Roman domination number of signed graphs

A function f : V → {0, 1, 2} on a signed graph S = (G, σ) P where G = (V, E) is a Roman dominating function (RDF) if f(N[v]) = f(v) + ∑u∈N(v) σ(uv)f(u) ≥ 1 for all v ∈ V and for each vertex v with f(v) = 0 there is a vertex u in N+(v) such that f(u) = 2. The weight of an RDF f is given by ω(f) = ∑v∈V f(v) and the minimum weight among all the RDFs on S is called the Roman domination number γR(S). Any RDF on S with the minimum weight is known as a γR(S)-function. In this article we obtain certain bounds for γR and characterise the signed graphs attaining small values for γR.