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

Consider a finite and simple graph $G=(V,E)$ with maximum degree $\Delta$. A strong Roman dominating function over the graph $G$ is understood as a map $f : V (G)\rightarrow \{0, 1,\ldots , \left\lceil \frac{\Delta}{2}\right\rceil+ 1\}$ which carries out the condition stating that all the vertices $v$ labeled $f(v)=0$ are adjacent to at least one another vertex $u$ that satisfies $f(u)\geq 1+ \left\lceil \frac{1}{2}\vert N(u)\cap V_0\vert \right\rceil$, such that $V_0=\{v \in V \mid f(v)=0 \}$ and the notation $N(u)$ stands for the open neighborhood of $u$. The total version of one strong Roman dominating function includes the additional property concerning the not existence of vertices of degree zero in the subgraph of $G$, induced by the set of vertices labeled with a positive value. The minimum possible value for the sum $\omega(f)=f(V)=\sum_{v\in V} f(v)$ (also called the weight of $f$), taken amongst all existent total strong Roman dominating functions $f$ of $G$, is called the total strong Roman domination number of $G$, denoted by $\gamma_{StR}^t(G)$. This 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. For instance, we establish upper bounds for such parameter, and relate it with several parameters related to vertex domination in graphs, from which we remark the standard domination number, the total version of the standard domination number and the (strong) Roman domination number. In addition, among other results, we show that for any tree $T$ of order $n(T)\ge 3$, with maximum degree $\Delta(T)$ and $s(T)$ support vertices, $\gamma_{StR}^t(T)\ge \left\lceil \frac{n(T)+s(T)}{\Delta(T)}\right\rceil+1$.