On Local Convexity in Graphs.

A chord of a path chi(0)chi(1)...chi(n) in a graph G is an edge chi(i)chi(j) of G where j>i+1. A chord of a cycle is defined similarly. A graph is chordal if every cycle of length greater than three has a chord. The class of chordal graphs has been studied extensively. It arises, for example, in the study of Gaussian elimination with no fill-in. It also arises in the study of abstract notions of convexity in graphs. Thus far in the study of convexity in graphs, two types of convexity have played a prominent role.