A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity
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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(LCM(L)· 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 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· l ·LCM(L) upper bound on the length required by any edge-periodic cycle to ensure that it is robber win, where l = 1 if LCM(L) ≥ 2·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 ·LCM(L) in the l=1 case and one with length 3·LCM(L) in the l=2 case.
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