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Power domination on triangular grids with triangular and hexagonal shape

Author

Listed:
  • Prosenjit Bose

    (Carleton University)

  • Valentin Gledel

    (Université de Lyon, Université Lyon 1)

  • Claire Pennarun

    (Univ. Montpellier)

  • Sander Verdonschot

    (Carleton University)

Abstract

The concept of power domination emerged from the problem of monitoring electrical systems. Given a graph G and a set $$S \subseteq V(G)$$ S ⊆ V ( G ) , a set M of monitored vertices is built as follows: at first, M contains only the vertices of S and their direct neighbors, and then each time a vertex in M has exactly one neighbor not in M, this neighbor is added to M. The power domination number of a graph G is the minimum size of a set S such that this process ends up with the set M containing every vertex of G. We show that the power domination number of a triangular grid $$H_k$$ H k with hexagonal-shaped border of length $$k-1$$ k - 1 is $$\left\lceil \dfrac{k}{3} \right\rceil $$ k 3 , and the one of a triangular grid $$T_k$$ T k with triangular-shaped border of length $$k-1$$ k - 1 is $$\left\lceil \dfrac{k}{4} \right\rceil $$ k 4 .

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:40:y:2020:i:2:d:10.1007_s10878-020-00587-z
    DOI: 10.1007/s10878-020-00587-z
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    References listed on IDEAS

    as
    1. 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.
    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. Boris Brimkov & Derek Mikesell & Logan Smith, 2019. "Connected power domination in graphs," Journal of Combinatorial Optimization, Springer, vol. 38(1), pages 292-315, July.
    4. Chung-Shou Liao, 2016. "Power domination with bounded time constraints," Journal of Combinatorial Optimization, Springer, vol. 31(2), pages 725-742, February.
    Full references (including those not matched with items on IDEAS)

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