A Sublinear Bound on the Cop Throttling Number of a Graph

01/25/2019

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by   Anthony Bonato, et al.

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We provide a sublinear bound on the cop throttling number of a connected graph. Related to the graph searching game Cops and Robbers, the cop throttling number, written th_c(G), is given by th_c(G)=_k{k+capt_k(G)}, in which capt_k(G) is the k-capture time, or the length of a game of Cops and Robbers with k cops on the graph G, assuming both players play optimally. No general sublinear bound was known on the cop throttling number of a connected graph. Towards a question asked by Breen et al., we prove that th_c(G)≤(2+o(1))n√(W((n)))/√((n)), where W=W(x) is the Lambert W function.