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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n$\end{document} and that γsR(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3$\end{document}, and we characterize the extremal graphs.