k-tuple total domination in cross products of graphs |
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Authors: | Michael A. Henning Adel P. Kazemi |
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Affiliation: | 1. Department of Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa 2. Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 179, Ardabil, Iran
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Abstract: | For k??1 an integer, a set S of vertices in a graph G with minimum degree at least?k is a k-tuple total dominating set of G if every vertex of G is adjacent to at least k vertices in S. The minimum cardinality of a k-tuple total dominating set of G is the k-tuple total domination number of G. When k=1, the k-tuple total domination number is the well-studied total domination number. In this paper, we establish upper and lower bounds on the k-tuple total domination number of the cross product graph G×H for any two graphs G and H with minimum degree at least?k. In particular, we determine the exact value of the k-tuple total domination number of the cross product of two complete graphs. |
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