Chasing a fast robber on planar graphs and random graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, that is, 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_∞(G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the tree-width of G. We show that if G is planar then c_∞(G) = Θ(tw(G)), and there is a polynomial-time constant-factor approximation algorithm for computing c∞(G).We also determine, up to constant factors, the value of c_∞(G) of the Erdo″s-Rényi random graph G = G(n, p) for all admissible values of p, and show that when the average degree is ω(1), c_∞(G) is typically asymptotic to the domination number.