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 Z+∪{∞} for any connected graph Γ, the weak cop numberwCop(Γ) and strong cop numbersCop(Γ). These invariants satisfy that sCop(Γ)≤wCop(Γ). 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 Δ is a quasi-retract of Γ then wCop(Δ)≤wCop(Γ) and sCop(Δ)≤sCop(Γ). We exhibit families of examples of graphs with arbitrary weak cop numbers (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.