Which claw-free graphs are strongly perfect?

AbstractA set S of pairwise nonadjacent vertices in an undirected graph G is called a stable transversal of G if S meets every maximal (with respect to set-inclusion) clique of G. G is called strongly perfect if all its induced subgraphs (including G itself) have stable transversals. A claw is a graph consisting of vertices a,b,c,d and edges ab,ac,ad. We characterize claw-free strongly perfect graphs by five infinite families of forbidden induced subgraphs. This result—whose validity had been conjectured by Ravindra [Research problems, Discrete Math. 80 (1990) 105–107]—subsumes the characterization of strongly perfect line-graphs that was discovered earlier by Ravindra [Strongly perfect line graphs and total graphs, Finite and Infinite Sets. Colloq. Math. Soc. János Bolyai 37 (1981) 621–633]