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 $v\in V(G)$ with‎ ‎$f(v)=\emptyset$ the condition $\bigcup_{u\in N(v)}f(u)=\{1,2\}$‎ ‎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 $\{w\in‎‎V(G)|f(w)=\emptyset\}$ is not a dominating set of $G$‎. ‎The‎ ‎weight of a maximal 2RDF $f$ is the value $\omega(f)=\sum_{v\in‎ ‎V}|f (v)|$‎. ‎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.

Keywords

• $2$-rainbow dominating function
• $2$-rainbow domination number
• maximal $2$-rainbow dominating function
• maximal $2$-rainbow domination number