Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

Abstract

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is δ-hyperbolic if every geodesic triangle is δ-thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize 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.

Mathematical Properties on the Hyperbolicity of Interval Graphs

“…The papers [6,9,28] prove, respectively, that chordal, k-chordal and edge-chordal graphs are hyperbolic; these results are improved in [23]. In addition, several authors have proved results on hyperbolicity for some particular classes of graphs (see, e.g., [21,[29][30][31]).…”

A note on isoperimetric inequalities of Gromov hyperbolic manifolds and graphs

“…Besides, Gromov hyperbolic spaces are an important class of metric spaces [14,16,19,32,54]. Gromov hyperbolicity has been thoroughly studied not only in Cayley graphs, but also in general graphs (see, e.g., [5,6,11,12,15,39,40,47,[49][50][51]55] and the references therein). Also, hyperbolicity has been applied in several areas such as the secure transmission of information and virus propagation on networks [35,36], real networks [1,2,24,38,45], or phylogenetics [25,26].…”

Vertex separators, chordality and virtually free groups

“…In [24] the second author studied some relations between vertex separator sets, certain chordality properties that generalize being chordal and conditions for a graph to be quasi-isometric to a tree. Some of these ideas can be easily translated to the language of Geometric Group Theory.…”

“…A graph Γ satisfies the bottleneck property (BP) if there exists some constant ∆ > 0 so that given any two distinct points x, y ∈ V (Γ) and a midpoint c ∈ Γ such that d(x, c) = d(y, c) = 1 2 d(x, y), then every xy-path intersects N ∆ (c). This is an equivalent definition to the original which was defined in [22], see Proposition 2 in [24].…”