Slightly triangulated graphs are perfect

A graph is triangulated if it has no chordless cycle with at least four vertices (8k 4; C k 6 6 G). These graphs have been generalized by R. Hayward with the weakly triangulated graphs (8k 5; C k ; C k 6 6 G). In this note we propose a new generalization of triangulated graphs. A graph G is slightly triangulated if it satisses 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.