Distance d-Domination Games

We study graph searching games where a number of cops try to capture a robber that is hiding in a system of tunnels modelled as a graph. While the current position of the robber is unknown to the cops, each cop can see a certain radius d around his position. For the case d = 1 these games have been studied by Fomin, Kratsch and Müller [7] under the name domination games. We are primarily interested in questions concerning the complexity and monotonicity of these games. We show that dominating games are computationally much harder than standard graph searching games where the cops only see their own vertex and establish strong non-monotonicity results for various notions of monotonicity which arise naturally in the context of domination games. Answering a question of [7], we show that there exists graphs for which the shortest winning strategy for a minimal number of cops must necessarily be of exponential length. On the positive side, we establish tractability results for graph classes of bounded degree