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KP-digraphs and CKI-digraphs satisfying the k-Meyniel's condition

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Galeana-Sánchez H., Neumann-Lara V.
Discussiones Mathematicae Graph Theory
|
1996
|
tom 16
|
nr 1
5-16
EN
A digraph D is said to satisfy the k-Meyniel's condition if each odd directed cycle of D has at least k diagonals. The study of the k-Meyniel's condition has been a source of many interesting problems, questions and results in the development of Kernel Theory. In this paper we present a method to construct a large variety of kernel-perfect (resp. critical kernel-imperfect) digraphs which satisfy the k-Meyniel's condition.
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New classes of critical kernel-imperfect digraphs

100%
Galeana-Sánchez H., Neumann-Lara V.
Discussiones Mathematicae Graph Theory
|
1998
|
tom 18
|
nr 1
85-89
EN
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.
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