The game of Cops and Robber on geodesic spaces

TL;DR

This paper extends the game of Cops and Robber from finite graphs to general geodesic spaces, analyzing how the cop number relates to the topology of surfaces and proposing it as a new geometric invariant.

Contribution

It introduces the game on arbitrary geodesic spaces, establishes its equivalence with the discrete game on graphs, and relates cop number to surface genus.

Findings

Cop number is bounded by a function of surface genus

Exact cop number is 3 for certain low-genus surfaces

Cop number grows as O(g) with genus g

Abstract

The game of Cops and Robber is traditionally played on a finite graph. The purpose of this paper is to introduce and analyse the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer $c$ such that $c$ cops can win the game against one robber, and $c$ only depends on the genus $g$ of the surface. It is shown that $c=3$ for orientable surfaces of genus $0$ or $1$ and nonorientable surfaces of crosscap number $1$ or $2$ (with any number of boundary components) and that $c=O(g)$ and that $c=\Omega(\sqrt{g})$ when the genus $g$ is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space.