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 here show that the power domination number of a triangular grid Tk with hexagonal-shape border of length k − 1 is exactly [ k3 ] .

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Conference 29th Canadian Conference on Computational Geometry, CCCG 2017
Citation
Bose, P, Pennarun, C. (Claire), & Verdonschot, S. (Sander). (2017). Power domination on triangular grids. In CCCG 2017 - 29th Canadian Conference on Computational Geometry, Proceedings (pp. 2–6).