Coarse geometry of the cops and robber game

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