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) $\sum_{x\in N^{-}(v)}f(x)\ge k$ for each‎ ‎$v\in V(D)$‎, ‎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 $\omega(f)=\sum_{v\in V (D)}f(v)$‎. ‎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 $G$‎.

Abstract licence: CC BY-SA

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