further results on maximal rainbow domination number

‎a 2-rainbow dominating function (2rdf) of a graph g is a‎ ‎function f f from the vertex set v(g) to the set of all subsets‎ ‎of the set {1,2} such that for any vertex v∈v(g) with‎ ‎f(v)=∅ the condition ⋃u∈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∈‎‎v(g)|f(w)=∅} is not a dominating set of g ‎. ‎the‎ ‎weight of a maximal 2rdf f is the value ω(f)=∑v∈‎‎v|f(v)| ‎. ‎the maximal 2 -rainbow domination number of a‎ ‎graph g ‎, ‎denoted by γm2r(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