Cops and Robber on butterflies, grids, and AT-free graphs

Abstract

Cops and Robber is a well-studied two player pursuit-evasion game played on a graph. In this game, a set of cops, controlled by the first player, tries to capture the position of a robber, controlled by the second player. The cop number of a graph is the minimum number of cops required to capture the robber in the graph. A group of cops guard a subgraph if they can ensure that the robber cannot enter the subgraph without getting captured immediately.We study the applications of guarding to provide new bounds and improve existing bounds for several graph classes. In particular, we show that the cop number for butterfly networks and for solid grids is two. We also construct a partial grid with cop number 3 establishing that partial grids have same cop number as their superclass planar graphs. We also consider three well-studied variants of Cops and Robber: Cops and Fast Robber, Cops and Attacking Robber, and Surrounding CnR. We improve the existing bounds for the cop number of multidimensional grids for both Cops and Attacking Robber and Surrounding CnR. Finally, we consider Cops and Fast Robber on AT-free graphs to improve the existing bounds on the cop number for this game.