Athanasios Kehagias, Georgios Konstantinidis
In this paper we study the two-player generalized Cops and Robber (GCR) games introduced by Bonato and MacGillivray. Our main goal is to present a full, self-contained game theoretic analysis of such games.
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In this paper, we consider a variant of the cops and robbers game on a graph, introduced by Kinnersley and Peterson, in which every time the robber uses an edge, it is removed from the graph, known as bridge-burning cops and robbers. In particular, we study the maximum time it takes the cops to capture the robber.
We investigate multiple variants of the game Cops and Robbers. Playing it on an $n \times n$ toroidal chess graph, the game is varied by defining moves for cops and robbers differently, always mimicking moves of certain chess pieces. In these cases, the cop number is completely determined.
We study a variation of the classical pursuit-evasion game of Cops and Robbers in which agents are required to move to an adjacent vertex on every turn. We explore how the minimum number of cops needed to catch the robber can change when this condition is added to the rules of the game. We study this `Fully Active Cops and Robbers' game for a number of classes of graphs and present some open problems for future research.
Seeing is not believing in limited visibility cops and robbers
本文研究有限能见度Cops and Robbers模型，证明看到强盗所需警察数可任意小于抓捕所需数，并探讨警力不足时的逼近程度及概率解释。
We consider the effect on the length of the game of Cops and Robbers when more cops are added to the game play. In Overprescribed Cops and Robbers, as more cops are added, the capture time (the minimum length of the game assuming optimal play) monotonically decreases. We give the full range of capture times for any number of cops on trees, and classify the capture time for an asymptotic number of cops on grids, hypercubes, and binomial random graphs. The capture time of planar graphs with a number of cops at and far above the cop number is considered.
We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber's neighbouring vertices. We denote by $σ(G)$ the {\em surrounding cop number} of $G$, namely the least number of cops required to surround a robber in the graph $G$. We present a number of results regarding this parameter, including general bounds as well as exact values for several classes of graphs. Particular classes of interest i…
Primal-Dual Cops and Robber
研究平面图上警察在相邻面移动、强盗在相邻顶点移动的博弈变体，证明当且仅当最大度 $Δ\leq 4$ 时常数名警察足以抓捕强盗。
Some remarks on cops and drunk robbers
在警察和强盗都是最优博弈的假设下，警察和强盗博弈已经得到了广泛的研究。在本文中，我们研究的问题，其中警察正在追逐一个喝醉了的强盗（即，强盗谁执行随机行走）上的一个图。我们的主要目标是描述“醉酒的代价。“具体来说，我们研究了最佳版本和醉酒抢劫者的预期捕获时间之比。我们还研究了算法方面的问题，也就是说，如何计算接近最佳的搜索时间表的警察。最后，我们提出了一个初步的调查无形的强盗游戏，并指出这种游戏和图搜索之间的差异。
Cops and robbers pebbling in graphs
本文提出警卫与强盗推石游戏，研究其推石数及图类中的上界、下界和精确值，并探讨相关猜想与新问题。
A bound for the cops and robber problem in terms of 2-component order connectivity
本文研究警察与强盗游戏中的警察数界，利用图的2-组件阶连通度刻画删除顶点后组件规模受限的结构，并给出警察数的上界。
In this short paper we study the game of Cops and Robbers, played on the vertices of some fixed graph $G$ of order $n$. The minimum number of cops required to capture a robber is called the cop number of $G$. We show that the cop number of graphs of diameter 2 is at most $\sqrt{2n}$, improving a recent result of Lu and Peng by a constant factor. We conjecture that this bound is still not optimal, and obtain some partial results towards the optimal bound.
We explore a variant of the game of Cops and Robber introduced by Bonato et al.~where the robber is invisible unless outside the common neighbourhood of the cops. The hyperopic cop number is analogous to the cop number and we investigate bounds on this quantity. We define a small common neighbourhood set and relate the minimum cardinality of this graph parameter to the hyperopic cop number. We consider diameter 2 graphs, particularly the join of two graphs, as well as Cartesian products.
The 'Cheating Robot' version of Cops and Robbers is played on a finite, simple, connected graph. The players move in the same time period. However, before moving, the robot observes to which vertices the cops are moving and it is fast enough to complete its move in the time period. The cops also know that the robot will use this information. More cops are required to capture a robot than to capture a robber. Indeed, the minimum degree is a lower bound on the number of cops required to capture a robot. Only on a tree is one cop guaranteed to capture a robot, although two cops are sufficient to …
$k$-Hyperopic Cops and Robber
本文提出$k$-远视警察与强盗游戏，定义$k$-远视警察数，刻画警察获胜图，并给出匹配数、树的直径及外平面图等情形下的上界。
The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on $\Vec{H}$-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept ($\Vec{H}^*$-free) to get a similar conclusion about the case of undirected graph.
We consider a variant of Cops and Robbers wherein each edge traversed by the robber is deleted from the graph. The focus is on determining the minimum number of cops needed to capture a robber on a graph $G$, called the {\em bridge-burning cop number} of $G$ and denoted $c_b(G)$. We determine $c_b(G)$ exactly for several elementary classes of graphs and give a polynomial-time algorithm to compute $c_b(T)$ when $T$ is a tree. We also study two-dimensional square grids and tori, as well as hypercubes, and we give bounds on the capture time of a graph (the minimum number of rounds needed for a si…
Cops and Robber -- When Capturing is not Surrounding
本文研究了经典Cops and Robber游戏的四种“包围”版本，通过关联不同版本的警察数，证明其均不受经典警察数与图最大度数的函数所界，从而反驳了Crytser等人的猜想。
We study the problem of cops and robbers on the grid where the robber is allowed to move faster than the cops. It is well known that two cops are necessary and sufficient to catch the robber on any finite grid when the robber has unit speed. Here, we prove that when the speed of the robber is a sufficiently large constant, the number of cops needed to catch the robber on an $n \times n$ grid is $\exp(Ω(\log n / \log \log n))$.
Guarding isometric subgraphs and Cops and Robber in planar graphs
本文引入子图中宽阴影概念刻画所有1-可保护图，并证明在最多两名警察同时移动的限制下3名警察可在任何平面图中捕获强盗，从而证明了Yang的猜想并加强了Aigner和Fromme的经典结果。
Cops against a cheating robber
本文研究警察与作弊机器人博弈，提出推挤数以刻画最少需多少警察登上强盗顶点，并建立其与包围式警察强盗博弈的关系，给出平面图界、多项式判定算法及图积界。
Limited-visibility Cops and Robbers on Hamming graphs
本文研究汉明图上的有限可见性警察与强盗游戏，给出捕获强盗所需警察数量的界，解决了Clarke等人提出的若干开放问题。
The game of cops and robbers is a well-known game played on graphs. In this paper we consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and we prove that their cop number is bounded by a constant. We also show that the cop number of every k-regularly oriented toroidal grid is at most 13.
Accelerated Cops and Robbers
本文研究了Cops and Robbers变体中双方每回合可遍历最多 $s$ 条边（$s \ge 2$）的新模型，给出了网格与超立方体的警察数界限，并确定了 $s=2$ 时警察获胜图的捕获时间。
We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays of the game. A positive integer $t$ is fixed, and the cops are required to capture the robber in at most $t$ time-steps in each play. The associated optimization parameter is the eternal cop number, denoted by $c_t^{\infty},$ which equals the eternal domination number in the case $t=1,$ and the cop number for sufficiently large $t.$ We study the complexity of Cops and Eternal Robbers, and show that game is NP-hard when $t$ is a fixed constant …
We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.
In this paper, we study the game of cops and robber on the class of graphs with no even hole (induced cycle of even length) and claw (a star with three leaves). The cop number of a graph $G$ is defined as the minimum number of cops needed to capture the robber. Here, we prove that the cop number of all claw-free even-hole-free graphs is at most two and, in addition, the capture time is at most $2n$ rounds, where $n$ is the number of vertices of the graph. Moreover, our results can be viewed as a first step towards studying the structure of claw-free even-hole-free graphs.
The game of Cops and Robbers is a well known game played on graphs. In this paper we consider the class of graphs of bounded diameter. We improve the strategy of cops and previously used probabilistic method which results in an improved upper bound for the cop number of graphs of bounded diameter. In particular, for graphs of diameter four, we improve the upper bound from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{3}{5}+o(1)}$ and for diameter three from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{4}{7}+o(1)}$.
Cops and Robbers for Graphs on Surfaces with Crossings
本文研究带交叉的曲面图上的警察与强盗博弈，提出距离d变体，证明当交叉边对之间存在短路径连接时，1-平面图、k-平面图等图类的警察数在较小d值下有界。
These are some personal notes about the pursuit game of Cops and Robbers that I made starting in 2007. More old and new problems (and some solutions) will be added in future versions of these notes.
The Cops & Robber game on series-parallel graphs
警察和强盗游戏是在无向有限图上进行的。$k$警察和一个强盗位于顶点上，并轮流沿边移动。如果在一次移动后，警察和强盗在同一个顶点上，警察就赢了。如果警察有一个获胜的策略，那么这个图就叫做$k$ -copwin。已知可平面图是3-copwin（Aigner & Fromme，1984），外可平面图是2-copwin（Clarke，2002）。在这篇短文中，我们证明了串并联（即，没有$K_4$次图的图是2-共赢的。在警察与强盗博弈的文献中，这是一个众所周知的技巧，定义了对警察可能的策略施加限制的博弈变体（参见Clarke，2002）。为了我们的证明，我们定义了“有出口的警察和强盗博弈”。我们的证据为警察提供了一个获胜的策略。
Cops and Robbers on 1-Planar Graphs
研究1-平面图上的警察与强盗追逃博弈，证明一般1-平面图警察数无界，极大1-平面图警察数为3，并刻画了外1-平面图的警察数。
We study relations between diameter $D(G)$, domination number $γ(G)$, independence number $α(G)$ and cop number $c(G)$ of a connected graph $G$, showing (i.) $c(G) \leq α(G)-\lfloor \frac{D(G)-3}{2} \rfloor$, and (ii.) $c(G) \leq γ(G) - \frac{D(G)}{3} + O (\sqrt{D(G)})$.
Cops and robber on variants of retracts and subdivisions of oriented graphs
研究定向图上 Cops and Robber 博弈的三种变体（强、普通、弱警察模型），证明强和普通警察数在相应收缩变体中保持不变，并证实所有变体在 2-degenerate 二部图上仍具计算难度。
The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= Ω(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as…
We consider a surrounding variant of cops and robbers on graphs of bounded genus. We obtain bounds on the number of cops required to surround a robber on planar graphs, toroidal graphs, and outerplanar graphs. We also obtain improved bounds for bipartite planar and toroidal graphs. We briefly consider general graphs of bounded genus and graphs with a forbidden minor.
The game of Cops and Robber is a pursuit-evasion game which is usually played on a connected graph. In the game, a set of cops and a robber move around the vertices of a graph along edges, where the cops aim to capture the robber, while the robber aims to avoid capture. Much research about this game have been done since the early 1980s. The game has a natural generalisation to being played on connected hypergraphs, where the cops and the robber may now move along hyperedges. In this paper, we shall provide a characterisation of all hypergraphs where one cop is sufficient to capture the robber.…
Pushing Cops and Robber on Graphs of Maximum Degree 4
研究在最大度数为4的图上，通过推动操作扩展了警官和小偷游戏的必胜策略条件。
A new bound for the cops and robbers problem
在这篇短文中，我们研究了在某个固定图$G$的顶点上进行的警匪博弈。警察和强盗被允许沿着$G$的边缘移动，警察的目标是抓住强盗。$G$的警察号码$c(G)$是赢得游戏所需的最低警察人数。Meyniel很久以前就证明了，对于$n$顶点上的任何连接的$G$，$O(\sqrt{n})$ cop就足够了。改进了已有的几个结果，证明了$n$ -顶点图的copnumber至多为$n 2^{-(1+o(1))\sqrt{\log n}}$ .
The robber strikes back
我们考虑一个新的警察和抢劫犯博弈，它与通常的警察和抢劫犯博弈相同，除了如果抢劫犯移动到一个包含一个警察的顶点，那么这个警察将从博弈中删除。我们研究了在图$G$上捕获一个抢劫犯所需的最少警察数，写为$cc(G)$。在二部图类和直径为2的无$K_{1,m}$图类中，我们给出了$cc(G)$的上界.
Cops and Robbers: A $\times$-homotopy Invariant Variant
本文引入了'狡猾主动警探与罪犯'变体游戏，证明在反射图中该游戏等同于经典游戏，并展示图的警探数在×-同调等价下保持不变。
Cops and Attacking Robbers with Cycle Constraints
本文研究警察与攻击型强盗博弈，刻画了警察数不超过2的无三角形图，证明二部平面图警察数至多为4且构造出达到4的紧例子，并首次构造出警察数差至少为3的图。
Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction which shows that a more sophisticated robber strategy can indefinitely evade three cops on a particular s…
In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph $G$. All players occupy vertices of $G$. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on $G$ is the cop number of $G$, denoted $c(G)$, and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an $n$-vertex graph with cop number $k$ is $O(n^{k+1})$. More recently, Bonato, Golovach, Hahn, and Kratochv…
In the ordinary version of the pursuit-evasion game "cops and robbers", a team of cops and a robber occupy vertices of a graph and alternately move along the graph's edges, with perfect information about each other. If a cop lands on the robber, the cops win; if the robber can evade the cops indefinitely, he wins. In the variant "lazy cops and robbers", the cops may only choose one member of their squad to make a move when it's their turn. The minimum number of cops (respectively lazy cops) required to catch the robber is called the "cop number" (resp. "lazy cop number") of G and is denoted $c…
We consider "Containment": a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by "containing" the robber---that is, by occupying all $°(v)$ of the edges incident with a vertex $v$ while the robber is at $v$. We develop bounds that relate the minimal number of cops, $ξ(G)$, required to contain a robber to the well-known "cop-number" $c(G)$ in the original game: in particular, $c(G) {\le} ξ(G) {\le} γ(G) Δ(G)$. We note that $ξ(G)…
Fast and Furious: A study on Monotonicity and Speed in Cops-and-Robber Games
该研究系统分析了不同可见性、行动模式与速度下警察与强盗博弈的单调性性质，明确了各类变体的警察宽度与路径宽度、树宽度的对应关系，并证明了强盗单调变体与非单调变体间的警察宽度差距有界。
Cops and robbers on $2K_2$-free graphs
本文证明了$2K_2$-自由图的警察数至多为2，并证实了$2K_1+K_2$-自由图警察数上界3的最优性，解决了相关猜想。
The Cops and Robber game on graphs with forbidden (induced) subgraphs
在无向有限图上进行两人完全信息博弈。一些警察和一个强盗被放置在顶点上，轮流沿着边滑动。如果在一次移动后，警察和强盗在同一个顶点上，警察就赢了。在图上抓住抢劫犯所需的最小警察数量称为该图的警察数量。本文研究了禁止一个或多个图为子图或诱导子图的图类中的cop数。在一个单一的禁止图的情况下，我们完全刻画（两种关系）的图形，迫使有界的copnumber。顺便说一下，我们用树的宽度来限制警察的数量。
The game of cops and robber is a turn based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say the team of cops win the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in each component of a graph is called the cop number of the graph. Sivaraman [Discrete Math. 342(2019), pp. 2306-2307] conjectured that for every $t\geq 5$, the cop number of a connected $P_t$-free graph is at most $t-3$, where $P_t$ denotes a path on…
We consider the well-studied cops and robbers game in the context of oriented graphs, which has received surprisingly little attention to date. We examine the relationship between the cop numbers of an oriented graph and its underlying undirected graph, giving a surprising result that there exists at least one graph $G$ for which every strongly connected orientation of $G$ has cop number strictly less than that of $G$. We also refute a conjecture on the structure of cop-win digraphs, study orientations of outerplanar graphs, and study the cop number of line digraphs. Finally, we consider some …
We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the …
Monotonicity of the cops and robber game for bounded depth treewidth
研究了刻画树宽的警察与强盗博弈变体，证明若k名警察有必胜策略则必有单调必胜策略，由此给出有界深度树宽的新刻画并肯定回答Fluck等人的开放问题。
Complexity results for a cops and robber game on directed graphs
该研究探究有向图警察与小偷博弈的警察数复杂性，证明单警察判定的NP难性与竞赛图的多项式可判定性，给出不同参数下的固定参数可处理性结论，并建立与反向图、矩阵死亡率问题的关联。
Cops and Robber Game with a Fast Robber on Interval, Chordal, and Planar Graphs
我们考虑一个由Fomin，Golovach，Kratochvil提出的警察和强盗博弈的变体，其中强盗具有无限的速度，也就是说，在她的回合中可以从她的顶点采取任何路径，但她不允许通过一个被警察占据的顶点。我们在区间图、弦图、可平面图和超立方体图上研究了这个博弈。令c_{\infty}（G）表示在该变体中捕获图G中的强盗所需的警察的数量。证明了如果G是区间图，则c_{\infty}（G）= O（sqrt（|V（G）|）），并给出了求区间图中c_{\infty}（G）的多项式时间3-近似算法.证明了对任意n，存在n-顶点弦图G，使c_{\infty}（G）= Omega（n / \log n）.令tw（G）和Delta（G）分别表示G的树宽和最大度。证明了对任意G，tw（G）+ 1 \leq（Delta（G）+ 1）c_{\infty}（G）.利用c_{\infty}（G）的这个下界，我们证明了两件事。第一个是，如果G是一个可平面图（或更一般地说，如果G没有固定顶点图作为子图），则c_{\infty}（G）= Theta（tw（G））。这立即导致O（1）-近似算法计算可平面图的c_{\infty}。第二个是如果G是m-超立方体图，则存在常数eta 1，eta 2>0使得（eta 1）2 ^m/（m sqrt（m））\leq c_{\infty}（G）\leq（eta 2）2 ^m/ m.
In previous work, we introduced median decompositions, a generalisation of tree decompositions where a graph can be modelled after any median graph, along with a hierarchy of $i$-medianwidth parameters $(mw_i)_{i\geq 1}$ starting from treewidth and converging to the clique number. We introduce another graph parameter based on the concept of median decompositions, to be called $i$-latticewidth and denoted by $lw_i$, for which we restrict the modelling median graph of a decomposition to be isometrically embeddable into the Cartesian product of $i$ paths. The sequence $(lw_i)_{i\geq 1}$ gives ris…
In many variants of the game of Cops and Robbers on graphs, multiple cops play against a single robber. In 2019, Cox and Sanaei introduced a variant of the game that gives the robber a more active role than simply evading the cop. In their version, the robber tries to damage as many vertices as possible and the cop attempts to minimize this damage. While the damage variant was originally studied with one cop and one robber, it was later extended to play with multiple cops by Carlson et. al in 2021. We take a different approach by studying the damage variant with multiple robbers against one co…
We survey results at the intersection of topological graph theory and the game of Cops and Robbers, focusing on results, conjectures, and open problems for the cop number of a graph embedded on a surface. After a discussion on results for planar graphs, we consider graphs of higher genus. In 2001, Schroeder conjectured that if a graph has genus $g,$ then its cop number is at most $g + 3.$ While Schroeder's bound is known to hold for planar and toroidal graphs, the case for graphs with higher genus remains open. We consider the capture time of graphs on surfaces and examine results for embeddin…
The capture time in the game of cops and many robbers
本文研究多名劫匪参与的警察与劫匪图博弈，给出了警察获胜图捕获时间的一般上界并证明其最佳性，同时揭示了捕获时间随劫匪数量增加的变化与0-可见性警察数量间的惊人关系。
Cops and robbers in random graphs
我们考虑有限的，连通的，无向图称为警察和强盗的追求和规避对策。Meyniel指出，对于每个n个顶点的图，有足够多的警察可以赢得比赛。我们证明，这举行了一个log（n）的随机图G（n，p）的因素，如果p不是很小，这是接近紧，除非图是非常密集的。我们分析了区域防御策略（Aigner在可平面图的情况下使用），并给出了它不能太有效的例子。
We present an algorithm of time complexity $O(kn^{k+2})$ deciding whether a graph $G$ on $n$ vertices is $k$-copwin. The fastest algorithm thus far had time complexity $O(n^{2k+2})$.
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