Geodesic
Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments
Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2.

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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. 
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     author = {Galeana-S\'anchez, H. and Olsen, M.},
     title = {Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2018},
     doi = {10.23638/DMTCS-20-2-16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-2-16/}
}
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Galeana-Sánchez, H.; Olsen, M. Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments. Discrete mathematics & theoretical computer science, Tome 20 (2018) no. 2. doi : 10.23638/DMTCS-20-2-16. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-20-2-16/

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