AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 54 (2012), Pages 133–140 Trees with strong equality between the Roman domination number and the unique response Roman domination number Nader Jafari Rad ∗ Department of Mathematics Shahrood University of Technology, Shahrood Iran n.jafarirad@shahroodut.ac.ir Chun-Hung Liu School of Mathematics Georgia Institute of Technology, Atlanta, GA 30332 U.S.A. cliu87@math.gatech.edu Abstract A Roman dominating function (RDF) on a graph G =(V,E) is a function f : V →{0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of an RDF f is the value f (V (G)) = ∑ u∈V (G) f (u). A function f : V (G) →{0, 1, 2} with the ordered partition (V 0 ,V 1 ,V 2 ) of V (G), where V i = {v ∈ V (G) | f (v)= i} for i =0, 1, 2, is a unique response Roman function if x ∈ V 0 implies |N (x) ∩ V 2 |≤ 1 and x ∈ V 1 ∪ V 2 implies that |N (x) ∩ V 2 | = 0. A function f : V (G) →{0, 1, 2} is a unique response Roman dominating function (or just URRDF) if it is a unique response Roman function and a Roman dominating function. The Roman domination number γ R (G) (respectively, the unique response Roman domination number u R (G)) is the minimum weight of an RDF (respectively, URRDF) on G. We say that γ R (G) strongly equals u R (G), denoted by γ R (G) ≡ u R (G), if every RDF on G of minimum weight is a URRDF. In this paper we provide a constructive characterization of trees T with γ R (T ) ≡ u R (T ). * Also at School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran. The research was in part supported by a grant from IPM (No.90050042).