicts the fact that $v\\in V(Q)$ and thus $\\dist_C(v, T_{branch}\\cap V(C))\\ge 20$. \nWe conclude that $j\\ge 2$ and thus $R_1$ contains $u_2$. Since $P_i$ meets (iv) as a $W_i$-extension, we have $u_2\\notin N[W_i]$.\nTherefore, $R_1$ satisfies all four conditions for being a $W_i$-extension.\n\\end{proofofclaim}\n\nNext, we show that $R$ has length exactly $3$. \nWhen $R$ has length $1$ or $2$, \nwe derive a contradiction from the fact that $P_i$ is taken as a $i$-th $W_i$-extension.\n\n\\begin{CLAIM}\\label{claim:length3}\n$s=3$; that is, $R$ has length $3$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nFirst assume that $R$ has length $1$.\nIf $j2$ such that $v_i$ has a neighbor that is a non-branching point of $W$. Clearly $2 i+1$ because we have $p_{i+1}\\notin N[C]$ due to the choice of $i$. \nObserve that $p_{\\ell}wp_i$ is an induced path with $w$ as an internal vertex and $w$ is not adjacent to any internal vertex \nof $p_iPp_{\\ell}$. Now Lemma~\\ref{lem:twopaths} applies, implying that $G[V(p_iPp_{\\ell})\\cup \\{w\\}]$ has a hole $H'$ containing $w$. \nBy Claim~\\ref{claim:nopoint}, $H'$ contains no point of $W$ other than $w$. \n\nObserve that $H'$ qualifies as an almost $W$-extension if $p_{\\ell}\\neq d$; especially we have $p_{i+1}\\notin N[W]$ by Claim~\\ref{claim:secondvertex}.\nTherefore $T_{almost}$ hits $H'$. On the other hand, $T_{almost}\\cap (V(H')\\setminus \\{w\\})\\subseteq T_{ext}\\cap V(H)=\\emptyset$, which \nimplies $w\\in T_{almost}$. Then by the construction of $T_{ext}$, we have $x\\in T_{ext}$, a contradiction. \nIf $p_{\\ell}=d$, then $H'-dw$ is a path certifying an edge in an auxiliary bipartite graph. Therefore either one of $\\{d,w\\}$ is contained in the vertex cover \nor $w=v_{5\\lfloor{\\frac{m}{5}}\\rfloor+a}$ with $0\\leq a \\leq 4$. In both cases, $x$ is included in $T_{trav:tulip}$, a contradiction.\nThis completes the proof. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Proof of our main result}\\label{subsec:final}\n\nWe prove Theorem~\\ref{thm:core}. \n\nWe apply Lemma~\\ref{lem:petalcover}, Proposition~\\ref{prop:hitsunflower}, and Proposition~\\ref{prop:skew}.\nOver all, we can in polynomial time either output $k+1$ vertex-disjoint holes or vertex sets $T_{petal}, T_{full}, T_{trav:sunf}$ hitting petals, full sunflowers, and $D$-avoiding sunflowers, respectively.\n\n\n\n\nWe construct $W$ with the set $T_{branch}$ of branching points as described in Subsection~\\ref{subsec:tuliphive}.\nBy Lemma~\\ref{lem:manybranching}, \nif $W$ has at least $s_{k+1}$ branching points, then there are $k+1$ vertex-disjoint holes and they can be detected in polynomial time. \nWe apply Proposition~\\ref{prop:almostpacking}.\nIf it outputs \n$k+1$ vertex-disjoint holes in $G$, then we are done.\nWe may assume it outputs\na vertex set $T_{almost}$ of at most $5k+4$ vertices where $T_{almost}$ hits all almost $W$-extensions.\n\nLet $T_{ext}$ be the union of $T_{petal}\\cup T_{full}\\cup T_{trav:sunf}\\cup T_{branch}\\cup T_{almost}$ and the $20$-neighborhood of $V(C)\\cap (T_{petal}\\cup T_{full} \\cup T_{trav:sunf} \\cup T_{branch}\\cup T_{almost})$.\n\nBy Proposition~\\ref{prop:Davoid}, we can in polynomial time either find $k+1$ vertex-disjoint holes or find a set \n$T_{avoid:tulip}\\subseteq V(G)\\setminus T_{ext}$ of at most $35(s_{k+1}+42k+26)$ vertices such that $T_{ext}\\cup T_{avoid:tulip}$ hits all $D$-avoiding tulips.\nBy Lemma~\\ref{lem:dominatingtulip2}, we can either find $k+1$ holes \nor find a set $T_{trav:tulip}\\subseteq V(G)\\setminus (T_{ext}\\cup T_{avoid:tulip})$ of size $25k+9$ such that \n$T_{ext}\\cup T_{avoid:tulip}\\cup T_{trav:tulip}$ hits all $D$-traversing tulips. \nTherefore, we can either find $k+1$ vertex-disjoint holes, or output a vertex set with at most \n\\begin{align*}\n&\\abs{T_{ext}\\cup T_{avoid:tulip}\\cup T_{trav:tulip}} \\\\\n&\\le 41(s_{k+1}+42k+26)+ 35(s_{k+1}+42k+26)+25k+9 \\\\\n\t\t\t\t\t\t\t\t\t\t%\n\t\t\t\t\t\t\t\t\t\t&\\le 76s_{k+1}+3217k+1985\n\t\t\t\t\t\t\t\t\t\t\\end{align*}\n\t\t\t\t\t\tvertices hitting all holes.\nThis completes the proof of Theorem~\\ref{thm:core}.\n\n\n\\section{Cycles of length at least $5$ do not have the Erd\\H{o}s-P\\'osa property under the induced subgraph relation}\\label{sec:lowerbound}\n\nIn this section, we show that the class of cycles of length at least $\\ell$ for every fixed $\\ell\\ge 5$ \nhas no Erd\\H{o}s-P\\'osa property under induced subgraph reltation.\n\nA \\emph{hypergraph} is a pair $(X,\\mathcal{E})$ such that $X$ is a set of elements and $\\mathcal{E}$ is a family of non-empty subsets of $X$, called \\emph{hyperedges}.\nA subset $Y$ of $X$ is called a \\emph{hitting set} if for every $F\\in \\mathcal{E}$, $Y\\cap F\\neq \\emptyset$.\nFor positive integers $a,b$ with $a\\geq b$, \nlet $(a,b)$-uniform hypergraph, denote it by $U_{a,b}$, be the hypergraph $(X, \\mathcal{E})$ such that \n$\\abs{X}=a$ and $\\mathcal{E}$ is the set of all subsets of $X$ of size $b$.\nIt is not hard to observe that in $U_{2k-1, k}$, every two hyperedges intersect and\nthe minimum size of a hitting set of $U_{2k-1, k}$ is precise $k$.\n\n\n\\begin{THMMAIN2}\nLet $\\ell\\ge 5$ be a positive integer. \nThen the class of cycles of length at least $\\ell$ has no Erd\\H{o}s-P\\'osa property under induced subgraph relation.\n\\end{THMMAIN2}\n\n\\begin{proof}\nSuppose for contradiction that there is a function $f_{\\ell}:\\mathbb{N}\\rightarrow \\mathbb{N}$ such that\nfor every graph $G$ and a positive integer $k$, either\n\\begin{itemize}\n\\item $G$ contains $k+1$ pairwise vertex-disjoint holes of length at least $\\ell$ or\n\\item there exists $T\\subseteq V(G)$ with $\\abs{T}\\le f_{\\ell}(k)$ such that $G- T$ contains no holes of length at least $\\ell$.\n\\end{itemize} \nLet $x=\\max \\{f_{\\ell}(1)+1, \\ell\\}$. From the hypergraph $U_{2x-1, x}=(X,\\mathcal{E})$, \nwe construct a graph $G$ on the vertex set $S\\uplus \\bigcup_{F\\in \\mathcal{E}}Y_F$, where\n\\begin{itemize}\n\\item $S=\\{s_v:v\\in X\\}$ is an independent set of size $\\abs{X}$,\n\\item $Y_F=\\{y_v:v\\in F\\}$ is an independent set of size $x$ for each $F\\in \\mathcal{E}$.\n\\end{itemize}\nThe edge set of $G$ is created as follows.\n\\begin{itemize}\n\\item For each hyperedge $F\\in \\mathcal{E}$ with $F=\\{v_i:1\\le i\\le x\\}$, we add the edge set\n\\[\\{y_{v_1}s_{v_1},s_{v_1}y_{v_2},\\ldots , y_{v_x}s_{v_x},s_{v_x}y_{v_1}\\}.\\]\n\\item For each pair of two distinct hyperedges $F_1, F_2\\in \\mathcal{E}$, we add all possible edges between $Y_{F_1}$ and $Y_{F_2}$.\n\\end{itemize}\nNote that for each $F\\in \\mathcal{E}$, $G[Y_F\\cup S]$ contains precisely one hole, which has length $2x(\\ge \\ell)$. We denote this hole as $C_F$.\nFigure~\\ref{fig:construction} depicts the construction.\n\n\nWe verify that every hole of length at least $\\ell$ is one of the holes in $\\{C_F:F\\in \\mathcal{E}\\}$.\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=0.7]\n \\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=4pt]\n\n\n \\foreach \\y in {0, 2, 4, 6, 8}{\n \\draw (\\y,0) node [w] (a\\y) {};\n }\n\n \\foreach \\y in {0, 2, 4}{\n \\draw (\\y-2,2.5) node [w] (b\\y) {};\n }\n\n \\foreach \\y in {4, 6, 8}{\n \\draw (\\y+2,2.5) node [w] (c\\y) {};\n }\n \n \\draw (b0)--(a0)--(b2)--(a2)--(b4)--(a4)--(b0);\n \\draw (c4)--(a4)--(c6)--(a6)--(c8)--(a8)--(c4);\n\n\t \\draw(b0) [in=150,out=30] to (c4); \n\t \\draw(b0) [in=150,out=30] to (c6); \n\t \\draw(b0) [in=150,out=30] to (c8); \n\t \\draw(b2) [in=150,out=30] to (c4); \n\t \\draw(b2) [in=150,out=30] to (c6); \n\t \\draw(b2) [in=150,out=30] to (c8); \n\t \\draw(b4) [in=150,out=30] to (c4); \n\t \\draw(b4) [in=150,out=30] to (c6); \n\t \\draw(b4) [in=150,out=30] to (c8); \n \n\\draw[rounded corners] (0,1)--(9.5,1)--(9.5,-1)--(-1.5,-1)--(-1.5,1)--(0,1);\n\n\\draw[rounded corners] (0,.5)--(4.5,.5)--(4.5,-.5)--(-.5,-.5)--(-.5,.5)--(0,.5);\n\\draw[rounded corners] (4,.7)--(8.5,.7)--(8.5,-.7)--(4-.5,-.7)--(4-.5,.7)--(4,.7);\n\n \\node at (-2, 0) {$S$};\n \n \\end{tikzpicture} \\caption{An illustration of two holes constructed from two hyperedges.}\\label{fig:construction}\n\\end{figure}\n\n\n\\begin{CLAIM}\\label{claim:chordlesscycle}\nEvery hole of length at least $\\ell$ is exactly one of the holes in $\\{C_F:F\\in \\mathcal{E}\\}$.\n\\end{CLAIM}\n\\begin{proofofclaim}\nSuppose $C$ is a hole of length at least $\\ell\\ge 5$.\nWe show that $V(C)\\subseteq V(C_F)$ for some $F\\in \\mathcal{E}$. Clearly, it implies the claim as each $C_F$ is a hole.\n\nSuppose for contradiction that $C$ is not contained in one of $\\{C_F:F\\in \\mathcal{E}\\}$. Then there are two distinct \nhyperedges $F, F'\\in \\mathcal{E}$ such that $V(C)\\cap Y_F\\neq \\emptyset$ and $V(C)\\cap Y_{F'}\\neq \\emptyset$. \nLet $v\\in V(C)\\cap Y_F$ and $v' \\in V(C)\\cap Y_{F'}$. Due to construction of $G$, we have $vv'\\in E(G)$. Furthermore, \nthis also implies that for every $F''\\in \\mathcal{E}\\setminus \\{F,F'\\}$, we have $V(C)\\cap Y_{F''}=\\emptyset$. \n\nSince $S$ is independent, among the vertices of $V(C)\\setminus \\{v,v'\\}$ there are at least $\\lfloor (\\abs{V(C)}-2)\/2 \\rfloor$ vertices \nof $Y_{F}\\cup Y_{F'}$. Suppose $V(C)\\setminus \\{v,v'\\} \\setminus S$ has two vertices $w$ and $w'$. \nIf both of $w$ and $w'$ are in $Y_F$, then $v'$ is adjacent to at least three vertices of $C$, a contradiction. \nTherefore, we may assume that $w\\in Y_F$ and $w'\\in Y_{F'}$. Then $G[\\{v,v',w,w'\\}]$ is a cycle of length four, \ncontradicting the assumption that $C$ is a hole of length at least $\\ell(\\ge 5)$. If $V(C)\\setminus \\{v,v'\\} \\setminus S$ \ncontains a unique vertex, say $w\\in Y_F$, observe that $\\abs{V(C)}=5$ and $wv'$ is a chord of $C$, a contradiction.\n\\end{proofofclaim}\n\nBy Claim~\\ref{claim:chordlesscycle}, $\\{C_F:F\\in \\mathcal{E}\\}$ is precisely the set of all holes of length at least $\\ell$ in $G$.\nOne can observe that two holes in $\\{C_F:F\\in \\mathcal{E}\\}$ intersect because $(X,\\mathcal{E})$ is the hypergraph $U_{2x-1, x}$, \nin which every two hyperedges intersect.\nTherefore, by the property of the function $f_{\\ell}$, \nthere exists a vertex subset $T\\subseteq V(G)$ with $\\abs{T}\\le f_{\\ell}(1)