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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate SchoolKing’s College LondonLondonUK
  2. 2.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia
  3. 3.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia
  4. 4.School of Mathematical SciencesNational Institute of Science Education and ResearchBhubaneswarIndia

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