Graphs with Large Total 2-Rainbow Domination Number

摘要

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 $$vin V(G)$$ v ∈ V ( G ) with $$f(v)=emptyset$$ f ( v ) = ∅ the condition $$bigcup _{uin N(v)}f(u)={1,2}$$ ⋃ 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 $$w(f)= sum _{vin V(G)} |f(v)|$$ w ( f ) = ∑ v ∈ V ( G ) | f ( v ) | . The total 2- rainbow domination number , $$gamma _{tr2}(G)$$ γ t r 2 ( 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.