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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, 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∞(G) denote the number of cops needed to capture the robber in a graph G in this va ..."
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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∞(G) denote the number of cops needed to capture the robber in a graph G in this variant, and let tw(G) denote the treewidth of G. We show that if G is planar then c∞(G) = Θ(tw(G)), and there is a constantfactor approximation algorithm for computing c∞(G). We also determine, up to constant factors, the value of c∞(G) of the ErdősRényi 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.