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

TL;DR

This paper characterizes kernel perfect and critical kernel imperfect digraphs within specific families of generalized tournaments and bipartite tournaments, using forbidden subdigraphs, and establishes polynomial-time algorithms for related decision problems.

Contribution

It provides a complete characterization of CKI and KP digraphs in certain classes and proves polynomial solvability of the kernel problem for these families.

Findings

Characterization of CKI and KP digraphs in locally in-/out-semicomplete families

Polynomial-time algorithms for determining CKI in these classes

Progress towards the conjecture on polynomial solvability for locally in-semicomplete digraphs

Abstract

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