$k$-Tuple Total Domination in Supergeneralized Petersen Graphs

Abstract

Total domination number of a graph without isolated vertex is the minimum cardinality of a total dominating set, that is, a set of vertices such that every vertex of the graph is adjacent to at least one vertex of the set. Henning and Kazemi in [4] extended this definition as follows: for any positive integer $k$, and any graph $G$ with minimum degree-$k$, a set $D $ of vertices is a $k$-tuple total dominating set of $G$ if each vertex of $G$ is adjacent to at least $k$ vertices in $D$. The $k$-tuple total domination number $\gamma _{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$. In this paper, we give some upper bounds for the $k$-tuple total domination number of the supergeneralized Petersen graphs. Also we calculate the exact amount of this number for some of them.