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Discussiones Mathematicae Graph Theory 31(2) (2011) 293-312
DOI: https://doi.org/10.7151/dmgt.1546

k-kernels in generalizations of transitive digraphs

Hortensia Galeana-Sánchez and César Hernández-Cruz Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria, México, D.F., C.P. 04510, México

Abstract

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.

A (k,l)-kernel N of D is a k-independent set of vertices (if u,v ∈ N, u ≠ v, then d(u,v), d(v,u) ≥ k) and l-absorbent (if u ∈ V(D)−N then there exists v ∈ N such that d(u,v) ≤ l). A k-kernel is a (k,k−1)-kernel. Quasi-transitive, right-pretransitive and left-pretransitive digraphs are generalizations of transitive digraphs. In this paper the following results are proved: Let D be a right-(left-) pretransitive strong digraph such that every directed triangle of D is symmetrical, then D has a k-kernel for every integer k ≥ 3; the result is also valid for non-strong digraphs in the right-pretransitive case. We also give a proof of the fact that every quasi-transitive digraph has a (k,l)-kernel for every integers k > l ≥ 3 or k = 3 and l = 2.

Keywords: digraph, kernel, (k,l)-kernel, k-kernel, transitive digraph, quasi-transitive digraph, right-pretransitive digraph, left-pretransitive digraph, pretransitive digraph

2010 Mathematics Subject Classification: 05C20.

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Received 12 November 2009
Revised 23 August 2010
Accepted 24 August 2010