# Further results on maximal rainbow domination number

Document Type: Research Paper

Author

Department of Mathematics, Babol Noshirvani University of Technology, Babol, I.R. Iran

10.22108/toc.2020.120014.1684

Abstract

A {em 2-rainbow dominating function} (2RDF) of 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 for any vertex $vin V(G)$ with
$f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$
is fulfilled, where $N(v)$ is the open neighborhood of $v$. A {em
maximal 2-rainbow dominating function} of a graph $G$ is a
2-rainbow dominating function $f$ such that the set ${win V(G)|f(w)=emptyset}$ is not a dominating set of $G$. The {em
weight} of a maximal 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The {em maximal $2$-rainbow domination number} of a
graph $G$, denoted by $gamma_{m2r}(G)$, is the minimum weight of a
maximal 2RDF of $G$. In this paper, we continue the study of maximal
2-rainbow domination {number} in graphs. Specially, we first characterize all graphs with large
maximal 2-rainbow domination number. Finally, we determine the maximal 2-
rainbow domination number in the sun and sunlet graphs.

Keywords

Main Subjects