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 $S \subseteq V(G)$, 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 $\left\lceil \dfrac{k}{3} \right\rceil$, and the one of a triangular grid $T_k$ with triangular-shaped border of length $k-1$ is $\left\lceil \dfrac{k}{4} \right\rceil$.