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

Power Domination in Graphs

“…When studying the triangular grid with hexagonal outer shape, Bose, Pennarun and Verdonschot [6] noticed a conection with that earlier result. They got a similar bound: Theorem 13 (Bose, Pennarun, Verdonschot [6]) For k ≥ 1, le T k be the triangular grid with hexagonal shape, whose side is made of k vertices. We have…”

Connected power domination number of product graphs

“…Further, some upper bound for the power domination number of graphs is obtained in [23]. Furthermore, the power domination number of some standard families of graphs and product graphs are studied in [5,6,8,9,14,15,[17][18][19][20][21][22]. Recently, Brimkvo et al [7] introduced the concept of connected power domination number of graph and obtained the exact value for trees, block graph, and cactus graph.…”