### $k$-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.

#### Keywords

$k$-tuple total domination number; Supergeneralized Petersen graph; Cartesian product graph

#### Full Text:

PDFDOI: http://dx.doi.org/10.26713%2Fcma.v2i1.130

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eISSN 0975-8607; pISSN 0976-5905