Table of Contents
ISRN Discrete Mathematics
Volume 2011, Article ID 681274, 13 pages
Research Article

π‘˜-Tuple Total Domination in Complementary Prisms

Department of Mathematics, University of Mohaghegh Ardabili, P.O. Box 5619911367, Ardabil, Iran

Received 29 September 2011; Accepted 30 October 2011

Academic Editor: W. Wang

Copyright © 2011 Adel P. Kazemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let π‘˜ be a positive integer, and let 𝐺 be a graph with minimum degree at least π‘˜. In their study (2010), Henning and Kazemi defined the π‘˜-tuple total domination number π›ΎΓ—π‘˜,𝑑(𝐺) of 𝐺 as the minimum cardinality of a π‘˜-tuple total dominating set of 𝐺, which is a vertex set such that every vertex of 𝐺 is adjacent to at least π‘˜ vertices in it. If 𝐺 is the complement of 𝐺, the complementary prism 𝐺𝐺 of 𝐺 is the graph formed from the disjoint union of 𝐺 and 𝐺 by adding the edges of a perfect matching between the corresponding vertices of 𝐺 and 𝐺. In this paper, we extend some of the results of Haynes et al. (2009) for the π‘˜-tuple total domination number and also obtain some other new results. Also we find the π‘˜-tuple total domination number of the complementary prism of a cycle, a path, or a complete multipartite graph.