Geodesic
k-Kernels and some operations in digraphs
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 39-49

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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 
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Galeana-Sanchez, Hortensia; Pastrana, Laura. k-Kernels and some operations in digraphs. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 39-49. http://geodesic.mathdoc.fr/item/DMGT_2009_29_1_a2/