Journal of the
Korean Mathematical Society JKMS

Let $k$ be a positive integer, and let $G$ be a simple graph with vertex set $V(G)$. A Roman $k$-dominating function on $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ such that every vertex $u$ for which $f(u)=0$ is adjacent to at least $k$ vertices $v_1,v_2,\ldots,v_k$ with $f(v_i)=2$ for $i=1,2,\ldots,k$. The weight of a Roman $k$-dominating function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman $k$-dominating function on a graph $G$ is called the Roman $k$-domination number $\gamma_{kR}(G)$ of $G$. Note that the Roman 1-domination number $\gamma_{1R}(G)$ is the usual Roman domination number $\gamma _R(G)$. In this paper, we investigate the properties of the Roman $k$-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Keywords: domination, Roman $k$-domination, Roman domination, $k$-domination

MSC numbers: 05C69