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, so any graph that is not perfect cannot be strongly perfect. However, the converse is not necessarily true.
Important classes of strongly perfect graphs include bipartite graphs, comparability graphs, chordal
graphs (also called triangulated graphs), graph
complements of chordal graphs, and 
-free graphs (i.e., graphs not containing the path
graph 
as a vertex-induced subgraph) (Berge and Duchet 1984, Ravindra 1999). In particular,
every cograph is strongly perfect.
Among cycle graphs, 
is strongly perfect exactly when 
or 
is even; odd cycles 
for 
are not strongly perfect since they are not perfect.
