## Galeana-Sánchez, H. and Olsen, M. - Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

dmtcs:4122 - Discrete Mathematics & Theoretical Computer Science, December 11, 2018, vol. 20 no. 2
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

Authors: Galeana-Sánchez, H. and Olsen, M.

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.

Source : oai:arXiv.org:1712.00138
DOI : 10.23638/DMTCS-20-2-16
Volume: vol. 20 no. 2
Section: Graph Theory
Published on: December 11, 2018
Submitted on: December 4, 2017
Keywords: Mathematics - Combinatorics,05C20, 05C69, 05C75