  We consider the cop-throttling number of a graph $G$ for the game of Cops and
Robbers, 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 of
rounds needed for $k$ cops to capture the robber on $G$ over all possible
games. We provide some tools for bounding the cop-throttling number, including
showing that the positive semidefinite (PSD) throttling number, a variant of
zero forcing throttling, is an upper bound for the cop-throttling number. We
also characterize graphs having low cop-throttling number and investigate how
large the cop-throttling number can be for a given graph. We consider trees,
unicyclic graphs, incidence graphs of finite projective planes (a Meyniel
extremal family of graphs), a family of cop-win graphs with maximum capture
time, grids, and hypercubes. All the upper bounds on the cop-throttling number
we obtain for families of graphs are $ O(\sqrt n)$.
