Nordhaus–Gaddum problems for power domination

Benson, Katherine F.; Ferrero, Daniela; Flagg, Mary; Furst, Veronika; Hogben, Leslie; Vasilevska, Violeta

doi:10.1016/j.dam.2018.06.004

Cited by 10 publications

(11 citation statements)

References 19 publications

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“…It is also interesting to point out some connections between our work and other well‐known graph‐theoretic problems. In particular, the so‐called zero‐forcing problem (and the strictly related concept of power domination ) are dynamical processes in graphs whose evolution can be described within our framework. More precisely, it can be easily seen that they correspond to an evangelization process (to use our language) in networks whose thresholds

t_{I} (v)

are all equal to one.…”

Section: What Is Already Known and What We Provementioning

confidence: 99%

Evangelism in social networks: Algorithms and complexity

et al. 2017

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We consider a population of interconnected individuals that, with respect to a piece of information, at each time instant can be subdivided into three (time-dependent) categories: agnostics, influenced, and evangelists. A dynamical process of information diffusion evolves among the individuals of the population according to the following rules. Initially, all individuals are agnostic. Then, a set of people is chosen from the outside and convinced to start evangelizing, i.e., to start spreading the information. When a number of evangelists, greater than a given threshold, communicate with a node v, the node v becomes influenced, whereas, as soon as the individual v is contacted by a sufficiently much larger number of evangelists, it is itself converted into an evangelist and consequently it starts spreading the information. The question is: How to choose a bounded cardinality initial set of evangelists so as to maximize the final number of influenced individuals? We prove that the problem is hard to solve, even in an approximate sense. On the positive side, we present exact polynomial time algorithms for trees and complete graphs. For general graphs, we derive exact parameterized algorithms. We also investigate the problem when the objective is to select a minimum number of evangelists capable of influencing the whole network. Our motivations to study these problems come from the areas of Viral Marketing and the analysis of quantitative models of spreading of influence in social networks.

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t_{I} (v)

are all equal to one.…”

Section: What Is Already Known and What We Provementioning

confidence: 99%

Evangelism in social networks: Algorithms and complexity

et al. 2017

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show abstract

“…This problem is studied for block graphs [34], circular-arc graphs [26], hypercubes [8,11], grids [16], generalized Petersen graphs [4,9,22,33], permutaion graphs [32], planar graphs with small diameter [37], maximal planar graphs [12], Knodel graphs and Hanoi graphs [20], de Bruijn graphs and Kautz graphs [24], regular claw-free graphs [28], and certain chemical graphs [30]. This problem is also discussed for Cartesian product of graphs in [4,23], tensor and strong product in [15], corona product and join of graphs in [35], and for some other graph products were discussed in [5]. The lower bounds for this problem is discussed in [17].…”

Section: Introductionmentioning

confidence: 99%

Optimal PMU placement problem in octahedral networks

Prabhu¹,

Deepa²,

Elavarasan

et al. 2022

RAIRO-Oper. Res.

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Power utilities must track their power networks to respond to changing demand and availability conditions to ensure effective and efficient operation. As a result, several power companies employ phase measuring units (PMUs) to check their power networks continuously. Supervising an electric power system with the fewest possible measurement equipment is precisely the vertex covering graph-theoretic problem, in which a set D is defined as a power dominating set (PDS) of a graph if it supervises every components (vertices and edges) in the system (with a couple of rules). The γp(G) is the minimal cardinality of a PDS of a graph G. In this present study, the PDS is identified for octahedral networks.

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“…Since it was formally introduced in [17], the power domination number has generated considerable interes; see, for example, [5,9,10,12,16,29].…”

Section: Introductionmentioning

confidence: 99%

Some Families of Graphs with Small Power Domination Number

Shahbaznejad,

Kazemi,

Pelayo

2021

Preprint

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Let G be a graph with the vertex set V (G) and S be a subset of V (G). Let cl(S) be the set of vertices built from S, by iteratively applying the following propagation rule: if a vertex and all of its neighbors except one of them are in cl(S), then the exceptional neighbor is also in cl(S). A set S is called a zero forcing set of G if cl(S) = V (G). The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set. Let cl(N [S]) be the set of vertices built from the closed neighborhood N [S] of S, by iteratively applying the previous propagation rule. A set S is called a power dominating set of G if cl(N [S]) = V (G). The power domination number γ p (G) of G is the minimum cardinality of a power dominating set. In this paper, we present some families of graphs that their power domination number is 1 or 2.

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Nordhaus–Gaddum problems for power domination

Cited by 10 publications

References 19 publications

Evangelism in social networks: Algorithms and complexity

Evangelism in social networks: Algorithms and complexity

Optimal PMU placement problem in octahedral networks

Some Families of Graphs with Small Power Domination Number

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