A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity

Abstract

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. In contrast to previous work, we study this classical combinatorial game on 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 known framework of reachability games, we obtain an $O(\textsf {LCM}(L)\cdot n^3)$ time algorithm to decide if a given n-vertex edge-periodic graph $G^\tau$ is cop-win 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). For the special case of edge-periodic cycles, we prove an upper bound of $2\cdot l \cdot \textsf {LCM}(L)$ on the minimum length required of any edge-periodic cycle to ensure that it is robber-win, where $l = 1$ if $\textsf {LCM}(L) \ge 2\cdot \max L$, and $l=2$ otherwise. Furthermore, we provide constructions of edge-periodic cycles that are cop-win and have length $1.5 \cdot \textsf {LCM}(L)$ in the $l=1$ case and length $3\cdot \textsf {LCM}(L)$ in the $l=2$ case.