Discussiones Mathematicae Graph Theory 29(1) (2009) 39-49
doi: 10.7151/dmgt.1431

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k-KERNELS AND SOME OPERATIONS IN DIGRAPHS

Hortensia Galeana-Sanchez

Instituto de Matemáticas
Universidad Nacional Autónoma de México
Ciudad Universitaria, México, D.F. 04510, México

Laura Pastrana

Facultad de Ciencias Universidad Nacional Autónoma de México Ciudad Universitaria, Circuito Exterior México, D.F. 04510, México

Abstract

Let D be a digraph. V(D) denotes the set of vertices of D; a set N ⊆ V(D) is said to be a k-kernel of D if it satisfies the following two conditions: for every pair of different vertices u,v ∈ N it holds that every directed path between them has length at least k and for every vertex x ∈ V(D)−N there is a vertex y ∈ N such that there is an xy-directed path of length at most k−1.

In this paper, we consider some operations on digraphs and prove the existence of k-kernels in digraphs formed by these operations from another digraphs.

Keywords: k-kernel, k-subdivision digraph, k-middle digraph and k-total digraph.

2000 Mathematics Subject Classification: Primary: 05C20; Secondary: 05C69.

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Received 12 September 2007
Revised 8 December 2007
Accepted 29 December 2008