Parity graphs are kernel-M-solvable

While the famous Berge’s Strong Perfect Graph Conjecture (see [l] for details on perfect graphs) remains a major unsolved problem in Graph Theory, an alternative characterization of Perfect Graphs was conjectured in 1982 by Berge and the author [3]. This second conjecture asserts the existence of kernels for a certain type of orientations of perfect graphs. Here we prove a weaker form of the conjecture for a well-known special class of perfect graphs, that generalizes bipartite graphs, namely parity graphs. Let us recall that a kernel of a digraph D = (X, U) is a subset of vertices KC X which is both independant (no vertex of K is adjacent to another vertex of K), and absorbing (every vertex of X/K has a successor in K). When every induced subdigraph of D has a kernel, the digraph D is said to be kernel-perfect [7]. Throughout this article, any digraph D is to be viewed as an orientation of its underlying undirected graph that we denote by D,. In particular, an orientation of a graph may contain reversible arcs, i.e., arcs whose reversal arc is also present. A subdigraph is said to be complete whenever its vertices are pairwise adjacent. The orientations D of perfect graphs we are interested in are normal orientations, i.e., they have the property