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.
Highlights
Throughout this note we consider simple graphs G with vertex set V (G)
A set D ⊆ 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 domination number of G is the minimum cardinality among all dominating sets of G and it is denoted by γ(G)
Summary
Throughout this note we consider simple graphs G with vertex set V (G). Given a vertex v ∈ V (G), N (v) denotes the open neighbourhood of v in G. The k-domination number of G, denoted by γk(G), is the minimum cardinality among all k-dominating sets of G. Given a graph G and a positive integer k ≤ δ(G) + 1, a k-dominating set D is said to be a k-tuple dominating set of G if degD(v) ≥ k − 1 for every v ∈ D. The k-tuple domination number of G, denoted by γ×k(G), is the minimum cardinality among all k-tuple dominating sets of G. We provide new bounds on the k-tuple domination number. Some of these bounds generalize other ones that have been given for the double domination number
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