Power domination on triangular grids with triangular and hexagonal shape

Abstract : The concept of power domination emerged from the problem of monitoring electrical systems. Given a graph G and a set 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 with hexagonal-shaped border of length k − 1 is the ceiling of k/3 , and the one of a triangular grid T_k with triangular-shaped border of length k − 1 is the ceiling of k/4 .
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2018
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Prosenjit Bose, Valentin Gledel, Claire Pennarun, Sander Verdonschot. Power domination on triangular grids with triangular and hexagonal shape. 2018. 〈hal-01814346〉

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