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)


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.


Power domination de Bruijn digraph Kautz digraph 



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


  1. 1.
    Aazami, A., Stilp, K.: Approximation algorithms and hardness for domination with propagation. SIAM J. Discret. Math. 23, 1382–1399 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    AIM Minimum Rank – Special Graphs Work Group: Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl. 428(7), 1628–1648 (2008)Google Scholar
  3. 3.
    Barioli, F., Barrett, W., Fallat, S.M., Hall, H.T., Hogben, L., Shader, B., van den Driessche, P., van der Holst, H.: Zero forcing parameters and minimum rank problems. Linear Algebra Appl. 433(2), 401–411 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barioli, F., Fallat, S.M., Hall, H.T., Hershkowitz, D., Hogben, L., van der Holst, H., Shader, B.: On the minimum rank of not necessarily symmetric matrices: a preliminary study. Electron. J. Linear Algebra 18, 126–145 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barioli, F., Barrett, W., Fallat, S.M., Hall, H.T., Hogben, L., Shader, B., van den Driessche, P., van der Holst, H.: Parameters related to tree-width, zero forcing, and maximum nullity of a graph. J. Graph Theory 72(2), 146–177 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Berman, A., Friedland, S., Hogben, L., Rothblum, U.G., Shader, B.: An upper bound for the minimum rank of a graph. Linear Algebra Appl. 429(7), 1629–1638 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chang, G.J., Dorbec, P., Montassier, P., Raspaud, A.: Generalized power domination of graphs. Discret. Appl. Math. 160, 1691–1698 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dorbec, P., Mollard, M., Klavzar, S., Spacapan, S.: Power domination in product graphs. SIAM J. Discret. Math. 22, 554–567 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dong, Y., Shan, E., Kang, L.: Constructing the minimum dominating sets of generalized de Bruijn digraphs. Discret. Math. 338, 1501–1508 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Edholm, C.J., Hogben, L., Huynh, M., LaGrange, J., Row, D.D.: Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph. Linear Algebra Appl. 436(12), 4352–4372 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T., Henning, M.A.: Domination in graphs applied to electric power networks. SIAM J. Discret. Math. 15(4), 519–529 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hogben, L., Huynh, M., Kingsley, N., Meyer, S., Walker, S., Young, M.: Propagation time for zero forcing on a graph. Discret. Appl. Math. 160(13–14), 1994–2005 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Huang, J., Xu, J.M.: The bondage numbers of extended de Bruijn and Kautz digraphs. Comput. Math. Appl. 51, 1137–1147 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Imase, M., Itoh, M.: A design for directed graphs with minimum diameter. IEEE Trans. Comput. 32, 782–784 (1983)CrossRefGoogle Scholar
  15. 15.
    Kuo, J., Wu, W.L.: Power domination in generalized undirected de Bruijn graphs and Kautz graphs. Discrete Math. Algorithm. Appl. 07, 2961–2973 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lu, L., Wu, B., Tang, Z.: Proof of a conjecture on the zero forcing number of a graph. arXiv:1507.01364 (2015)
  17. 17.
    Severini, S.: Nondiscriminatory propagation on trees. J. Phys. A: Math. Theor. 41(48), 482002 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Stephen, S., Rajan, B., Ryan, J., Grigorious, C., William, A.: Power domination in certain chemical structures. J. Discret. Algorithms 33, 10–18 (2015)MathSciNetCrossRefGoogle Scholar

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

Personalised recommendations