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 ⊆ 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 x N -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 . be a graph and α = ( α u ) 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 ) α u , and then for each x ∈ V ( α w ) and y ∈ V ( α v ) with [ w , v ] ∈ E ( G ) we put at least one of the two arcs ( x , y ) or ( y , x ) in σ ( α , G ) . in result of this paper is the following theorem which provides a new characterization of perfect graphs. m. A graph G is perfect if and only if for any family α = ( α v ) v ∈ V ( G ) of mutually disjoint asymmetric kernel-perfect digraphs, any digraph constructed as a sum of α over G , σ ( α , G ) and oriented by sinks is kernel-perfect.