We consider "surrounding" versions of the classic Cops and Robber game. The game is played on a connected graph in which two players, one controlling a number of cops and the other controlling a robber, take alternating turns. In a turn, each player may move each of their pieces: The robber always moves between adjacent vertices. Regarding the moves of the cops we distinguish four versions that differ in whether the cops are on the vertices or the edges of the graph and whether the robber may move on/through them. The goal of the cops is to surround the robber, i.e., occupying all neighbors (vertex version) or incident edges (edge version) of the robber's current vertex. In contrast, the robber tries to avoid being surrounded indefinitely. Given a graph, the so-called cop number denotes the minimum number of cops required to eventually surround the robber. We relate the different cop numbers of these versions and prove that none of them is bounded by a function of the classical cop number and the maximum degree of the graph, thereby refuting a conjecture by Crytser, Komarov and Mackey [Graphs and Combinatorics, 2020].

翻译：我们考虑经典“警察与小偷”游戏中的“包围”版本。游戏在一个连通图上进行，两名玩家（一名控制若干警察，另一名控制一个小偷）交替行动。在每个回合中，每位玩家可以移动其所有棋子：小偷总是在相邻顶点之间移动。关于警察的移动，我们区分了四个版本，这些版本的区别在于警察位于图的顶点还是边上，以及小偷是否可以移动至/通过它们。警察的目标是包围小偷，即占据小偷当前顶点的所有邻居（顶点版本）或关联边（边版本）。相反，小偷试图避免被无限期包围。给定一个图，所谓的警察数表示最终包围小偷所需的最少警察数量。我们建立了这些版本中不同警察数之间的关系，并证明它们中没有一个受经典警察数和图的最大度的函数所限制，从而反驳了Crytser、Komarov和Mackey [Graphs and Combinatorics, 2020]的一个猜想。