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 \(\left\lceil \dfrac{k}{4} \right\rceil \).

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具有三角形和六边形的三角形网格上的功率控制

功率支配的概念来自监视电气系统的问题。给定一个图G和一个集合\（S \ subseteq V（G）\），按如下方法构建一组受监视的顶点M：首先，M仅包含S的顶点及其直接相邻的顶点，然后每次M中的顶点恰好有一个邻居不在M中，该邻居被添加到M中。图G的幂控制数是集合S的最小大小，因此该过程以包含每个顶点的集合M结束摹。我们显示，边长为（k-1 \）的六边形边界的三角形网格\（H_k \）的幂控制数为\（\ left \ lceil \ dfrac {k} {3} \ right \ rceil \ ），而三角形边框边界为\（k-1 \）的三角形网格\（T_k \）之一是\（\ left \ lceil \ dfrac {k} {4} \ right \ rceil \）。