Cop throttling number: Bounds, values, and variants
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The cop throttling number th_c(G) of a graph G for the game of Cops and Robbers is the minimum of k + capt_k(G), where k is the number of cops and capt_k(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games in which both players play optimally. In this paper, we answer in the negative a question from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418--2430.] about whether the cop throttling number of any graph is O(√(n)) by constructing a family of graphs having th_c(G)= Ω(n^2/3). We establish a sublinear upper bound on the cop throttling number and show that the cop throttling number of chordal graphs is O(√(n)). We also introduce the product cop throttling number th_c^×(G) as a parameter that minimizes the person-hours used by the cops. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph G, th_c^×(G)=1+rad(G).
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