%0 Journal Article %T $k$-tuple total restrained domination/domatic in graphs %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A P. Kazemi, Adel %D 2014 %\ 06/01/2014 %V 40 %N 3 %P 751-763 %! $k$-tuple total restrained domination/domatic in graphs %K $k$-tuple total domination number‎ %K ‎$k$-tuple total domatic number‎ %K ‎$k$-tuple total restrained domination number‎ %K ‎$k$-tuple total restrained domatic number‎ %R %X ‎For any integer $kgeq 1$‎, ‎a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-‎tuple total dominating set of $G$ if any vertex‎ ‎of $G$ is adjacent to at least $k$ vertices in $S$‎, ‎and any vertex‎ ‎of $V-S$ is adjacent to at least $k$ vertices in $V-S$‎. ‎The minimum number of vertices of such a set‎ ‎in $G$ we call the $k$-tuple total restrained domination number of $G$‎. ‎The maximum number of classes of a partition of $V$ such that its‎ all classes are $k$-tuple total restrained dominating sets in $G$ we call‎ ‎the $k$-tuple total restrained domatic number of $G$‎.   ‎In this paper‎, ‎we give some sharp bounds for the $k$-tuple‎ ‎total restrained domination number of a graph‎, ‎and also calculate it‎ ‎for some of the known graphs‎. ‎Next‎, ‎we mainly present basic properties of the‎ ‎$k$-tuple total restrained domatic number of a graph‎. %U http://bims.iranjournals.ir/article_529_f5f373afe7aa443a872393ba1dd1ad50.pdf