Generalized chordality, vertex separators and hyperbolicity on graphs

Let $G$ be a graph with the usual shortest-path metric. A graph is $\delta$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. In this paper we study the relation between vertex separator sets, some chordality properties which are natural generalizations of being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons.Comment: 16 pages, 3 figure

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'<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

Ricci Curvature of the Internet Topology

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin, etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201