A new characterization of perfect graphs

Abstract

Let D=(V(D),A(D)) be a digraph; a kernel N of D is a set of vertices N⊆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 us recall that a graph G is perfect iff every induced subdigraph H satisfies α(H)=θ(H), where α(G) denotes the stability number of G (i.e. the maximum cardinality of an independent set of vertices of G) and θ(G) denotes the minimum number of cliques needed to cover the vertex-set of G.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 ⋃u∈V(G)αu, and then for each x∈V(αw) and y∈V(αv) with [w,v]∈E(G) we put 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 graphGis perfect if and only if for any familyα=(αv)v∈V(G)of mutually disjoint asymmetric kernel-perfect digraphs, any digraph constructed as a sum ofαoverG, σ(α,G)and oriented by sinks is kernel-perfect.