$k$-Tuple total domination and mycieleskian graphs

Document Type: Research Paper

Author

UMA (University of Mohaghegh Ardabili)

Abstract

‎Let $k$ be a positive integer‎. ‎A subset $S$ of $V(G)$ in a graph $G$‎ ‎is a $k$-tuple total dominating set of $G$ if every vertex of $G$‎ ‎has at least $k$ neighbors in $S$‎. ‎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 for a‎ ‎given graph $G$ with minimum degree at least $k$‎, ‎we find some sharp‎ ‎lower and upper bounds on the $k$-tuple total domination number of the $m$‎ -‎Mycieleskian graph $\mu _{m}(G)$ of $G$ in terms on $k$ and $\gamma‎ ‎_{\times k,t}(G)$‎. ‎Specially we give the sharp bounds $\gamma‎ ‎_{\times k,t}(G)+1$ and $\gamma _{\times k,t}(G)+k$ for $\gamma‎ ‎_{\times k,t}(\mu _1(G))$‎, ‎and characterize graphs with $\gamma‎ ‎_{\times k,t}(\mu _1(G))=\gamma _{\times k,t}(G)+1$‎.

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