k-tuple total restrained domination and k-tuple total restrained domatic in graphs

Abstract

Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$ vertices of $V(G)-S$. The minimum number of vertices of a such sets in $G$ are the $k$-tuple total restrained domination number $\gamma_{\times k,t}^{r}(G)$ of $G$. The maximum number of classes of a partition of $V(G)$ such that its all classes are $k$-tuple total restrained dominating sets in $G$, is called the $k$-tuple total restrained domatic number of $G$. In this manuscript, we first find $\gamma_{\times k,t}^{r}(G)$, when $G$ is complete graph, cycle, bipartite graph and the complement of path or cycle. Also we will find bounds for this number when $G$ is a complete multipartite graph. Then we will know the structure of graphs $G$ which $\gamma_{\times k,t}^{r}(G)=m$, for some $m\geq k+1$ and give upper and lower bounds for $\gamma_{\times k,t}^{r}(G)$, when $G$ is an arbitrary graph. Next, we mainly present basic properties of the $k$-tuple total restrained domatic number of a graph and give bounds for it. Finally we give bounds for the $k$-tuple total restrained domination number of the complementary prism $G\bar{G}$ in terms on the similar number of $G$ and $\bar{G}$ when $G$ is a regular graph or an arbitrary graph. And then we calculate it when $G$ is cycle or path.