A sharp upper bound on the independent 2-rainbow domination in graphs with minimum degree at least two
An independent 2-rainbow dominating function (I$2$-RDF) on a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that $\{x\in V\mid f(x)\neq \emptyset \}$ is an independent set of $G$ and for any vertex $v\in V(G)$ with $f(v)=\emptyset $ we h...
Main Authors: | , , , |
---|---|
Format: | Article |
Language: | English |
Published: |
Institute of Mathematics and Computer Science of the Academy of Sciences of Moldova
2020-12-01
|
Series: | Computer Science Journal of Moldova |
Subjects: | |
Online Access: | http://www.math.md/files/csjm/v28-n3/v28-n3-(pp373-388).pdf |
Summary: | An independent 2-rainbow dominating function (I$2$-RDF) on a graph $G$ is a
function $f$ from the vertex set $V(G)$ to the set of all subsets of the set
$\{1,2\}$ such that $\{x\in V\mid f(x)\neq \emptyset \}$ is an independent
set of $G$ and for any vertex $v\in V(G)$ with $f(v)=\emptyset $ we have $%
\bigcup_{u\in N(v)}f(u)=\{1,2\}$. The \emph{weight} of an I$2$-RDF $f$ is the
value $\omega (f)=\sum_{v\in V}|f(v)|$, and the independent $2$-rainbow
domination number $i_{r2}(G)$ is the minimum weight of an I$2$-RDF on $G$. In
this paper, we prove that if $G$ is a graph of order $n\geq 3$ with minimum
degree at least two such that the set of vertices of degree at least $3$ is
independent, then $i_{r2}(G)\leq \frac{4n}{5}$. |
---|---|
ISSN: | 1561-4042 |