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 \(n(T)\ge 3\), maximum degree \(\Delta (T)\) and s(T) support vertices, the total strong Roman domination number is bounded below by \(\left\lceil \frac{n(T)+s(T)}{\Delta (T)}\right\rceil +1\).

中文翻译：

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关于图中罗马支配函数的完整版本和强版本

在这项研究中介绍了罗马支配数的总和强形式（用于图形），因此开始了其数学性质的研究。我们为此类参数建立上限，并将其与涉及图形中顶点控制的几个参数相关联。另外，除其他结果外，我们显示对于阶数\（n（T）\ ge 3 \），最大度数\（\ Delta（T）\）和s（T）支持顶点的任何树T，总强罗马统治号码由\（\ left \ lceil \ frac {n（T）+ s（T）} {\ Delta（T）} \ right \ rceil +1 \）限制。