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 conditions‎‎that (i) $sum_{xin N^{-}(v)}f(x)ge k$ for each‎‎$vin V(D)$‎, ‎where $N^{-}(v)$ consists of all vertices of $D$ from‎‎which 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 we‎‎initiate the study of the signed total Roman $k$-domination number‎‎of digraphs‎, ‎and we present different bounds on $gamma^{k}_{stR}(D)$‎.‎In addition‎, ‎we determine the signed total Roman $k$-domination‎‎number of some classes of digraphs‎. ‎Some of our results are extensions‎‎of known properties of the signed total Roman $k$-domination‎‎number $gamma^{k}_{stR}(G)$ of graphs $G$‎.