On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles

‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)}|f(v)|$‎. ‎The 2-rainbow‎‎domination number of G‎, ‎denoted by Υr2 (G)‎, ‎is the‎‎minimum weight of a 2-RDF of G‎. ‎A 2-RDF f is called an outer independent 2-rainbow dominating function ‎(or OI2-RDF) of G if‎‎the set of all v ∈ V (G) with f(v) = ∅ is an‎ ‎independent set‎. ‎The outer independent 2-rainbow domination number Υoir2  (G) is‎‎the minimum weight of an OI2-RDF of G‎. ‎In this paper‎, ‎we obtain the‎‎outer independent 2-rainbow domination number of Pm□Pn‎ ‎and‎ Pm□Cn‎. ‎Also we determine the value of Υoir2  (Cm2Cn) when m or n is even‎.