Cop and robber game and hyperbolicity

TL;DR

This paper establishes a connection between cop and robber game dynamics with different speeds and the hyperbolicity of graphs, providing new characterizations and approximation methods for Gromov hyperbolicity.

Contribution

It proves that cop-win graphs with different speeds are -hyperbolic with a quadratic bound and introduces a game-theoretic characterization of Gromov hyperbolicity.

Findings

All cop-win graphs with s'<s are -hyperbolic with (s^2) bound.

Linear dependency between  and s under certain conditions.

A simple O(n^2) approximation algorithm for hyperbolicity based on distance matrices.

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'<s, are \delta-hyperbolic with \delta=O(s^2). We also show that the dependency between \delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger condition. This solves an open question from the paper (J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and s'=r+2\delta for any r>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.