Signed Roman domination in graphs

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V→{?1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=?1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that $\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n$ and that γ _sR(G)≥(3n?4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that $\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3$ , and we characterize the extremal graphs.