AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 52 (2012), Pages 11–18 Properties of independent Roman domination in graphs ∗ M. Adabi E. Ebrahimi Targhi N. Jafari Rad † M. Saied Moradi Department of Mathematics Shahrood University of Technology Shahrood Iran Abstract A Roman dominating function on a graph G is a function f : V (G) → {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 a Roman dominating function is the value f (V (G)) = ∑ u∈V (G) f (u). The Roman domination number of G, γ R (G), is the minimum weight of a Roman dominating function on G. In this paper, we study independent Roman domination in graphs and obtain some properties, bounds and characterizations for the independent Roman domination number of a graph. 1 Introduction Let G =(V (G),E(G)) be a simple graph of order n. We denote the open neigh- borhood of a vertex v of G by N G (v), or just N (v), and its closed neighborhood by N [v]. For a vertex set S ⊆ V (G), N (S )= ∪ v∈S N (v) and N [S ]= ∪ v∈S N [v]. The degree deg(x) of a vertex x denotes the number of neighbors of x in G. A set of vertices S in G is a dominating set, if N [S ]= V (G). The domination number, γ(G) of G, is the minimum cardinality of a dominating set of G. For a graph G and a subset of vertices S we denote by G[S ] the subgraph of G induced by S . A subset S of vertices is independent if G[S ] has no edge. A set S ⊆ V (G) is an independent dominating set if S is independent and dominating set. The minimum cardinality of such a set is the independent domination number i(G). For notation and graph theory terminology in general we follow [2]. ∗ The research is supported by Shahrood University of Technology. † Corresponding author; email: n.jafarirad@shahroodut.ac.ir