Conditional resolvability in graphs: a survey

Saenpholphat, Varaporn; Zhang, Ping

doi:10.1155/s0161171204311403

Cited by 57 publications

(46 citation statements)

References 21 publications

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“…A resolving set of is connected if the subgraph induced by W is a nontrivial connected subgraph of . The minimum cardinality of a connected resolving set is called connected resolving number and it is denoted by [18]. In this paper we introduce a new resolving parameter called star resolving number.…”

Section: An Overview Of the Papermentioning

confidence: 99%

On Certain Connected Resolving Parameters of Hypercube Networks

Rajan¹,

William²,

Rajasingh³

et al. 2012

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Given a graph , a set is a resolving set if for each pair of distinct vertices there is a vertex such that . A resolving set containing a minimum number of vertices is called a minimum resolving set or a basis for . The cardinality of a minimum resolving set is called the resolving number or dimension of and is denoted by . A resolving set is said to be a star resolving set if it induces a star, and a path resolving set if it induces a path. The minimum cardinality of these sets, denoted respectively by

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Section: An Overview Of the Papermentioning

confidence: 99%

On Certain Connected Resolving Parameters of Hypercube Networks

Rajan¹,

William²,

Rajasingh³

et al. 2012

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“…Resolving sets have since been widely investigated [2,3,4,5,6,7,9,17,19,20,21,22,23,24,25,26,27,29,31,32,33], and arise in diverse areas including coin weighing problems [10,14,16,18,30], network discovery and verification [1], robot navigation [17,27], connected joins in graphs [26], the Djoković-Winkler relation [3], and strategies for the Mastermind game [8,11,12,13,16].…”

Section: Vertices In Smentioning

confidence: 99%

Extremal Graph Theory for Metric Dimension and Diameter

Hernando

Mora

Pelayo

et al. 2007

Electronic Notes in Discrete Mathematics

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Abstract. A set of vertices S resolves a connected graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. Let G β,D be the set of graphs with metric dimension β and diameter D. It is well-known that the minimum order of a graph in G β,D is exactly β + D. The first contribution of this paper is to characterise the graphs in G β,D with order β + D for all values of β and D. Such a characterisation was previously only known for D ≤ 2 or β ≤ 1. The second contribution is to determine the maximum order of a graph in G β,D for all values of D and β. Only a weak upper bound was previously known. IntroductionLet G be a connected graph 1 . A vertex x ∈ V (G) resolves 2 a pair of vertices v, w ∈ V (G) if dist(v, x) = dist(w, x). A set of vertices S ⊆ V (G) resolves G, and S is a resolving set of G, if every pair of distinct vertices of G are resolved by some vertex in S. Informally, S resolves G if every vertex of G is uniquely determined by its vector of distances to the 2000 Mathematics Subject Classification. 05C12 (distance in graphs), 05C35 (extremal graph theory). Key words and phrases. graph, distance, resolving set, metric dimension, metric basis, diameter, order. The research of Carmen Hernando, Mercè Mora, Carlos Seara, and David Wood is supported by the projects MEC MTM2006-01267 and DURSI 2005SGR00692. The research of Ignacio Pelayo is supported by the projects MTM2005-08990-C02-01 and SGR2005-00412. The research of David Wood is supported by a Marie Curie Fellowship of the European Community under contract 023865.1 Graphs in this paper are finite, undirected, and simple. The vertex set and edge set of a graph G are denoted by V (G) and, is the length (that is, the number of edges) in a shortest path between v and w in G. The eccentricity of a vertex v in G is eccG(v) := max{distG(v, w) : w ∈ V (G)}. We drop the subscript G from these notations if the graph G is clear from the context. The diameter2 It will be convenient to also use the following definitions for a connected graph G. A vertex x ∈ V (G)resolves a set of vertices T ⊆ V (G) if x resolves every pair of distinct vertices in T . A set of vertices S ⊆ V (G) resolves a set of vertices T ⊆ V (G) if for every pair of distinct vertices v, w ∈ T , there exists a vertex x ∈ S that resolves v, w. [2,3,4,5,6,7,9,17,19,20,21,22,23,24,25,26,27,29,31,32,33], and arise in diverse areas including coin weighing problems [10,14,16,18,30], network discovery and verification [1], robot navigation [17,27], connected joins in graphs [26], the Djoković-Winkler relation [3], and strategies for the Mastermind game [8,11,12,13,16].For positive integers β and D, let G β,D be the class of connected graphs with metric dimension β and diameter D. Consider the following two extremal questions:• What is the minimum order of a graph in G β,D ?• What is the maximum order of a graph in G β,D ?The first question was independently answered by Yushmanov [33], Khuller ...

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“…Applications of the metric dimension to the navigation of robots in networks are discussed in [4] and applications to chemistry in [5,6]. This invariant was studied further in a number of other papers including, for instance [7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

The Simultaneous Local Metric Dimension of Graph Families

et al. 2017

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Abstract:In a graph G = (V, E), a vertex v ∈ V is said to distinguish two vertices x and y if v, y). A set S ⊆ V is said to be a local metric generator for G if any pair of adjacent vertices of G is distinguished by some element of S. A minimum local metric generator is called a local metric basis and its cardinality the local metric dimension of G. A set S ⊆ V is said to be a simultaneous local metric generator for a graph family G = {G 1 , G 2 , . . . , G k }, defined on a common vertex set, if it is a local metric generator for every graph of the family. A minimum simultaneous local metric generator is called a simultaneous local metric basis and its cardinality the simultaneous local metric dimension of G. We study the properties of simultaneous local metric generators and bases, obtain closed formulae or tight bounds for the simultaneous local metric dimension of several graph families and analyze the complexity of computing this parameter.

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Conditional resolvability in graphs: a survey

Cited by 57 publications

References 21 publications

On Certain Connected Resolving Parameters of Hypercube Networks

On Certain Connected Resolving Parameters of Hypercube Networks

Extremal Graph Theory for Metric Dimension and Diameter

The Simultaneous Local Metric Dimension of Graph Families

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