Coarse geometry of the Cops and robber game

TL;DR

This paper introduces two new invariants for graphs based on variations of the cops and robber game, exploring their properties, invariance under graph perturbations, and behavior on special classes of graphs.

Contribution

It defines weak and strong cop numbers, proves their invariance under quasi-isometries, and analyzes their values on various graph families, including hyperbolic and vertex-transitive graphs.

Findings

Hyperbolic graphs have strong cop number one.

Finite or tree graphs have strong cop number one.

Some graphs have arbitrarily large weak or strong cop numbers.

Abstract

We introduce two variations of the cops and robber game on graphs. These games yield two invariants in $\mathbb{Z}_+\cup\{\infty\}$ for any connected graph $\Gamma$, the {weak cop number $\mathsf{wcop}(\Gamma)$} and the {strong cop number $\mathsf{scop}(\Gamma)$}. These invariants satisfy that $\mathsf{scop}(\Gamma)\leq\mathsf{wcop}(\Gamma)$. Any graph that is finite or a tree has strong cop number one. These new invariants are preserved under small local perturbations of the graph, specifically, both the weak and strong cop numbers are quasi-isometric invariants of connected graphs. More generally, we prove that if $\Delta$ is a quasi-retract of $\Gamma$ then $\mathsf{wcop}(\Delta)\leq\mathsf{wcop}(\Gamma)$ and $\mathsf{scop}(\Delta)\leq\mathsf{scop}(\Gamma)$. We exhibit families of examples of graphs with arbitrary weak cop number (resp. strong cop number). We prove that hyperbolic graphs have strong cop number one. We also prove that one-ended non-amenable locally-finite vertex-transitive graphs have infinite weak cop number. We raise the question of whether there exists a connected vertex transitive graph with finite weak (resp. strong) cop number different than one.