On the power domination number of de Bruijn and Kautz digraphs

Abstract

Let $G=(V,A)$ be a directed graph without parallel arcs, and let $S\subseteq V$ be a set of vertices. Let the sequence $S=S_0\subseteq S_1\subseteq S_2\subseteq \cdots$ be defined as follows: $S_1$ is obtained from $S_0$ by adding all out-neighbors of vertices in $S_0$. For $k⩾2$, $S_k$ is obtained from $S_{k-1}$ by adding all vertices $w$ such that for some vertex $v\in S_{k-1}$, $w$ is the unique out-neighbor of $v$ in $V\setminus S_{k-1}$. We set $M(S)=S_0\cup S_1\cup \cdots$, and call $S$ a \emph{power dominating set} for $G$ if $M(S)=V(G)$. The minimum cardinality of such a set is called the \emph{power domination number} of $G$. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs.