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)=∅, the condition ⋃u∈ng(v)f(u)={1,2} is fulfilled, where ng(v) is the open neighborhood of v. the weight of 2-rdf f of g is the value ω(f):=∑v∈v(g)|f(v)|. the 2-rainbow domination number of g, denoted by γr2(g), is the minimum 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 if the set of all v∈v(g) with f(v)=∅ is an independent set. the outer independent 2-rainbow domination number γoir2(g) is the minimum weight of an oi2-rdf of g. in this paper, we obtain the outer independent 2-rainbow domination number of pm□pn and pm□cn. also we determine the value of γoir2(cm□cn) when m or n is even.