$k$-tuple total restrained domination/domatic in graphs

Abstract

‎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 $G$‎. <br />‎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 <br />‎$k$-tuple total restrained domatic number of a graph‎.