A new characterization of perfect graphs

Let D = (V (D). A(D)) be a digraph; a kernel N of D is a set of vertices N subset of V (D) such that N is independent (for any x, y is an element of N, there is no arc between them) and N is absorbent (for each x is an element of V (D) - N, there exists an xN-arc in D). A digraph D is said to be kernel-perfect whenever each one of its induced subdigraphs has a kernel. A digraph Disoriented by sinks when every semicomplete subdigraph of D has at least one kernel. Let us recall that a graph G is perfect iff every induced subdigraph H satisfies alpha(H) = theta(H), where alpha(G) denotes the stability number of G (i.e. the maximum cardinality of an independent set of vertices of G) and theta(G) denotes the minimum number of cliques needed to cover the vertex-set of G. Let G be a graph and alpha = (alpha(u))(u is an element of v(G)) a family of mutually disjoint digraphs; a sum of alpha over G, denoted by sigma (alpha, G) is a digraph defined as follows. Take U-u is an element of v(G), alpha(u), and then for each x is an element of V(alpha(w)) and y is an element of V(alpha(nu)) with vertical bar w, upsilon vertical bar is an element of E(G) we put at least one of the two arcs (x. y) or (y, x) in a (alpha, G). The main result of this paper is the following theorem which provides a new characterization of perfect graphs. Theorem. A graph G is perfect 'land only iffor any family alpha = (alpha)(upsilon is an element of V(G)) of mutually disjoint asymmetric kernel-perfect digraphs, any digraph constructed as a sum of alpha over G, sigma (alpha, G) and oriented by sinks is kernel-perfect. (C) 2012 Elsevier B.V. All rights reserved.