Coarse geometry of the Cops and robber game

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.