Further results on maximal rainbow domination number

Abstract

‎A  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 ‎ ‎maximal 2-rainbow dominating function of a graph $G$ is a ‎‎$‎‎2‎$‎-rainbow dominating function $f$ such that the set ${win‎‎V(G)|f(w)=emptyset}$ is not a dominating set of $G$‎. ‎The‎ ‎weight of a maximal 2RDF $f$ is the value $omega(f)=sum_{vin‎ ‎V}|f (v)|$‎. ‎The  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.