On the hyperbolicity of edge-chordal and path-chordal graphs

Abstract

If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3 } is the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the other two sides, for every geodesic triangle T in X. An important problem in the study of hyperbolic graphs is to relate the hyperbolicity with some classical properties in graph theory. In this paper we find a very close connection between hyperbolicity and chordality: we extend the classical definition of chordality in two ways, edge-chordality and path-chordality, in order to relate this property with Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph with small path-chordality constant is hyperbolic.

Several extremal problems on graphs involving the circumference, girth, and hyperbolicity constant

“…The papers [11,52,5,13] prove, respectively, that chordal, k-chordal, edge-chordal and join graphs are hyperbolic. Moreover, in [5] it is shown that hyperbolic graphs are path-chordal graphs. These results relating chordality and hyperbolicity are improved in [34].…”

Hyperbolicity of Direct Products of Graphs

“…Thus, a way to approach the problem is to study hyperbolicity for particular types of graphs. In this line, many researches have studied the hyperbolicity of several classes of graphs: chordal graphs [7,14,40,54], vertex-symmetric graphs [15], bipartite and intersection graphs [22], bridged graphs [37], expanders [39] and some products of graphs: Cartesian product [41], strong product [16], corona and join product [17].…”

Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

“…They prove that k-chordal graphs are hyperbolic where a graph is k-chordal if every induced cycle has at most k edges. In [1], the authors define the more general properties of being (k, m)-edge-chordal and (k, k 2 )-path-chordal and prove that every (k, m)-edge-chordal graph is hyperbolic and that every hyperbolic graph is (k, k 2 )-path-chordal. In [32], we continue this work and define being ε-densely (k, m)-path-chordal and ε-densely k-path-chordal.…”

“…In [32], we continue this work and define being ε-densely (k, m)-path-chordal and ε-densely k-path-chordal. In [1] and [32], edges where aloud to have any finite length but in this work we assume that all edges have length one. Therefore, the distinction between edge and path is unnecessary and these properties are referred as (k, m)-chordal and ε-densely k-chordal.…”