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Restricted power domination and zero forcing problems

Author

Listed:
  • Chassidy Bozeman

    (Iowa State University)

  • Boris Brimkov

    (Rice University)

  • Craig Erickson
  • Daniela Ferrero

    (Texas State University)

  • Mary Flagg

    (University of St. Thomas)

  • Leslie Hogben

    (Iowa State University
    American Institute of Mathematics)

Abstract

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.

Suggested Citation

  • Chassidy Bozeman & Boris Brimkov & Craig Erickson & Daniela Ferrero & Mary Flagg & Leslie Hogben, 2019. "Restricted power domination and zero forcing problems," Journal of Combinatorial Optimization, Springer, vol. 37(3), pages 935-956, April.
    • Handle: RePEc:spr:jcomop:v:37:y:2019:i:3:d:10.1007_s10878-018-0330-6
    DOI: 10.1007/s10878-018-0330-6
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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. Wayne Goddard & Michael A. Henning, 2007. "Restricted domination parameters in graphs," Journal of Combinatorial Optimization, Springer, vol. 13(4), pages 353-363, May.
    3. Boting Yang, 2017. "Lower bounds for positive semidefinite zero forcing and their applications," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 81-105, January.
    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. Shuchao Li & Wanting Sun, 2020. "On the zero forcing number of a graph involving some classical parameters," Journal of Combinatorial Optimization, Springer, vol. 39(2), pages 365-384, February.
    2. 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.
    3. 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.
    4. Boris Brimkov & Derek Mikesell & Illya V. Hicks, 2021. "Improved Computational Approaches and Heuristics for Zero Forcing," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1384-1399, October.

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