SLIGHTLY TRIANGULATED GRAPHS ARE PERFECT

A graph is triangulated if it has no chordless cycle with at least four vertices (FOR-ALL k greater-than-or-equal-to 4, C(k) not-subset-of-or-equal-to G). These graphs have been generalized by R. Howard with the weakly triangulated graphs (FOR-ALLk greater-than-or-equal-to 5, C(k), CBAR(k) not-subset-of-or-equal-to G). In this note we propose a new generalization of triangulated graphs. A graph G is slightly triangulated if it satisfies the two following conditions; 1. G contains no chordless cycle with at least 5 vertices. 2. For every induced subgraph H of G, there is a vertex in H the neighbourhood of which in H contains no chordless path of 4 vertices. We shall prove that these graphs are perfect, and compare them with other classical families of perfect graphs.