2-rainbow domination number of the subdivision of graphs

Let $g$ be a simple graph and $f : v (g) rightarrow p({1,2})$ be a function where for each vertex $v in v (g)$ with $f(v)= emptyset$ we have $bigcup_{u in n_{g}(v)} f(u) = {1,2}.$ then $f$ is a $2$-rainbow dominating function (a $2rdf$) of $g.$ the  weight of $f$ is $omega(f)=sum_{v in v(g)} |f(v)|.$ the minimum weight among all of $2-$rainbow dominating functions is $2-$rainbow domination number  and is denoted by $gamma_{r2}(g)$. in this paper,  we provide some bounds for the $2-$rainbow domination number of the subdivision graph $s(g)$ of  a graph $g$. also, among some other interesting results, we determine the exact value of $gamma_{r2}(s(g))$ when $g$ is a tree, a bipartite graph, $k_{r,s}$, $k_{n_1,n_2,dots,n_k}$ and $k_n$.