On transversals in minimal imperfect graphs

Chvatal (1984) proved that no minimal imperfect graph has a small transversal, that is, a set of vertices of cardinality at most alpha + omega - 1 which meets every omega-clique and every alpha-stable set. In this paper we prove that a slight generalization of this notion of small transversal leads to a conjecture which is as strong as Berge's strong perfect graph conjecture for a very large class of graphs, namely for those graphs whose diameter does not exceed 6.