On the Power Dominating Sets of Hypercubes

Dean, Nathaniel; Ilic, Alexandra; Ramírez, Ignacio Méndez; Shen, Jian; Tian, Kevin

doi:10.1109/cse.2011.89

Cited by 21 publications

(10 citation statements)

References 10 publications

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“…Actually, in the graph G K 2 , dominating one copy of G is enough to power dominate the whole graph G K 2 . Therefore, we get that γ P (G K 2 ) ≤ γ(G) for any graph G. For the hypercube, this was observed by Dean et al in [9]. They further conjectured that the domination number of Q n was equal to the power domination number of Q n+1 .…”

Section: Theorem 8 (Dorfling Henning [15]) the Power Domination Number Of Thementioning

confidence: 51%

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Power Domination in Graphs

Dorbec

2020

Topics in Domination in Graphs

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In this chapter, we are interested in power domination in graphs. Power domination is a variation of domination introduced to address a physical problem of monitoring a network with phasor measurement units. The originality of this variation is that some propagation happens, and the set of covered vertices results from an iterative process. We present a survey of known results on this specific parameter.

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Section: Theorem 8 (Dorfling Henning [15]) the Power Domination Number Of Thementioning

confidence: 51%

“…Indeed, taking the vertices of a power dominating set plus all but one neighbor of each of them, one gets a zero forcing set of size at most γ P (G)∆(G). This was explicitly stated by Dean et al in [9], where the first link between the two parameters was probably made.…”

Section: Relation With Zero Forcing Setsmentioning

confidence: 84%

Power Domination in Graphs

Dorbec

2020

Topics in Domination in Graphs

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“…It has been shown that the PDSP is NP‐hard even on specific types of graphs like cubic graphs (Binkele‐Raible & Fernau, 2012). Another direction of research has been in finding the optimal size or bounds for specific graphs like grid graphs (Dorfling & Henning, 2006; Pai, Chang, & Wang, 2007), hyper cubes (Dean, Ilic, Ramirez, Shen, & Tian, 2011), generalised Petersen graphs (Koh & Soh, 2016; Lai, Chien, Chou, & Kao, 2012), cylinders (Koh & Soh, 2016), circular‐arc graphs (Liao & Lee, 2013), and tori (Koh & Soh, 2016). The PDSP has been generalised in the form of the k ‐power dominating set problem ( k ‐PDSP) which analyses the potential extensions of the propagation rule in the original problem (Chang, Dorbec, Montassier, & Raspaud, 2012).…”

Section: Introductionmentioning

confidence: 99%

The fixed set search applied to the power dominating set problem

Jovanović

Voß

2020

Expert Systems

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In this article, we focus on solving the power dominating set problem and its connected version. These problems are frequently used for finding optimal placements of phasor measurement units in power systems. We present an improved integer linear program (ILP) for both problems. In addition, a greedy constructive algorithm and a local search are developed. A greedy randomised adaptive search procedure (GRASP) algorithm is created to find near optimal solutions for large scale problem instances. The performance of the GRASP is further enhanced by extending it to the novel fixed set search (FSS) metaheuristic. Our computational results show that the proposed ILP has a significantly lower computational cost than existing ILPs for both versions of the problem. The proposed FSS algorithm manages to find all the optimal solutions that have been acquired using the ILP. In the last group of tests, it is shown that the FSS can significantly outperform the GRASP in both solution quality and computational cost.

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“…In this paper we begin the study of reconfiguration for zero forcing sets. Zero forcing is a coloring process on a graph that has seen much recent attention (see [12] and the references therein) in part because of its connections to linear algebra [1], power domination [4,8], and control of quantum systems [5]. The color change rule is: A blue vertex u can change the color of a white vertex w to blue if w is the unique white neighbor of u.…”

Section: Introductionmentioning

confidence: 99%

Reconfiguration graphs of zero forcing sets

Geneson¹,

Haas²,

Hogben³

2020

Preprint

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This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph G, its zero forcing graph, Z (G), is the graph whose vertices are the minimum zero forcing sets of G with an edge between vertices B and B of Z (G) if and only if B can be obtained from B by changing a single vertex of G. It is shown that the zero forcing graph of a forest is connected, but that many zero forcing graphs are disconnected. We characterize the base graphs whose zero forcing graphs are either a path or the complete graph, and show that the star cannot be a zero forcing graph. We show that computing Z (G) takes 2 Θ(n) operations in the worst case for a graph G of order n.

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On the Power Dominating Sets of Hypercubes

Cited by 21 publications

References 10 publications

Power Domination in Graphs

Power Domination in Graphs

The fixed set search applied to the power dominating set problem

Reconfiguration graphs of zero forcing sets

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