Connected power domination in graphs

Brimkov, Boris; Mikesell, Derek; Smith, Logan Pearsall

doi:10.48550/arxiv.1712.02388

Cited by 3 publications

(4 citation statements)

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“…A closely related problem to zero forcing is power domination, where given a set S of initially colored vertices, the zero forcing color change rule is applied to N [S] instead of to S. Integer programming formulations for power domination and its variants have been explored in [1,18].…”

Section: Introductionmentioning

confidence: 99%

Computational approaches for zero forcing and related problems

Brimkov

Fast

Hicks

2019

European Journal of Operational Research

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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.Connected zero forcing is a variant of zero forcing in which the initially colored set of vertices induces a connected subgraph. The connected zero forcing number of a graph is the cardinality of the smallest connected set of initially colored vertices which forces the entire graph to be colored (i.e., the smallest connected zero forcing set). Applications and various structural and computational aspects of connected zero forcing have been investigated in [15,16,17]; in particular, it can be used for modeling the spread of ideas or diseases originating from a single connected source in a network, or for power network monitoring accounting for the cost of supporting infrastructure. Other variants of zero forcing, such as positive semidefinite zero forcing [11,33,44,57], fractional zero forcing, signed zero forcing [41], and k-forcing [5,48] have also been studied. These are typically obtained by modifying the zero forcing color change rule, or adding certain restrictions to a zero forcing set. The number of timesteps in the zero forcing process after which a graph becomes colored is also a problem of interest (see, e.g., [14,21,28,43,57]). Connected variants of other graph problems -such as connected domination and connected power domination [25,31,36,40] -have been extensively studied as well.A closely related problem to zero forcing is power domination, where given a set S of initially colored vertices, the zero forcing color change rule is applied to N [S] instead of to S. Integer programming formulations for power domination and its variants have been explored in [1,18].The power domination problem is derived from the phase measurement unit (PMU) placement problem in electrical engineering, which has also been studied extensively; see, e.g., [47,49] and the bibliographies therein for various integer programming models and combinatorial algorithms for the PMU placement problem. Another closely related problem to zero forcing is the target set selection problem, where given a set S of initially colored vertices and a threshold function θ : V (G) → Z, all uncolored vertices v that have at least θ(v) colored neighbors become colored. Thus, the zero forcing problem constrains the infectors, but the target set selection problem constrai...

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Section: Introductionmentioning

confidence: 99%

Computational approaches for zero forcing and related problems

Brimkov

Fast

Hicks

2019

European Journal of Operational Research

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“…This PMU placement problem has been explored extensively in the electrical engineering literature; see [4,5,14,30,35,36,37,38], and the bibliographies therein for various placement strategies and computational results. The PMU placement literature also considers various other properties of power dominating sets, such as redundancy, controlled islanding, and connectedness, and optimizes over them in addition to the cardinality of the set (see, e.g., [3,13,34,41]).…”

Section: Introductionmentioning

confidence: 99%

“…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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“…[12]). The PMU placement literature also considers power dominating sets with various additional properties, such as redundancy, controlled islanding, and connectedness, and optimizes over them in addition to the cardinality of the set (see, e.g., [5,15,41,49]). In order to study the collection of power dominating sets of a graph in a more general framework, we introduce the power domination polynomial, which counts the number of distinct power dominating sets of a given size.…”

Section: Introductionmentioning

confidence: 99%

Power domination polynomials of graphs

Brimkov,

Patel,

Suriyanarayana

et al. 2018

Preprint

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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 P(G; x) = n i=1 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.

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Connected power domination in graphs

Cited by 3 publications

References 0 publications

Computational approaches for zero forcing and related problems

Computational approaches for zero forcing and related problems

Power domination throttling

Power domination polynomials of graphs

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