Signed total Roman k-domination in directed graphs

Document Type: Original paper

Authors

1 Sirjan University of Technology, Sirjan 78137, Iran

2 Lehrstuhl II fur Mathematik, RWTH Aachen University, 52056 Aachen, Germany

Abstract

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 $G$‎.

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