Characterization of asymmetric CKI- and KP-digraphs with covering number at most 3

A set N ⊆ V ( D ) is said to be a kernel if N is an independent set and for every vertex x ∈ ( V ( D ) ∖ N ) there is a vertex y ∈ N such that x y ∈ A ( D ) . Let D be a digraph such that every proper induced subdigraph of D has a kernel. D is said to be kernel perfect digraph (KP-digraph) if the digraph D has a kernel and critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. In this paper we characterize the asymmetric CKI-digraphs with covering number at most 3 . Moreover, we prove that the only asymmetric CKI-digraphs with covering number at most 3 are: C ⃗ 3 , C ⃗ 5 and C ⃗ 7 ( 1 , 2 ) . Several interesting consequences are obtained.