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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 here show that the power domination number of a triangular grid T_k with hexagonal-shape border of length k -- 1 is exactly $lceil k/3 rceil.
Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measur
The domination number of a graph $G = (V,E)$ is the minimum cardinality of any subset $S subset V$ such that every vertex in $V$ is in $S$ or adjacent to an element of $S$. Finding the domination numbers of $m$ by $n$ grids was an open problem for ne
A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $Vbackslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by $gamma_{[1,2]}(G)$, is call
The power domination problem seeks to find the placement of the minimum number of sensors needed to monitor an electric power network. We generalize the power domination problem to hypergraphs using the infection rule from Bergen et al: given an init
In this paper we continue the study of the edge intersection graphs of one (or zero) bend paths on a rectangular grid. That is, the edge intersection graphs where each vertex is represented by one of the following shapes: $llcorner$,$ulcorner$, $urco