Graphs with Large Total 2-Rainbow Domination Number

Let G = (V,E) be a simple graph with no isolated vertex. A 2-rainbow dominating function (2RDF) of G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2} such that for any vertex v is an element of V(G) with f(v) = phi the condition upsilon(u is an element of N(v))f(u) = {1,2} is fulfilled, where N(v) is the open neighborhood of v. A 2-rainbow dominating functionfis called a total 2-rainbow dominating function (T2RDF) if the subgraph of G induced by {v is an element of V(G) | f(v) is an element of phi} has no isolated vertex. The weight of a T2RDFfis defined as w(f) = Sigma(v is an element of v(G) )|f(v)|. The total 2-rainbow domination number, gamma(tr2)(G), is the minimum weight of a total 2-rainbow dominating function on G. In this paper, we characterize all graphs G whose total 2-rainbow domination number is equal to their order minus one.