Connected power domination in graphs

Abstract

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...

Computational approaches for zero forcing and related problems

“…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.…”

“…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].…”

Power domination throttling

“…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]).…”

“…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].…”

The fixed set search applied to the power dominating set problem

“…In this article, we focus on finding optimal and near optimal solutions for the PDSP and the CPDSP. In case of finding optimal solutions, a new ILP is introduced similar to the one presented in (Brimkov et al, 2017). The main difference to this model is that the two rules for observing nodes (domination and propagation) are treated separately.…”