A Covering Pursuit Game
Benjamin Gillott
TL;DR
This paper investigates the Covering pursuit game on grids, proving a subquadratic upper bound of $n^{1.999}$ cops and a lower bound of $n^{1.357}$ for large $n$ on $[n]^2$, and extends the discussion to higher dimensions via exact capture-time analysis. It introduces a phantom-cop/tiling strategy and a dynamic 'hole' mechanism to achieve the upper bound, while a density-based sub-square argument yields the lower bound. The authors further develop a time-constrained capture theory in $[n]^d$ showing that the required cops grow between $c_1 n^{d/2}$ and $c_2 n^{d/2+1}$, and connect these results to related games such as Rugby and Catching a Fast Robber. The work highlights subquadratic cop numbers as a meaningful regime in pursuit-evasion on grids and opens multiple avenues for refinement, especially in higher dimensions and for variations of robber speed.
Abstract
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 $[n]^2$. Bollobás 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, Bollobás, Narayanan and Shaw for this game.