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.