Connected power domination in graphs

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.Comment: 21 page

Power domination throttling

A power dominating set of a graph $G=(V,E)$ is a set $S\subset V$ that colors every vertex of $G$ according to the following rules: in the first timestep, every vertex in $N[S]$ becomes colored; in each subsequent timestep, every vertex which is the only non-colored neighbor of some colored vertex becomes colored. The power domination throttling number of $G$ is the minimum sum of the size of a power dominating set $S$ and the number of timesteps it takes $S$ to color the graph. In this paper, we determine the complexity of power domination throttling and give some tools for computing and bounding the power domination throttling number. Some of our results apply to very general variants of throttling and to other aspects of power domination.Comment: 19 page

Computational Approaches for Zero Forcing and Related Problems

In this paper, we propose computational approaches for the zero forcing problem, the connected zero forcing problem, and the problem of forcing a graph within a specified number of timesteps. Our approaches are based on a combination of integer programming models and combinatorial algorithms, and include formulations for zero forcing as a dynamic process, and as a set-covering problem. We explore several solution strategies for these models, test them on various types of graphs, and show that they are competitive with the state-of-the-art algorithm for zero forcing. Our proposed algorithms for connected zero forcing and for controlling the number of zero forcing timesteps are the first general-purpose computational methods for these problems, and are superior to brute force computation.Comment: 37 pages, 4 tables; computer code available in GitHu

Optimal Sensor Placement in Power Grids: Power Domination, Set Covering, and the Neighborhoods of Zero Forcing Forts

To monitor electrical activity throughout the power grid and mitigate outages, sensors known as phasor measurement units can installed. Due to implementation costs, it is desirable to minimize the number of sensors deployed while ensuring that the grid can be effectively monitored. This optimization problem motivates the graph theoretic power dominating set problem. In this paper, we propose a novel integer program for identifying minimum power dominating sets by formulating a set cover problem. This problem's constraints correspond to neighborhoods of zero forcing forts; we study their structural properties and show they can be separated, allowing the proposed model to be solved via row generation. The proposed and existing methods are compared in several computational experiments in which the proposed method consistently exhibits an order of magnitude improvement in runtime performance.Comment: 26 pages, 7 figure

Power domination polynomials of graphs

A power dominating set of a graph is a set of vertices that observes every vertex in the graph by combining classical domination with an iterative propagation process arising from electrical circuit theory. In this paper, we study the power domination polynomial of a graph $G$ of order $n$, defined as $\mathcal{P}(G;x)=\sum_{i=1}^n p(G;i) x^i$, where $p(G;i)$ is the number of power dominating sets of $G$ of size $i$. We relate the power domination polynomial to other graph polynomials, present structural and extremal results about its roots and coefficients, and identify some graph parameters it contains. We also derive decomposition formulas for the power domination polynomial, compute it explicitly for several families of graphs, and explore graphs which can be uniquely identified by their power domination polynomials.Comment: 23 page