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Chordality properties and hyperbolicity on graphs

Let $G$ be a graph with the usual shortest-path metric. A graph is $δ$-hyperbolic if for every geodesic triangle $T$, any side of $T$ is contained in a $δ$-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 $δ$-hyperbolic. Moreover, we obtain a characterization of being hyperbolic on terms of a chordality property on the triangles.