Throttling adversaries on trees
share
For the cop versus robber game, the throttling number th_c(G) of a graph G is the minimum possible value of k + capt_k(G) over all positive integers k, where capt_k(G) is the number of rounds needed for k cops to capture the robber on G if the cops and robber play optimally. We improve an upper bound on _T th_c(T) for all trees T of order n that was proved in [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430] from 2 √(n) to √(14)/2√(n) + O(1), and we show the same upper bound holds for chordal graphs. As a corollary, we obtain the same upper bound on throttling numbers for positive semidefinite (PSD) zero forcing on trees. We also use the same method to improve on previous throttling upper bounds for the cop versus gambler game. Furthermore, we exhibit a family of trees T of order n that have th_c(T) > 1.4502 √(n) for all n sufficiently large, improving on the previous lower bound of √(2n)-1/2 + 1 on _T th_c(T) for trees T of order n.
READ FULL TEXT