﻿ Signed total Roman k-domination in directed graphs

# Signed total Roman k-domination in directed graphs

##### نویسندگان
• Lutz Volkmann Lehrstuhl II fur Mathematik, RWTH Aachen University, 52056 Aachen, Germany
##### چکیده

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\$‎.

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عنوان ژورنال:

دوره 1  شماره 2

صفحات  165- 178

تاریخ انتشار 2016-12-30

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