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 is an element of V with f(v) = empty set the condition U-u is an element of 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 omega(f) = Sigma v is an element of V vertical bar f(v)vertical bar. The 2-rainbow domination number gamma(r2)(G) (respectively, the independent 2-rainbow domination number i(r2)(G)) is the minimum weight of a 2RDF (respectively, I2RDF) on G. We say that gamma(r2)(G) is strongly equal to i(r2)(G) and denote by gamma(r2)(G) = i(r2)(G), if every 2RDF on G of minimum weight is an I2RDF. In this paper, we provide a constructive characterization of trees T with gamma(r2)(T) equivalent to i(r2)(T).