Combinatorial designs related to the strong perfect graph conjecture

Our graphs are “Michigan” except that they have vertices and edges rather than points and lines. If G is a graph, then y1 = y1 (G) denotes the number of its vertices, ar = (Y(G) denotes the size of its largest stable (independent) set of vertices and o = o(G) denotes the size of its largest clique. The graphs that we are interested in have the following three properties: (i) n =0X0+1, (ii) every vertex is in precisely (Y stable sets of size (Y and in precisely o cliques of size w, (iii) the yt stable sets of size (x may be enumerated as S1, S2, . . . , S, and the n cliques of size o may be enumerated as C1, C2, . . . , C,, in such a way that Si n Ci = $9 for all i but Si fI Cj # $J whenever i f i. We shall call then (a, o)-graphs. This concept, contrived as it may seem at first, arises quite naturally in the investigations of imperfect graphs; we are about to explain how.