outer-independent total 2-rainbow dominating functions in graphs

Let $g=(v,e)$ be a simple graph with vertex set $v$ and edge set $e$. an {outer-independent total $2$-rainbow dominating function of a graph $g$ is a function $f$ from $v(g)$ to the set of all subsets of ${1,2}$ such that the following conditions hold: (i) for any vertex $v$ with $f(v)=emptyset$ we have $bigcup_{uin n_g(v)} f(u)={1,2}$, (ii) the set of all vertices $vin v(g)$ with $f(v)=emptyset$ is independent and (iii) ${vmid f(v)neqemptyset}$ has no isolated vertex. the outer-independent total $2$-rainbow domination number of $g$, denoted by ${gamma}_{oitr2}(g)$, is the minimum value of $omega(f)=sum_{vin v(g)} |f(v)|$ over all such functions $f$. in this paper, we study the outer-independent total $2$-rainbow domination number of $g$ and classify all graphs with outer-independent total $2$-ainbow domination number belonging to the set ${2,3,n}$. among other results, we present some sharp bounds concerning the invariant.