On the total and strong version for Roman dominating functions in graphs

The total and strong version of the Roman domination number (for graphs) is introduced in this research, and the study of its mathematical properties is therefore initiated. We establish upper bounds for such a parameter, and relate it with several parameters concerning vertex domination in graphs. In addition, among other results, we show that for any tree T of order n(T)≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n(T)\ge 3$$\end{document}, maximum degree Δ(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (T)$$\end{document} and s(T) support vertices, the total strong Roman domination number is bounded below by n(T)+s(T)Δ(T)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\lceil \frac{n(T)+s(T)}{\Delta (T)}\right\rceil +1$$\end{document}.