The capture time of grids

TL;DR

This paper determines the exact capture time for the game of Cops and Robber on grid graphs, showing it equals half the diameter for two cops, with specific results for m x n grids.

Contribution

It provides a precise formula for the 2-capture time on grid graphs, linking it to the graph's diameter, which was previously unknown.

Findings

2-capture time equals half the diameter of the graph

For m x n grids, the 2-capture time is floor((m+n-2)/2)

The results are based on perfect play assumptions

Abstract

We consider the game of Cops and Robber played on the Cartesian product of two trees. Assuming the players play perfectly, it is shown that if there are two cops in the game, then the length of the game (known as the 2-capture time of the graph) is equal to half the diameter of the graph. In particular, the 2-capture time of the m x n grid is proved to be floor ((m+n-2)/2).