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�, 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 steps. Utilising the known framework of reachability games, we obtain an time algorithm to decide if a given n-vertex edge-periodic graph 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, and denotes the least common multiple of the set L). For the special case of edge-periodic cycles, we prove an upper bound of on the minimum length required of any edge-periodic cycle to ensure that it is robber-win, where if, and otherwise. Furthermore, we provide constructions of edge-periodic cycles that are cop-win and have length in the case and length in the case.