A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

TL;DR

This paper improves bounds on the minimum number of cops needed to catch a robber in the Cops & Robber game on graphs embedded in non-orientable surfaces by using covering space techniques, refining previous estimates.

Contribution

It introduces a reduction method to orientable surfaces for non-orientable cases, leading to tighter bounds on cop numbers for graphs on such surfaces.

Findings

Maximum cop number for graphs in the projective plane is 3.

Cop number for graphs in the Klein Bottle is at most 4.

Upper bounds for cop numbers on non-orientable surfaces of genus g are improved.

Abstract

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.