Geodesic
New classes of critical kernel-imperfect digraphs
Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 85-89

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A kernel of a digraph D is a subset N ⊆ V(D) which is both independent and absorbing. When every induced subdigraph of D has a kernel, the digraph D is said to be kernel-perfect. We say that D is a critical kernel-imperfect digraph if D does not have a kernel but every proper induced subdigraph of D does have at least one. Although many classes of critical kernel-imperfect-digraphs have been constructed, all of them are digraphs such that the block-cutpoint tree of its asymmetrical part is a path. The aim of the paper is to construct critical kernel-imperfect digraphs of a special structure, a general method is developed which permits to build critical kernel-imperfect-digraphs whose asymmetrical part has a prescribed block-cutpoint tree. Specially, any directed cactus (an asymmetrical digraph all of whose blocks are directed cycles) whose blocks are directed cycles of length at least 5 is the asymmetrical part of some critical kernel-imperfect-digraph.
Keywords: digraphs, kernel, kernel-perfect, critical kernel-imperfect, block-cutpoint tree 
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Galeana-Sánchez, Hortensia; Neumann-Lara, V. New classes of critical kernel-imperfect digraphs. Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 85-89. http://geodesic.mathdoc.fr/item/DMGT_1998_18_1_a6/