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

TL;DR

This paper introduces and analyzes the concepts of k-tuple total restrained domination and domatic numbers in graphs, providing exact values, bounds, and structural characterizations for various graph classes and their complements.

Contribution

It defines new graph parameters related to k-tuple total restrained domination and domatic numbers, and derives their exact values, bounds, and structural properties for multiple classes of graphs.

Findings

Exact values of k-tuple total restrained domination number for complete graphs, cycles, bipartite graphs, and complements of paths or cycles.

Bounds for the k-tuple total restrained domatic number in various graph classes.

Structural characterizations of graphs where the domination number reaches certain bounds.

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.