Signed total Roman k-domination in directed graphs

Abstract

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