UNICYCLIC GRAPHS WITH STRONG EQUALITY BETWEEN THE 2-RAINBOW DOMINATION AND INDEPENDENT 2-RAINBOW DOMINATION NUMBERS

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) equivalent to i(r2)(G), if every 2RDF on G of minimum weight is an I2RDF. In this paper we characterize all unicyclic graphs G with gamma(r2)(G) equivalent to i(r2)(G).