Graphs with Large Total 2-Rainbow Domination Number

Abstract

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 \(v\in V(G)\) with \(f(v)=\emptyset\) the condition \(\bigcup _{u\in N(v)}f(u)=\{1,2\}\) 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 \(\{v \in V(G) \mid f (v) \not =\emptyset \}\) has no isolated vertex. The weight of a T2RDF f is defined as \(w(f)= \sum _{v\in V(G)} |f(v)|\). 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.