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# Title: Generalized chordality, vertex separators and hyperbolicity on 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.
 Comments: 16 pages, 3 figures Subjects: Combinatorics (math.CO) MSC classes: Primary: 05C63, 05C75. Secondary: 05C38, 05C12 Cite as: arXiv:1708.06153 [math.CO] (or arXiv:1708.06153v1 [math.CO] for this version)

## Submission history

From: Alvaro Martínez-Pérez [view email]
[v1] Mon, 21 Aug 2017 11:21:52 GMT (282kb,D)