Restricted power domination and zero forcing problems

Bozeman, Chassidy; Brimkov, Boris; Erickson, Craig; Ferrero, Daniela; Flagg, Mary; Hogben, Leslie

doi:10.1007/s10878-018-0330-6

Cited by 19 publications

(22 citation statements)

References 24 publications

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“…In this section, we present a technique for computing the connected power domination numbers of graphs with cut vertices in terms of the connected power domination numbers of their nontrivial blocks. We first introduce several definitions and lemmas which will be used in the main result of this section; some of these are adapted from [12,32].…”

Section: Cut Vertex Decompositionmentioning

confidence: 99%

Connected power domination in graphs

2019

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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.vertices; the physical laws by which PMUs can observe a network give rise to the following color change rules (see [17]): 1) Every neighbor of an initially colored vertex becomes colored.2) Whenever there is a colored vertex with exactly one uncolored neighbor, that neighbor becomes colored.S is a power dominating set of G if all vertices in G become colored after applying rule 1) once, and rule 2) as many times as possible (i.e. until no more vertices can change color). The power domination number of G, denoted γ P (G), is the cardinality of a minimum power dominating set. S is a zero forcing set of G if all vertices in G become colored after applying rule 2) as many times as possible (and not applying rule 1) at all). The zero forcing number of G, denoted Z(G), is the cardinality of a minimum zero forcing set. The process of zero forcing was introduced independently in combinatorial matrix theory [5] and in quantum control theory [18].In this paper, we study a variant of power domination which requires every set of initially colored vertices to induce a connected subgraph. Given a connected graph G = (V, E), a set S ⊂ V is a connected power dominating set of G if S is a power dominating set and G[S] is connected. The connected power domination number, denoted γ P,c (G), is the cardinality of a minimum connected power dominating set. Requiring a power dominating set to be connected is motivated by the application in monitoring electrical networks: the data from PMUs is relayed by high-speed communication infrastructure to processing stations which collect and manage this data; thus, in addition to minimizing the production costs of the PMUs, an electric power company may seek to place all PMUs in a compact, connected region in the network in order to reduce the number of processing stations and related infrastructure required to collect the data.Connected power domination was explored from a computational perspective in [27] (although the p...

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Section: Cut Vertex Decompositionmentioning

confidence: 99%

Connected power domination in graphs

2019

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“…Several variations of power domination have also been proposed. These include versions in which the initial colored set must induce a connected graph [10,18] or must contain certain vertices [8,28]. The computational complexity of power domination has also been a topic of interest.…”

Section: Introductionmentioning

confidence: 99%

“…A third infection IP has also been introduced by Fan and Watson [18]; however, their model considered a slightly different problem than the standard power dominating set problem and cannot be directly compared with the other methods. In contrast to the infection models, Bozeman et al [8] proposed a computational method for the closely related restricted power dominating set problem using a set cover formulation and delayed constraint generation based on structures called forts, which originate from the related zero forcing set problem [9].…”

Section: Introductionmentioning

confidence: 99%

“…As the main results of this paper, we introduce new constraint generation method for the power domination set cover formulation based on the neighborhoods of forts and detail several general computational enhancements which can be applied to our methodology as well as previous methods. Our proposed constraint generation method is based on a novel IP formulation which directly considers the neighborhoods of forts, in contrast to the method proposed by Bozeman et al [8] which only considers the forts themselves. To formulate this IP, we study the structural properties of fort neighborhoods, provide necessary and sufficient conditions for vertex sets to be the neighborhoods of forts, and give a fast polynomial procedure by which special instances of these neighborhoods can be identified.…”

Section: Introductionmentioning

confidence: 99%

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New computational approaches for the power dominating set problem: Set covering and the neighborhoods of zero forcing forts

Smith

Hicks

2021

Networks

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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 method for computing minimum power dominating sets via a set cover IP formulation and a novel constraint generation procedure. The set cover problem's constraints correspond to neighborhoods of zero forcing forts; we study their structural properties and show they can be separated with delayed row generation. In addition, we offer several computation enhancements which be be applied to our methodology as well as existing methods. The proposed and existing methods are evaluated in several computational experiments. In many of the larger test instances considered, the proposed method exhibits an order of magnitude runtime performance improvement.

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“…Power domination has also been widely studied from a purely graph theoretic perspective. See, e.g., [6,10,13,20,21,29,42,44] for various structural and computational results about power domination and related variants. The power propagation time of a graph has previously been studied in [1,19,24,31].…”

Section: Introductionmentioning

confidence: 99%

Power domination throttling

Brimkov

Carlson

Hicks

et al. 2019

Theoretical Computer Science

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A power dominating set of a graph G = (V, E) is a set S ⊂ 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.

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Restricted power domination and zero forcing problems

Cited by 19 publications

References 24 publications

Connected power domination in graphs

Connected power domination in graphs

New computational approaches for the power dominating set problem: Set covering and the neighborhoods of zero forcing forts

Power domination throttling

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