Cop-Win Graphs: Optimal Strategies and Corner Rank

TL;DR

This paper introduces corner ranking to characterize cop-win graphs, enabling determination of capture time and optimal strategies for both cop and robber in the game.

Contribution

It provides a new corner ranking method for cop-win graphs, linking it to dismantling orderings and defining optimal strategies for both players.

Findings

Corner rank characterizes cop-win graphs.

Capture time can be determined from corner rank.

Optimal strategies are defined as Lower Way and Higher Way strategies.

Abstract

We investigate the game of cops and robber, played on a finite graph, between one cop and one robber. 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 $\infty$ to each vertex. We give a characterization of cop-win in terms of corner rank and also show that the well-known characterization of cop-win 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.