A graph is strongly perfect if every induced subgraph has an independent vertex set meeting all maximal cliques
of
(Berge and Duchet 1984, Ravindra 1999).
Every strongly perfect graph is perfect, but the
converse is not necessarily true.
Every -free
graph (i.e., every graph not containing the path graph as a vertex-induced subgraph), is
strongly perfect (Ravindra 1999).
Berge, C. and Duchet, P. "Strongly Perfect Graphs." Ann. Disc. Math.21, 57-61, 1984.Ravindra, G. "Some
Classes of Strongly Perfect Graphs." Disc. Math.206, 197-203,
1999.Wang, H. Y. "Which Claw-Free Graphs Are Strongly Perfect?"
Disc. Math.306, 2602-2629, 2006.
has an independent vertex set meeting all maximal cliques
of 
(Berge and Duchet 1984, Ravindra 1999).
-free
graph (i.e., every graph not containing the path graph 
as a vertex-induced subgraph), is
strongly perfect (Ravindra 1999).
