Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximum degree $\Delta$. A signed strong total Roman dominating function on a graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil \frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) for every vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where $N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ for which $f(v)=-1$ is adjacent to at least one vertex $w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where $V_{-1}=\{v\in V: f(v)=-1\}$. The minimum of the values $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strong total Roman dominating functions $f$ of $G$, is called the signed strong total Roman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$. In this paper, we initiate signed strong total Roman domination number of a graph and give several bounds for this parameter. Then, among other results, we determine the signed strong total Roman domination number of special classes of graphs.
