k-Tuple Total Domination in Complementary Prisms

Let k be a positive integer, and let G be a graph with minimum degree at least k . In their study (2010), Henning and Kazemi defined the k -tuple total domination numberγ × k , tGof 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. IfG ̅ is the complement of G , the complementary prism G G ̅ of G is the graph formed from the disjoint union of G andG ̅ by adding the edges of a perfect matching between the corresponding vertices of G andG ̅ . In this paper, we extendsome of the results of Haynes et al. (2009) for the k -tuple total domination numberand also obtain some other new results. Also we find the k -tuple total domination number of thecomplementary prism of a cycle, a path, or a complete multipartite graph.