\(k\)-tuple total domination in complementary prisms (Q410666)

Summary: Let \(k\) be a positive integer, and let \(G\) be a graph with minimum degree at least \(k\). In their study \textit{M. A. Henning} and \textit{A. P. Kazemi} [Discrete Appl. Math. 158, No. 9, 1006--1011 (2010; Zbl 1210.05097)] defined the \(k\)-tuple total domination number \(\gamma_{\times k,t}(G)\) of \(G\) as the minimum cardinality of a \(k\)-tuple total dominating set of \(G\), which is a vertex set such that every vertex of \(G\) is adjacent to at least \(k\) vertices in it. If \(\overline G\) is the complement of \(G\), the complementary prism \(G \overline G\) of \(G\) is the graph formed from the disjoint union of \(G\) and \(\overline G\) by adding the edges of a perfect matching between the corresponding vertices of \(G\) and \(\overline G\). In this paper, we extend some of the results of \textit{T. W. Haynes} et al. [Bull. Inst. Comb. Appl. 51, 21--30 (2007; Zbl 1155.05051)] for the \(k\)-tuple total domination number and also obtain some other new results. Also we find the \(k\)-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.