Chasing a Fast Robber on Planar 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 ∞ (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 constant-factor approximation algorithm for computing c ∞ (G). We also determine, up to constant factors, the value of c ∞ (G) of the Erdős-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.

Lazy Cops and Robbers played on random graphs and graphs on surfaces

“…A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer [8,9], play on edges [15], allow one or both players to move with different speeds [2,17] or to teleport, allow the robber to capture the cops [10], or make the robber invisible or drunk [20,21]. See Chapter 8 of [11] for a non-comprehensive survey of variants of Cops and Robbers.…”

Network Decontamination with a Single Agent

“…A simple approach to fully decontaminate a p × q mesh would be to place our agent 271 at v (1,1) at t = 0, proceed to visit all vertices in the column till we reach v (1,p) , move 272 right one step to v (2,p) and proceed all the way down to v (2,1) . This process may now 273 be continued by moving the agent to v (3,1) and going on to decontaminate the entire 274 graph column by column until we reach the last vertex. Clearly an temporal immunity 275 of 2p − 1 is enough for this strategy to monotonically decontaminate the entire graph.…”

“…Set the temporal immunity τ = p and start with the agent at v (1,1) . Proceed all the way up to v (1,p) , move the agent to the next column onto v (2,p) , and then start traversing down the column until we reach v (2, p 2 +1) .…”

“…Proceed all the way up to v (1,p) , move the agent to the next column onto v (2,p) , and then start traversing down the column until we reach v (2, p 2 +1) . Note that the vertices of C 1 had started getting recontaminated when the agent reached v (2,p−1) because the exposure time of v (1,1) became equal to τ at that point. Now move the agent back to C 1 onto v (1, p 2 +1) and proceed all the way down back to v (1,1) .…”

“…Note that the vertices of C 1 had started getting recontaminated when the agent reached v (2,p−1) because the exposure time of v (1,1) became equal to τ at that point. Now move the agent back to C 1 onto v (1, p 2 +1) and proceed all the way down back to v (1,1) . We declare that C 1 has been decontaminated and none of its vertices will be recontaminated during the course of decontamination of the rest of the graph.…”