RT - Journal Article T1 - Roman k-Tuple Domination in Graphs JF - IJMSI YR - 2020 JO - IJMSI VO - 15 IS - 2 UR - http://ijmsi.ir/article-1-1140-en.html SP - 101 EP - 115 K1 - Roman $k$-tuple domination number K1 - Roman $k$-tuple graph K1 - $k$-Tuple domination number K1 - $k$-Tuple total domination number K1 - Mycieleskian of a graph. AB - For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominating function $f$ on $G$ is called the Roman $k$-tuple domination number of the graph where the weight of $f$ is $f(V)=sum_{vin V}f(v)$‎. ‎In this paper‎, ‎we initiate to study the Roman $k$-tuple‎ ‎domination number of a graph‎, ‎by giving some sharp bounds for the Roman $k$-tuple domination number of a garph‎, ‎the Mycieleskian of a graph‎, ‎and the corona graphs‎. ‎Also finding the Roman $k$-tuple domination number of some known graphs is our other goal‎. ‎Some of our results extend these one‎ ‎given by Cockayne and et al‎. ‎cite{CDHH04} in 2004 for the Roman‎ ‎domination number‎. LA eng UL http://ijmsi.ir/article-1-1140-en.html M3 ER -