A new characterization of perfect graphs

LetD = (V (D), A(D)) be a digraph; a kernelN ofD is a set of verticesN ⊆ V (D) such that N is independent (for any x, y∈N there is no arc between them) and N is absorbent (for each x ∈ 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 D is oriented by sinks when every semicomplete subdigraph of D has at least one kernel. Let G be a graph and α = (αu)u∈V (G) a family of mutually disjoint digraphs; a sum of α over G, denoted by σ(α,G) is a digraph defined as follows: Take ⋃ v∈V (G) αu, and then for each x ∈ V (αu) and y ∈ V (αv) with [u, v] ∈ E(G) we have at least one of the two arcs (x, y) or (y, x) in σ(α,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 if and only if for any family α = (αv)v∈V (G) of mutually disjoint asymmetric kernel-perfect digraphs any sum of α over G, σ(α,G) oriented by sinks is kernel-perfect. Mathematics Subject Classification: 05C20.