A note on the $k$-tuple total domination number of a graph

Abstract

For every positive integer $k$ and every graph $G=(V,E)$ with minimum degree at least $k$, a vertex set $S$ is a $k$-tuple total dominating set (resp. $k$-tuple dominating set) of $G$, if for every vertex $v\in V$, $|N_{G}(v)\cap S|\geq k$, (resp. $|N_{G}[v]\cap S|\geq k$). The $k$-tuple total domination number $\gamma _{\times k,t}(G)$ (resp. $k$-tuple domination number $\gamma _{\times k}(G)$) is the minimum cardinality of a $k$-tuple total dominating set (resp. $k$-tuple dominating set) of $G$.

In this paper, we first prove that if $m$ is a positive integer, then for which graphs $G$, $\gamma _{\times k,t}(G)=m$ or $\gamma _{\times k}(G)=m$, and give a necessary and sufficient condition for $\gamma _{\times k,t}(G)=\gamma _{\times (k+1)}(G)$. Then we show that if $G$ is a graph of order $n$ with $\delta (G)\geq k+1\geq 2$, then $\gamma _{\times k,t}(G)$ has the lower bound$\ 2\gamma _{\times (k+1)}(G)-n$ and characterize graphs that equality holds for them. Finally we present two upper bounds for the $k$-tuple total domination number of a graph in terms of its order, minimum degree and $k$.