On Uni Chord Free Graphs

A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show that such a graph can be recognized in polynomial time. Further, we show that the chromatic number of unichord free graphs is one of (2,3, \omega(G)). We also present a polynomial-time algorithm to produce a coloring with \omega(G) colors.