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Connected power domination in graphs

Author

Listed:
  • Boris Brimkov

    (Rice University)

  • Derek Mikesell

    (Rice University)

  • Logan Smith

    (Rice University)

Abstract

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.

Suggested Citation

  • Boris Brimkov & Derek Mikesell & Logan Smith, 2019. "Connected power domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 292-315, July.
    • Handle: RePEc:spr:jcomop:v:38:y:2019:i:1:d:10.1007_s10878-019-00380-7
    DOI: 10.1007/s10878-019-00380-7
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    References listed on IDEAS

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    1. Daniela Ferrero & Leslie Hogben & Franklin H. J. Kenter & Michael Young, 2017. "Note on power propagation time and lower bounds for the power domination number," Journal of Combinatorial Optimization, Springer, vol. 34(3), pages 736-741, October.
    2. Ashkan Aazami, 2010. "Domination in graphs with bounded propagation: algorithms, formulations and hardness results," Journal of Combinatorial Optimization, Springer, vol. 19(4), pages 429-456, May.
    3. Li, Yuchao & Yang, Zishen & Wang, Wei, 2017. "Complexity and algorithms for the connected vertex cover problem in 4-regular graphs," Applied Mathematics and Computation, Elsevier, vol. 301(C), pages 107-114.
    4. Chung-Shou Liao, 2016. "Power domination with bounded time constraints," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 725-742, February.
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    Cited by:

    1. Prosenjit Bose & Valentin Gledel & Claire Pennarun & Sander Verdonschot, 0. "Power domination on triangular grids with triangular and hexagonal shape," Journal of Combinatorial Optimization, Springer, vol. 0, pages 1-19.
    2. Prosenjit Bose & Valentin Gledel & Claire Pennarun & Sander Verdonschot, 2020. "Power domination on triangular grids with triangular and hexagonal shape," Journal of Combinatorial Optimization, Springer, vol. 40(2), pages 482-500, August.
    3. Joanna Cyman & Joanna Raczek, 2022. "Application of Doubly Connected Dominating Sets to Safe Rectangular Smart Grids," Energies, MDPI, vol. 15(9), pages 1-12, April.

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