Critical Kernel Imperfect Problem in Generalizations of Bipartite Tournaments

Kernel is an important topic in digraphs. A digraph such that every proper induced subdigraph has a kernel is said to be critical kernel imperfect (CKI, for short) if the digraph does not have a kernel. Galeana-Sanchez and Olsen characterized the CKI-digraphs for the following families of digraphs: asymmetric arc-locally in-/out-semicomplete digraphs, asymmetric 3-quasi-transitive digraphs and asymmetric 3-anti-quasi-transitive \(TT_3\)-free digraphs. In this paper, we shall completely characterize the above four classes CKI-digraphs without any restriction on arcs and \(TT_3\)-free subdigraphs.