Catching an infinitely fast robber on a grid

TL;DR

This paper investigates a variant of the Cops and Robbers game where the robber can move infinitely fast on grid graphs, providing bounds on the number of cops needed for various grid structures.

Contribution

It introduces a new model allowing the robber unlimited movement per turn and determines cop numbers for multiple grid classes, including asymptotic bounds for higher dimensions.

Findings

Cop numbers for 2D grids and tori are determined up to an additive constant.

Asymptotic bounds for cop numbers of higher-dimensional grids.

Results extend understanding of pursuit-evasion dynamics on grid graphs.

Abstract

We consider a variant of Cops and Robbers in which the robber may traverse as many edges as he likes in each turn, with the constraint that he cannot pass through any vertex occupied by a cop. We study this model on several classes of grid-like graphs. In particular, we determine the cop numbers for two-dimensional Cartesian grids and tori up to an additive constant, and we give asymptotic bounds for the cop numbers of higher-dimensional grids and hypercubes.