Cop and Robber Game and Hyperbolicity

In this note, we prove that all cop-win graphs $G$ in the game in which the robber and the cop move at different speeds $s$ and $s'$ with $s' 0$, this establishes a new---game-theoretical---characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between $\delta$ and $s$ is linear for any $s'<s$. Using these results, we describe a simple constant-factor approximation of the hyperbolicity $\delta$ of a graph on $n$ vertices in $O(n^2)$ time when the graph is given by its distance matrix.

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