Variations of cops and robbers game on grids

Abstract

Cops and robber is a two player turn based game played on a graph where the cops try to catch the robber. The cop number of a graph is the minimum number of cops required to catch the robber. We consider two variants of this game, namely cops and attacking robber, and lazy cops and robber. In cops and attacking robber, the robber can attack a cop and remove him from the game, whereas in lazy cops and robber only one cop is allowed to move during the cop’s turn. We prove that the cop number for both these variants in finite square grids is two. We show that the cop number for cops and attacking robber in n-dimensional hypercube is at most n, and the cop number in the same version of a 3-dimensional grid is three.