Bulletin of the
Korean Mathematical Society BKMS

Let $G=(V,E)$ be a graph and $k$ be a positive integer. A $k$-dominating set of $G$ is a subset $S\subseteq V$ such that each vertex in $V\setminus S$ has at least $k$ neighbors in $S$. A Roman $k$-dominating function on $G$ is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to at least $k$ vertices $v_1,v_2,\ldots, v_k$ with $f(v_i)=2$ for $i=1,2,\ldots, k$. In the paper titled ``Roman $k$-domination in graphs" (J. Korean Math. Soc. {\bf 46} (2009), no. 6, 1309--1318) K. Kammerling and L. Volkmann showed that for any graph $G$ with $n$ vertices, $\gamma_{kR}(G)+ \gamma_{kR}(\overline{G})\geq\ \mbox{min}\ \{2n,4k+1\}$, and the equality holds if and only if $n\le 2k$ or $k\ge 2$ and $n=2k+1$ or $k=1$ and $G$ or $\overline{G}$ has a vertex of degree $n-1$ and its complement has a vertex of degree $n-2$. In this paper we find a counterexample of Kammerling and Volkmann's result and then give a correction to the result.

Keywords: dominating set, Roman $k$-dominating function, correction

MSC numbers: 05C69