A set D of vertices in a graph G = (V (G), E(G)) is an open neighborhood locating-dominating set (OLD-set) for G if for every two vertices u, v of V (G) the sets N (u) ∩ D and N (v) ∩ D are non-empty and different. The open neighborhood locating-dominating number OLD(G) is the minimum cardinality of an OLD-set for G. In this paper, we characterize graphs G of order n with OLD(G) = 2, 3, or n and graphs with minimum degree δ(G) ≥ 2 that are C 4 -free with OLD(G) = n − 1.

In this work, we continue to survey what has been done on the Roman domination. More precisely, we will present in two sections several variations of Roman dominating functions as well as the signed version of some of these functions. It should be noted that a first part of this survey comprising 9 varieties is published as a chapter book in "Topics in domination in graphs" edited by T.W. Haynes, S.T. Hedetniemi and M.A. Henning. We recall that a function f : V ! f0, 1, 2g is a Roman dominating function (or just RDF) if every vertex u for which f(u) ¼ 0 is adjacent to at least one vertex v for which f(v) ¼ 2. The Roman domination number of a graph G, denoted by c R ðGÞ, is the minimum weight of an RDF on G.

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any| over all such functions f . Let G be a connected graph of order |V (G)| = n ≥ 3. We prove that γ r2 (G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of γ r2 (G) in terms of diameter are also given.

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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 is the value f (V (G)) = u∈V (G) f (u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number γ R (G) (respectively, the independent Roman domination number i R (G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that γ R (G) strongly equals i R (G), denoted by γ R (G) ≡ i R (G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with γ R (T ) ≡ i R (T ).