Geodesic
Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 2, pp. 171-182

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A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V(D)-N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel-perfect digraph. In this paper I investigate some sufficient conditions for a digraph to have a kernel by asking for the existence of certain diagonals or symmetrical arcs in each odd directed cycle whose length is at most 2α(D)+1, where α(D) is the maximum cardinality of an independent vertex set of D. Previous results are generalized.
Keywords: kernel, kernel-perfect, critical kernel-imperfect 
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     title = {Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel},
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Galeana-Sánchez, Hortensia. Some sufficient conditions on odd directed cycles of bounded length for the existence of a kernel. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 2, pp. 171-182. http://geodesic.mathdoc.fr/item/DMGT_2004_24_2_a1/