A note on the k-tuple domination number of graphs

TL;DR

This paper investigates the $k$-tuple domination number in graphs, providing new bounds and improvements on existing lower bounds, especially for the case $k=2$, advancing theoretical understanding of graph domination parameters.

Contribution

The paper introduces new bounds for the $k$-tuple domination number and enhances existing lower bounds, generalizing previous results for specific cases like $k=2$.

Findings

Derived new bounds on the $k$-tuple domination number.

Generalized bounds for the case $k=2$.

Improved two well-known lower bounds.

Abstract

In a graph $G$, a vertex dominates itself and its neighbours. A set $D\subseteq V(G)$ is said to be a $k$-tuple dominating set of $G$ if $D$ dominates every vertex of $G$ at least $k$ times. The minimum cardinality among all $k$-tuple dominating sets is the $k$-tuple domination number of $G$. In this paper, we provide new bounds on this parameter. Some of these bounds generalize other ones that have been given for the case $k=2$. In addition, we improve two well-known lower bounds on the $k$-tuple domination number.