Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

Abstract

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c_{infty}(G)=1, and give an O(|V(G)|^2) algorithm for their detection. We prove a lower bound for c_{infty} of expander graphs, and use it to prove three things. The first is that if np > 4.2 log n then the random graph G = G(n,p) asymptotically almost surely has e1/p < c_{infty}(G) < e2 log (np)/p, for suitable constants e1 and e2. The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has c_{infty}(G) = Theta(n). The third is that if G is a Cartesian product of m paths, then n / 4km^2 < c_{infty}(G) < n / k, where n=|V(G)| and k is the number of vertices of the longest path.