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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Research supported in part by the South African National Research Foundation.
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Henning, M.A., Kazemi, A.P. k-tuple total domination in cross products of graphs. J Comb Optim 24, 339–346 (2012). https://doi.org/10.1007/s10878-011-9389-z
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DOI: https://doi.org/10.1007/s10878-011-9389-z