Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 8 2 2023 06 01 Outer-independent total 2-rainbow dominating functions in graphs 431 444 14401 10.22049/cco.2022.27753.1344 EN Akram Mahmoodi Department of Mathematics Payame Noor University I.R. Iran Lutz Volkmann RWTH Aachen University Journal Article 2022 03 30 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 $\bigcup_{u\in N_G(v)} f(u)=\{1,2\}$, (ii) the set of all vertices $v\in V(G)$ with $f(v)=\emptyset$ is independent and (iii) $\{v\mid f(v)\neq\emptyset\}$ has no isolated vertex. The outer-independent total $2$-rainbow domination number of $G$, denoted by ${\gamma}_{oitr2}(G)$, is the minimum value of $\omega(f)=\sum_{v\in V(G)} |f(v)|$ 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. https://comb-opt.azaruniv.ac.ir/article_14401_49dd07ea1663862635713568046d41f8.pdf