A note on Roman $k$-tuple domination number

Abstract

‎‎For an integer $k\geq 2$‎, ‎a Roman $k$-tuple dominating function‎, ‎(or just RkDF)‎, ‎in a graph $G$ is a function $f \colon V(G) \rightarrow \{0‎, ‎1‎, ‎2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least $k$ vertices $v$ for which $f(v) = 2$‎, ‎and every vertex $u$ for which $f(u) \neq 0$ is adjacent to at least $k-1$ vertices $v$ for which $f(v) = 2$‎. ‎The Roman $k$-tuple domination number of ‎$‎G‎$‎‎ ‎is the minimum weight of an RkDF in $G$. ‎In this note we settle two problems posed in [Roman $k$-tuple Domination in Graphs‎, ‎Iranian J‎. ‎Math‎. ‎Sci‎. ‎Inform‎. ‎15 (2020)‎, ‎101--115]‎.