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 xy∈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: C⃗3, C⃗5 and C⃗7(1,2). Several interesting consequences are obtained.