Cop and robber game and hyperbolicity

Abstract

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 δand s is linear for any s'<s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity δof a graph on n vertices in O(n^2) time when the graph is given by its distance-matrix.