Cop-Win Graphs: Optimal Strategies and Corner Rank

The game of cops and robber is a two-player game, played on a graph, between a cop and a robber. To begin the game, the cop and robber each choose a vertex to occupy, with the cop choosing first. Play then alternates between the cop and the robber, with the cop moving first; on a turn a player may move to an adjacent vertex or stay still. If the cop and robber ever occupy the same vertex, the cop wins. If the cop can force a win on a graph, the graph is called cop-win. We describe a procedure we call corner ranking, performed on a graph, which assigns a positive integer or ∞ to each vertex. We give a characterization of cop-win in terms of corner rank and also show that the well-known characterization of copwin via dismantling orderings follows from our work. From the corner rank we can determine the capture time of a graph, i.e. the number of turns the cop needs to win. We describe a class of optimal cop strategies we call Lower Way strategies, and a class of optimal robber strategies we call Higher Way strategies. Roughly speaking, in a Lower Way strategy, the cop pushes the robber down to lower ranked vertices, while in a Higher Way strategy, the robber moves to a highest rank vertex that is “safe.” While interesting in their own right, the strategies are themselves tools in our proofs. We investigate various properties of the Lower Way strategies.