Strong Equality Between the 2-Rainbow Domination and Independent 2-Rainbow Domination Numbers in Trees

A 2- rainbow dominating function (2RDF) on a graph G = ( V , E ) is a function f from the vertex set V to the set of all subsets of the set { 1 , 2 } such that for any vertex v ∈ V with f ( v ) = ∅ the condition ⋃ u ∈ N ( v ) f ( u ) = { 1 , 2 } is fulfilled. A 2RDF f is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value ω ( f ) = ∑ v ∈ V | f ( v ) | . The 2- rainbow domination number γ r 2 ( G ) (respectively, the independent 2-rainbow domination number i r 2 ( G ) ) is the minimum weight of a 2RDF (respectively, I2RDF) on G . We say that γ r 2 ( G ) is strongly equal to i r 2 ( G ) and denote by γ r 2 ( G ) ≡ i r 2 ( G ) , if every 2RDF on G of minimum weight is an I2RDF. In this paper, we provide a constructive characterization of trees T with γ r 2 ( T ) ≡ i r 2 ( T ) .