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Power domination with bounded time constraints

Author

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  • Chung-Shou Liao

    (National Tsing Hua University)

Abstract

Based on the power observation rules, the problem of monitoring a power utility network can be transformed into the graph-theoretic power domination problem, which is an extension of the well-known domination problem. A set $$S$$ S is a power dominating set (PDS) of a graph $$G=(V,E)$$ G = ( V , E ) if every vertex $$v$$ v in $$V$$ V can be observed under the following two observation rules: (1) $$v$$ v is dominated by $$S$$ S , i.e., $$v \in S$$ v ∈ S or $$v$$ v has a neighbor in $$S$$ S ; and (2) one of $$v$$ v ’s neighbors, say $$u$$ u , and all of $$u$$ u ’s neighbors, except $$v$$ v , can be observed. The power domination problem involves finding a PDS with the minimum cardinality in a graph. Similar to message passing protocols, a PDS can be considered as a dominating set with propagation that applies the second rule iteratively. This study investigates a generalized power domination problem, which limits the number of propagation iterations to a given positive integer; that is, the second rule is applied synchronously with a bounded time constraint. To solve the problem in block graphs, we propose a linear time algorithm that uses a labeling approach. In addition, based on the concept of time constraints, we provide the first nontrivial lower bound for the power domination problem.

Suggested Citation

  • Chung-Shou Liao, 2016. "Power domination with bounded time constraints," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 725-742, February.
  • Handle: RePEc:spr:jcomop:v:31:y:2016:i:2:d:10.1007_s10878-014-9785-2
    DOI: 10.1007/s10878-014-9785-2
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    References listed on IDEAS

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    1. 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.
    2. Guangjun Xu & Liying Kang, 2011. "On the power domination number of the generalized Petersen graphs," Journal of Combinatorial Optimization, Springer, vol. 22(2), pages 282-291, August.
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    Cited by:

    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. 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 & Logan Smith, 2019. "Connected power domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 292-315, July.
    5. 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.

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