Power domination on triangular grids with triangular and hexagonal shape

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 k 3 , and the one of a triangular grid T k with triangular-shaped border of length k - 1 is k 4 .