Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

Abstract

A digraph such that every proper induced subdigraph has a kernel is said tobe \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKIfor short) resp.) if the digraph has a kernel (does not have a kernel resp.).The unique CKI-tournament is $\overrightarrow{C}_3$ and the uniqueKP-tournaments are the transitive tournaments, however bipartite tournamentsare KP. In this paper we characterize the CKI- and KP-digraphs for thefollowing families of digraphs: locally in-/out-semicomplete, asymmetricarc-locally in-/out-semicomplete, asymmetric $3$-quasi-transitive andasymmetric $3$-anti-quasi-transitive $TT_3$-free and we state that the problemof determining whether a digraph of one of these families is CKI is polynomial,giving a solution to a problem closely related to the following conjectureposted by Bang-Jensen in 1998: the kernel problem is polynomially solvable forlocally in-semicomplete digraphs.Comment: 13 pages and 5 figure