A 2-rainbow dominating function (2-rD function) of a graph G=(V,E) is a function f:V(G)→{∅,{1},{2},{1,2}} having the property that if f(x)=∅, then f(N(x))={1,2}. The 2-rainbow domination number γr2(G) is the minimum weight of ∑v∈V(G)|f(v)| taken over all 2-rainbow dominating functions f. An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph G is a 2-rD function f for which the set of all v∈V(G) with f(v)=∅ is independent. The outer independent 2-rainbow domination number γoir2(G) is the minimum weight of an OI2-rD function of G. In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on γoir2(G). Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair (a,b) is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if a+1≤b≤2a.
