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We consider the cop-throttling number of a graph $G$ for the game of Cops andRobbers, which is defined to be the minimum of $(k + \text{capt}_k(G))$, where$k$ is the number of cops and $\text{capt}_k(G)$ is the minimum number ofrounds needed for $k$ cops to capture the robber on $G$ over all possiblegames. We provide some tools for bounding the cop-throttling number, includingshowing that the positive semidefinite (PSD) throttling number, a variant ofzero forcing throttling, is an upper bound for the cop-throttling number. Wealso characterize graphs having low cop-throttling number and investigate howlarge the cop-throttling number can be for a given graph. We consider trees,unicyclic graphs, incidence graphs of finite projective planes (a Meynielextremal family of graphs), a family of cop-win graphs with maximum capturetime, grids, and hypercubes. All the upper bounds on the cop-throttling numberwe obtain for families of graphs are $ O(\sqrt n)$.

Throttling for the game of Cops and Robbers on graphs