On Closed Graphs I

A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.
Comment: The paper "On Closed Graphs" (1306.5149v1) has been divided into two papers, "On Closed Graphs I", which is this arXiv submission, and "On Closed Graphs II", which will be a separate arXiv submission