Strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers in trees J. Amjadi, M. Falahat, S.M. Sheikholeslami * Department of Mathematics Azarbaijan Shahid Madani University Tabriz, I.R. Iran j-amjadi;s.m.sheikholeslami@azaruniv.edu N. Jafari Rad Department of Mathematics Shahrood University of Technology Shahrood, I.R. Iran n.jafarirad@gmail.com Abstract 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 γr2(G) (respectively, the independent 2- rainbow domination number ir2(G) ) is the minimum weight of a 2RDF (respectively, I2RDF) on G. We say that γr2(G) is strongly equal to ir2(G) and denote by γr2(G) ≡ ir2(G), if every 2RDF on G of minimum weight is an I2RDF. In this paper we provide a constructive characterization of trees T with γr2(T ) ≡ ir2(T ). Keywords: 2-rainbow domination number, independent 2-rainbow domination number, strong equality, tree. MSC 2000: 05C69 1 Introduction Let G be a simple graph with vertex set V = V (G) and edge set E = E(G). For every vertex v ∈ V , the open neighborhood N (v) is the set {u ∈ V | uv ∈ E} and the closed neighborhood of v is the set N [v]= N (v) ∪{v}. The degree of a vertex v ∈ V is deg G (v) = deg(v)= |N (v)|. If A ⊆ V (G), then G[A] is the subgraph induced by A. A vertex of degree one is called a leaf , and its neighbor is called a support vertex. If v is a support vertex, then L v will denote the set of all leaves adjacent to v. A support vertex v is called strong support vertex if |L v | > 1. For r, s ≥ 1, a double star S(r, s) is a tree with exactly two vertices that are not leaves, with one adjacent to r leaves and the other to s leaves. For a vertex v in a rooted tree T , let C(v) denote the set of children of v, D(v) denote the set of descendants of v and D[v]= D(v) ∪{v}, and the depth of v, depth(v), is the largest distance from v to a vertex in D(v). The maximal subtree at v is the subtree of T induced by D(v) ∪{v}, and is denoted by T v . For terminology and notation on graph theory not given here, the reader is referred to [14]. * Corresponding author 1