A Covering Pursuit Game

In the `Covering' pursuit game on a graph, a robber and a set of cops play alternately, with the cops each moving to an adjacent vertex (or not moving) and the robber moving to a vertex at distance at most 2 from his current vertex. The aim of the cops is to ensure that, after every one of their turns, there is a cop at the same vertex as the robber. How few cops are needed? Our main aim in this paper is to consider this problem for the two-dimensional grid $[n]^2$. Bollob\'{a}s and Leader asked if the number of cops needed is $o(n^2)$. We answer this question by showing that $n^{1.999}$ cops suffice. We also consider some applications. In particular we study the game `Catching a Fast Robber', concerning the number of cops needed to catch a fast robber of speed $s$ on the two-dimensional grid $[n]^2$. We improve the bounds proved by Balister, Bollob\'{a}s, Narayanan and Shaw for this game.