Signed total Roman k-domination in directed graphs

Let $D$ be a finite and simple digraph with vertex set $V(D)$‎. ‎A signed total Roman $k$-dominating function (STR$k$DF) on‎ ‎$D$ is a function $f:V(D)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎ ‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎ ‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎ ‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u)=-1$ has an inner neighbor $v$ for which $f(v)=2$‎. ‎The weight of an STR$k$DF $f$ is $omega(f)=sum_{vin V (D)}f(v)$‎. ‎The signed total Roman $k$-domination number $gamma^{k}_{stR}(D)$‎ ‎of $D$ is the minimum weight of an STR$k$DF on $D$‎. ‎In this paper we‎ ‎initiate the study of the signed total Roman $k$-domination number‎ ‎of digraphs‎, ‎and we present different bounds on $gamma^{k}_{stR}(D)$‎. ‎In addition‎, ‎we determine the signed total Roman $k$-domination‎ ‎number of some classes of digraphs‎. ‎Some of our results are extensions‎ ‎of known properties of the signed total Roman $k$-domination‎ ‎number $gamma^{k}_{stR}(G)$ of graphs $G$‎.