具有周期边连通度的图上的警匪对策
A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity
摘要：
本文考虑了一个博弈，其中一个警察和一个强盗轮流沿着给定图$G$的边移动。如果存在一个策略，使警察最终能够定位在同一个顶点的强盗，那么$G$被称为警察赢，和强盗赢否则。我们研究这个经典的组合游戏在一个新的背景下，扩大类潜在的游戏竞技场，包括边周期图。这些是具有由离散时间步长组成的无限寿命的图，使得每个边缘$e$被分配长度为$l_e$的位模式，其中模式的第$i$位置中的1指示边缘$e$在$l_e$步长的每个连续块的第$i$步长中的存在。利用已经开发的框架的可达性游戏，我们扩展现有的技术，以获得，除其他结果外，$O(\textsf{LCM}(L)\cdot n^3)$上界的时间，以决定如果一个给定的$n$ -顶点边周期图$G^τ$是警察或强盗赢，以及计算一个战略，为获胜的球员（这里，$L$是所有边缘图案长度$l_e$的集合，并且$\textsf{LCM}(L)$表示集合$L$的最小公倍数）。另外，把我们的注意力转向边周期循环图，我们证明了$2\cdot l \cdot \textsf{LCM}(L)$上界的长度所需的任何边周期循环，以确保它是强盗赢，其中$l = 1$如果$\textsf{LCM}(L) \geq 2\cdot \max L $，和$l=2$否则。此外，我们提供了一个下界结构的形式cop-win边周期循环：一个长度为$1.5 \cdot \textsf{LCM}(L)$的$l=1$的情况下，一个长度为$3\cdot \textsf{LCM}(L)$的$l=2$的情况下。
英文摘要：
This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph $G$. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then $G$ is called cop-win, and robber-win otherwise. We study this classical combinatorial game in a novel context, broadening the class of potential game arenas to include the edge-periodic graphs. These are graphs with an infinite lifetime comprised of discrete time steps such that each edge $e$ is assigned a bit pattern of length $l_e$, with a 1 in the $i$-th position of the pattern indicating the presence of edge $e$ in the $i$-th step of each consecutive block of $l_e$ steps. Utilising the already-developed framework of reachability games, we extend existing techniques to obtain, amongst other results, an $O(\textsf{LCM}(L)\cdot n^3)$ upper bound on the time required to decide if a given $n$-vertex edge-periodic graph $G^τ$ is cop or robber win as well as compute a strategy for the winning player (here, $L$ is the set of all edge pattern lengths $l_e$, and $\textsf{LCM}(L)$ denotes the least common multiple of the set $L$). Separately, turning our attention to edge-periodic cycle graphs, we give proof of a $2\cdot l \cdot \textsf{LCM}(L)$ upper bound on the length required by any edge-periodic cycle to ensure that it is robber win, where $l = 1$ if $\textsf{LCM}(L) \geq 2\cdot \max L $, and $l=2$ otherwise. Furthermore, we provide lower bound constructions in the form of cop-win edge-periodic cycles: one with length $1.5 \cdot \textsf{LCM}(L)$ in the $l=1$ case and one with length $3\cdot \textsf{LCM}(L)$ in the $l=2$ case.
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