Chordality properties and hyperbolicity on graphs

TL;DR

This paper explores the relationship between hyperbolicity and chordality properties in graphs, providing new characterizations and generalizations that deepen understanding of graph geometric structures.

Contribution

It introduces new chordality properties related to hyperbolicity and offers a characterization of hyperbolic graphs through chordality conditions on triangles.

Findings

Identifies chordality properties that are weaker and stronger than hyperbolicity.

Provides a new characterization of hyperbolic graphs based on chordality.

Establishes connections between hyperbolicity and generalized chordality properties.

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. In this paper we study the relation between the hyperbolicity of the graph and some chordality properties which are natural generalizations of being chordal. We find chordality properties that are weaker and stronger than being $\delta$-hyperbolic. Moreover, we obtain a characterization of being hyperbolic on terms of a chordality property on the triangles.