Abstract
Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.
Highlights
The focus of the first works on Gromov hyperbolic spaces were finitely generated groups [1].Initially, the main application of hyperbolic spaces were the automatic groups.This concept appears in some algorithmic problems.Besides, they are useful in the study of secure transmission of information on the internet [4].In [5], the equivalence of the hyperbolicity of graphs and negatively curved surfaces was proved
We would like to mention that Ref. [38] collects very rich results, especially those concerning path properties, about interval graphs and unit interval graphs. It is well-known that interval graphs and indifference graphs are hyperbolic
One of the main results in this paper is Theorem 8, which provides a sharp upper bound of the hyperbolicity constant of interval graphs verifying a very weak hypothesis
Summary
The focus of the first works on Gromov hyperbolic spaces were finitely generated groups [1]. If a graph G has edges with different lengths, we assume that it is locally finite These properties guarantee that any connected component of G is a geodesic metric space. One of the main results in this paper is Theorem 8, which provides a sharp upper bound of the hyperbolicity constant of interval graphs verifying a very weak hypothesis. This result allows for obtaining bounds for the hyperbolicity constant of every indifference graph (Corollary 6) and the hyperbolicity constant of every interval graph with edges of length 1 (Corollary 7). The main result in this paper is Theorem 9, which allows for computing the hyperbolicity constant of every interval graph with edges of length 1, by using geometric criteria
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