Graphs with Large Total 2-Rainbow Domination Number

Let $$G=(V,E)$$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 $$\{1,2\}$${1,2} such that for any vertex $$v\in V(G)$$v∈V(G) with $$f(v)=\emptyset$$f(v)=∅ the condition $$\bigcup _{u\in N(v)}f(u)=\{1,2\}$$⋃u∈N(v)f(u)={1,2} is fulfilled, where N(v) is the open neighborhood of v. A 2-rainbow dominating function f is called a total 2-rainbow dominating function (T2RDF) if the subgraph of G induced by $$\{v \in V(G) \mid f (v) \not =\emptyset \}$${v∈V(G)∣f(v)≠∅} has no isolated vertex. The weight of a T2RDF f is defined as $$w(f)= \sum _{v\in V(G)} |f(v)|$$w(f)=∑v∈V(G)|f(v)|. The total 2-rainbow domination number, $$\gamma _{tr2}(G)$$γtr2(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.