On 2-rainbow domination number of functigraph and its complement

Let G be a graph and ƒ : V(G) → P({1, 2}) be a function where for every vertex v ∈ V(G), with ƒ (v) = ∅ we have [formula]. Then ƒ is a 2-rainbow dominating function or a 2RDF of G. The weight of ƒ is[formula]. The minimum weight of all 2-rainbow dominating functions is 2-rainbow domination number of G, denoted by [formula]. Let G 1 and G2 be two copies of a graph G with disjoint vertex sets V(G 1) and V(G2), and let σ be a function from V(G 1) to V(G2). We define the functigraph C(G,σ) to be the graph that has the vertex set V(C(G, ,σ)) = V(G 1) U V(G2), and the edge set [formula]. In this paper, 2-rainbow domination number of the functigraph of C(G, ,σ) and its complement are investigated. We obtain a general bound for [formula] and we show that this bound is sharp.