New Results on Minimal Strongly Imperfect Graphs

The characterization of strongly perfect graphs by a restricted list of forbidden induced subgraphs has remained an open question for a long time. The minimal strongly imperfect graphs which are simultaneous imperfect are only odd holes and odd antiholes ( E. Olaru, [9]), but the entire list is not known, in spite of a lot of particular results in this direction. In this paper we give some new properties of the minimal strongly imperfect graphs. Thus we introduce the notion of critical (co-critical) pair of vertices, and we prove that any vertex of a minimal s-imperfect (minimal c-imperfect) graph is contained in a critical (co-critical) pair, and, in a minimal s-imperfect graph different from a cycle of length at least 5, any vertex is the center of a star cutset, or, if not, it belongs to a house or a domino. Also, we characterize the triangle-free minimal s-imperfect graphs. (By s-perfect we mean the complement of a strongly perfect graph.)