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Discussiones Mathematicae Graph Theory 24(2) (2004) 171-182
DOI: https://doi.org/10.7151/dmgt.1223

SOME SUFFICIENT CONDITIONS ON ODD DIRECTED CYCLES OF BOUNDED LENGTH FOR THE EXISTENCE OF A KERNEL

Hortensia Galeana-Sánchez

Instituto de Matemáticas, UNAM
Circuito Exterior, Ciudad Universitaria
04510 México, D.F. MEXICO
e-mail: hgaleana@matem.unam.mx

Abstract

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.

2000 Mathematics Subject Classification: 05C20.

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Received 6 May 2002
Revised 21 January 2004