On the power domination number of de Bruijn and Kautz digraphs

TL;DR

This paper determines the minimum size of power dominating sets in de Bruijn and Kautz digraphs, which are important in network monitoring and control, by analyzing their structural properties.

Contribution

It provides exact values for the power domination numbers of de Bruijn and Kautz digraphs, advancing understanding of their controllability.

Findings

Exact power domination numbers for de Bruijn digraphs

Exact power domination numbers for Kautz digraphs

Enhanced understanding of network monitoring in these graphs

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 $k\geqslant 2$, $S_k$ 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.