我们考虑了一个变形的警察和强盗的游戏，其中强盗可以移动t边的时间，并表明，在这个变形中，d-正则图的周长大于2 t +2的警察数是Ω（d^t）。通过已知的笼阶的上界，这意味着当t>1时，连通n-顶点图的cop数可以大到Omega（n^{2/3}），当t>3时，可以大到Omega（n^{4/5}）。当1<t<7时，这改进了Frieze、Krivelevich和Loh（Variations on Cops and Robbers，J.图论，2011）的Omega（n^{（t-3）/（t-2）}）下界。我们还猜想了这个变体中的cop数的一般上界O（n^{t/t+1}），推广了Meyniel猜想.
英文摘要：
We consider a variant of the Cops and Robbers game where the robber can move t edges at a time, and show that in this variant, the cop number of a d-regular graph with girth larger than 2t+2 is Omega(d^t). By the known upper bounds on the order of cages, this implies that the cop number of a connected n-vertex graph can be as large as Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, 2011) when 1<t<7. We also conjecture a general upper bound O(n^{t/t+1}) for the cop number in this variant, generalizing Meyniel's conjecture.
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.…
我们考虑警察和强盗游戏的一个变体，称为懒惰的警察和强盗，其中最多一个警察可以在任何一轮中移动。我们研究了这个博弈中警察号码的类似物，我们称之为懒惰警察号码。Lazy Cops and Robbers最近由Offner和Ojakian提出，他们给出了超立方体的Lazy Cops数的渐近上界和下界。通过研究扩张性质，我们给出了二项随机图$\mathcal{G}(n,p)$的懒惰copnumber的渐近几乎必然界，并给出了$p=p(n)$的广泛范围.通过将概率方法与一个势函数参数相结合，我们还改进了现有的超立方体的懒拷贝数的下界。最后，利用Gilbert-Hutchinson-Tarjan分离子定理，给出了亏格为$g$的图的懒惰copnumber的一个上界.
Nowakowski和Winkler在他们关于Cops and Robbers游戏的开创性论文中提供了cop-win图的关系特征。作为该特征的副产品，每个cop-win图被分配一个唯一的序数，我们称之为CR-序数。对于有限图，CR-序数对应于假设最优博弈的博弈长度，其中警察在最不利的初始位置开始博弈。然而，对于无限图，CR-序数的可能值在文献中没有被考虑，直到目前的工作。我们将cop-win树的CR序数分为有限序数和$α+ ω$形式的CR序数，其中$α$是极限序数.对于一般的无限cop-win图，我们提供了一个例子，其CR-序数不是这种形式。最后，我们讨论了在一般情况下的cop-win图的CR-序数的特征问题。
Chasing robbers on random geometric graphs---an alternative approach
我们研究了\emph{Cops and Robbers}的顶点追踪博弈，其中警察试图在图的顶点上抓住一个强盗。在给定的图$G$上获胜所需的最小cops数称为$G$的cops数。本文研究了随机几何图$G_{d}(n,r)$，其中$n$的顶点是独立于$[0,1]^d$的随机均匀选取的，如果两个顶点之间的欧氏距离至多为$r$，则两个顶点相邻.主要的结果是，如果$r^{3d-1}>c_d \frac{\log n}{n}$，则cop数是$1$，随着$n$趋于无穷大，概率趋于$1$。病例$d=2$在\cite{bdfm}中使用不同的方法较早且独立地得到证实。我们的方法提供了一个严格的$O(1/r^2)$上限的轮数需要抓住强盗。
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.
University of Primorska（波希米亚大学）；Université Caen Normandie, ENSICAEN, CNRS, Normandie Univ, GREYC UMR 6072（卡昂诺曼底大学，ENSICAEN，法国国家科学研究中心，诺曼底大学，GREYC UMR 6072）；Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院数学与信息学系）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）
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 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…
The localization game is played by two players: a Cop with a team of $k$ cops, and a Robber. The game is initialised by the Robber choosing a vertex $r \in V$, unknown to the Cop. Thereafter, the game proceeds turn based. At the start of each turn, the Cop probes $k$ vertices and in return receives a distance vector. If the Cop can determine the exact location of $r$ from the vector, the Robber is located and the Cop wins. Otherwise, the Robber is allowed to either stay at $r$, or move to $r'$ in the neighbourhood of $r$. The Cop then again probes $k$ vertices. The game continues in this fashi…
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.
In this paper we analyze and model three open problems posed by Harris, Insko, Prieto-Langarica, Stoisavljevic, and Sullivan in 2020 concerning the tipsy cop and robber game on graphs. The three different scenarios we model account for different biological scenarios. The first scenario is when the cop and robber have a consistent tipsiness level though the duration of the game; the second is when the cop and robber sober up as a function of time; the third is when the cop and robber sober up as a function of the distance between them. Using Markov chains to model each scenario we calculate the…
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 …
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 $O(n^{2k+2})$.
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.
