Generalized chordality, vertex separators and hyperbolicity on graphs

TL;DR

This paper explores the relationships between graph chordality, vertex separators, and hyperbolicity, providing characterizations of hyperbolic graphs and their properties related to tree-like structures.

Contribution

It introduces generalized chordality concepts and characterizes hyperbolic graphs through vertex separators and quasi-isometry to trees.

Findings

Characterization of hyperbolic graphs via chordality and vertex separators

Identification of conditions for graphs to be quasi-isometric to trees

Link between hyperbolicity and stable geodesics in graphs

Abstract

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.