IWOCA 2017: Combinatorial Algorithms pp 264-272

# On the Power Domination Number of de Bruijn and Kautz Digraphs

• Cyriac Grigorious
• Thomas Kalinowski
• Sudeep Stephen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10765)

## Abstract

Let $$G=(V,A)$$ be a directed graph, 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 power dominating set for G if $$M(S)=V(G)$$. The minimum cardinality of such a set is called the power domination number of G. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs.

## Keywords

Power domination de Bruijn digraph Kautz digraph

## Notes

### Acknowledgement

The authors would like to thank Dr. Joe Ryan for his valuable comments and suggestions to improve the quality of the paper.

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