## 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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