Further results on maximal rainbow domination number

‎A  ۲-rainbow dominating function (۲RDF) of a graph $G$ is a‎ ‎function $f$ from the vertex set $V(G)$ to the set of all subsets‎ ‎of the set $\{۱,۲\}$ such that for any vertex $v\in V(G)$ with‎ ‎$f(v)=\emptyset$ the condition $\bigcup_{u\in N(v)}f(u)=\{۱,۲\}$‎ ‎is fulfilled‎, ‎where $N(v)$ is the open neighborhood of $v$‎. ‎A ‎ ‎maximal ۲-rainbow dominating function of a graph $G$ is a ‎‎$‎‎۲‎$‎-rainbow dominating function $f$ such that the set $\{w\in‎‎V(G)|f(w)=\emptyset\}$ is not a dominating set of $G$‎. ‎The‎ ‎weight of a maximal ۲RDF $f$ is the value $\omega(f)=\sum_{v\in‎ ‎V}|f (v)|$‎. ‎The  maximal $۲$-rainbow domination number of a‎ ‎graph $G$‎, ‎denoted by $\gamma_{m۲r}(G)$‎, ‎is the minimum weight of a‎ ‎maximal ۲RDF of $G$‎. ‎In this paper‎, ‎we continue the study of maximal‎ ‎۲-rainbow domination {number} in graphs‎. ‎Specially‎, ‎we first characterize all graphs with large‎ ‎maximal ۲-rainbow domination number‎. ‎Finally‎, ‎we determine the maximal ‎$‎۲‎$‎‎-‎rainbow domination number in the sun and sunlet graphs‎.

Keywords:

$۲$-rainbow dominating function , $۲$-rainbow domination number , maximal $۲$-rainbow dominating function , maximal $۲$-rainbow domination number