  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.
