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

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 is assigned a bit pattern of length le, with a 1 in the i-th position of the pattern indicating the presence of edge in the i-th step of each consecutive block of le steps. Utilising the known framework of reach-ability games, we obtain an O(LCM(L)·n3) time algorithm to decide if a given n-vertex edge-periodic graph Gτ 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 le, and 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·l·LCM(L) on the minimum length required of any edge-periodic cycle to ensure that it is robber-win, where l= 1 if LCM(L)≥2·maxL, and l= 2 otherwise. Furthermore, we provide constructions of edge-periodic cycles that are cop-win and have length1.5·LCM(L) in the l= 1 case and length 3·LCM(L) in the l= 2 case.