On Necessary and Sufficient Number of Cops in the Game of Cops and Robber in Multidimensional Grids

TL;DR

This paper analyzes the Cops and Robber game in multidimensional grids, establishing the minimum number of cops needed and providing strategies for effective capture, including a special case in 2D grids.

Contribution

It introduces the first theoretical analysis of the game in multidimensional grids, determining necessary and sufficient cop numbers and strategies.

Findings

At least n cops are necessary in n-dimensional grids.

n cops are sufficient with the proposed strategies.

A single cop can catch the robber in 2D grids under certain conditions.

Abstract

We theoretically analyze the Cops and Robber Game for the first time in a multidimensional grid. It is shown that for an $n$-dimensional grid, at least $n$ cops are necessary to ensure capture of the robber. We also present a set of cop strategies for which $n$ cops are provably sufficient to catch the robber. Further, for two-dimensional grid, we provide an efficient cop strategy for which the robber is caught even by a single cop under certain conditions.