Resolving-power dominating sets

Stephen, Sudeep; Rajan, Bharati; Grigorious, Cyriac; William, Albert

doi:10.1016/j.amc.2015.01.037

Cited by 9 publications

(6 citation statements)

References 7 publications

(8 reference statements)

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“…For example, resolving sets serve as a tool for combinatorial optimization [39], game theory [20], and pharmaceutical chemistry [9]; and dominating sets are helpful to analyze computer networks [38], design codes [12], and model biological networks [23]. Although metric-locating-dominating sets are hard to handle, for entailing the complexity of the other two concepts, they have been studied in several papers, for instance [5,26,27], and further generalized in other works such as [35,44].…”

Section: Introductionmentioning

confidence: 99%

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

González

Hernando

Mora

2018

Applied Mathematics and Computation

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A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of S, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them concerning parameters that have not been related so far.

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Section: Introductionmentioning

confidence: 99%

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

González

Hernando

Mora

2018

Applied Mathematics and Computation

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“…A set S ⊆ V (G) is a resolving dominating set, if it is both resolving and dominating. The resolving domination number η(G) of G is the minimum cardinality of a resolving dominating set of G. Resolving dominating sets were introduced in [1], and also independently in [17] (in this last work they were called metric-locating-dominating sets), being further studied in [3,11,15,19,20,27].…”

Section: Dominating Partition Dimensionmentioning

confidence: 99%

Resolving dominating partitions in graphs

Hernando

Mora

Pelayo

2019

Discrete Applied Mathematics

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A partition Π = {S 1 , . . . , S k } of the vertex set of a connected graph G is called a resolving partition of G if for every pair of vertices u and v, d(u, S j ) = d(v, S j ), for some part S j . The partition dimension β p (G) is the minimum cardinality of a resolving partition of G. A resolving partition Π is called resolving dominating if for every vertex v of G, d(v, S j ) = 1, for some part S Many variations of location in graphs have since been defined (see survey [23]). For example, in 2000, Chartrand, Salehi and Zhang study the resolvability of graphs in terms of partitions [6], as a generalization of resolving sets when the vertices are classified in different types. A few years later, resolving dominating sets were introduced by Brigham, Chartrand, Dutton and Zhang [1] and independently by Henning and Oellermann [17] as metric-locating-dominating sets, combining the usefulness of resolving sets and dominating sets. Resolving dominating sets have been further studied in [3,11,19]. In this paper, following the ideas of these works, we introduce the resolving dominating partitions, as a way for distinguishing the vertices of a graph by using on the one hand partitions, and on the other hand, both domination and location.

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“…In the second one, frequently called the l‐round PDS, the objective is to find the minimal PDS when the number of times the propagation rule can be applied is limited to l (Aazami & Stilp, 2009; Liao, 2016). In Stephen, Rajan, Grigorious, and William (2015), a new problem called the resolving‐power DS is introduced; in it the PDS also needs to be a resolving set. The newly defined problem is used as a model for finding intruders or faults in electric networks.…”

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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Resolving-power dominating sets

Cited by 9 publications

References 7 publications

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

Resolving dominating partitions in graphs

The fixed set search applied to the power dominating set problem

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