Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21281220161201Signed total Roman k-domination in directed graphs1651781357610.22049/cco.2016.13576ENNasrinDehgardiSirjan University of Technology, Sirjan 78137, IranLutzVolkmannLehrstuhl II fur Mathematik,
RWTH Aachen University,
52056 Aachen, Germany0000-0003-3496-277XJournal Article20160913Let $D$ 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 $G$.http://comb-opt.azaruniv.ac.ir/article_13576_afdcd0fac389c7cc1b729f716dbbce32.pdf