On the damage number of graphs

We study a variant of Cops and Robbers in which the robber attempts to visit as many vertices of the graph as possible without being captured, while the cop aims to keep the robber confined to a small set of vertices. The $\mathit{\text{damage}}\mathit{\text{number}}$ of a graph $G$, introduced by Cox and Sanaei in 2019, is the maximum number of vertices the robber can visit in a game of Cops and Robbers on $G$. In this paper, we determine damage numbers for several classes of graphs, including hypercubes, Hamming graphs, Johnson graphs, incidence graphs of projective planes, and Erdős-Rényi random graph $G(n,p)$, for $p\gg {\log }^{2\mathrm{/}5}(n)\mathrm{/}{n}^{1\mathrm{/}5}$. We also show that the problem of determining the damage number of a graph is {\sf PSPACE}-complete.