Throttling numbers for adversaries on connected graphs

In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430]. 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 $G$. One of the problems from [Breen et al., 2018] was to determine whether there exists a family of trees $T$ of order $n$ for which $th_c(T)$ is asymptotically equal to $2 \sqrt{n}$. We show that such a family cannot exist by improving the upper bound on $\displaystyle \max_{T} th_c(T)$ for all trees $T$ of order $n$ from $2 \sqrt{n}$ to $\frac{\sqrt{14}}{2} \sqrt{n} + O(1)$. We prove this bound by deriving a more general throttling bound for connected graphs that applies to multiple graph adversaries, including the robber and the gambler. This also improves the best known upper bounds on $th_c(G)$ for chordal graphs and unicyclic graphs $G$, as well as throttling numbers for positive semidefinite (PSD) zero forcing on trees. In addition to the results about cop versus robber, we use our general throttling bound to improve previous upper bounds on throttling numbers for the cop versus gambler game on connected graphs. Another open problem from [Breen et al., 2018] was to obtain a bound on $th_c(G)$ for cactus graphs $G$. We prove an $O(\sqrt{n})$ bound for all cactus graphs $G$ of order $n$. Furthermore, we exhibit a family of trees $T$ of order $n$ that have $th_c(T) > 1.4502 \sqrt{n}$ for all $n$ sufficiently large, improving on the previous lower bound of $\lceil \sqrt{2n}-\frac{1}{2} \rceil + 1$ on $\displaystyle \max_{T} th_c(T)$ for trees $T$ of order $n$.