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.