Resolvability in graphs and the metric dimension of a graph

Chartrand, Gary; Eroh, Linda; Johnson, Mark A.; Oellermann, Ortrud R.

doi:10.1016/s0166-218x(00)00198-0

Cited by 814 publications

(938 citation statements)

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“…Observe that if W is a resolving set of a connected graph G and W ⊆ W , then W is also a resolving set of G. It was shown in [2] that the path of order n ≥ 2 is the only connected graph of order n with dimension 1. Thus for a connected graph G that is not a path, if W is a Steiner basis of G and T is a Steiner W -tree, then V (T ) is a connected resolving set for G. These observations yield an upper bound for the connected resolving number of a nontrivial connected graph that is not a path in terms of the Steiner distances of its bases.…”

Section: An Upper Bound For the Connected Resolving Number Of A Graphmentioning

confidence: 99%

“…In this section we show that for every integer k ≥ 2, there exists a graph with a unique cr-set of cardinality k. For positive integers d and n with d < n, define f (n, d) as the least positive integer k such that k + d k ≥ n. It was shown [2] that if G is a connected graph of order n ≥ 2 and diameter d, then dim(G) ≥ f (n, d). Since cr(G) ≥ dim(G) for every graph G, we have the following.…”

Section: Graphs With a Unique Cr-set Or Various Cr-setsmentioning

confidence: 99%

“…Thus, a graph-theoretic interpretation of this problem is to provide representations for the vertices of a graph in such a way that distinct vertices have distinct representations. This is the subject of the papers [1,2,4,8]. It was noted in [5, p.204] that determining the dimension of a graph is an NP-complete problem.…”

Section: Introductionmentioning

confidence: 99%

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Connected Resolvability of Graphs

Saenpholphat

Zhang

2003

Czech Math J

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For an ordered set W = {w 1 , w 2 , · · · , w k } of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the, where d(x, y) represents the distance between the vertices x and y. The set W is a connected resolving set for G if distinct vertices of G have distinct representations with respect to W and the subgraph W induced by W is a nontrivial connected subgraph of G. The minimum cardinality of a connected resolving set in a graph G is its connected resolving number cr(G). A connected resolving set in G of cardinality cr(G) is called a cr-set of G. An upper bound for the connected resolving number of a connected graph that is not a path is presented. We study how the connected resolving number of a connected graph is affected by adding a vertex to the graph. It is shown that for every integer k ≥ 2, there exists a connected graph with a unique cr-set. Moreover, for every pair k, r of integers with k ≥ 2 and 0 ≤ r ≤ k, there exists a connected graph G with connected resolving number k such that there are exactly r vertices in G that belong to every cr-set of G.

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Section: An Upper Bound For the Connected Resolving Number Of A Graphmentioning

confidence: 99%

Section: Graphs With a Unique Cr-set Or Various Cr-setsmentioning

confidence: 99%

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Connected Resolvability of Graphs

Saenpholphat

Zhang

2003

Czech Math J

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“…The first question was independently answered by Yushmanov [33], Khuller et al [17], and Chartrand et al [5], who proved that the minimum order of a graph in G β,D is β + D (see Lemma 2.2). Thus it is natural to consider the following problem:…”

Section: Introductionmentioning

confidence: 99%

“…Previously, only a weak upper bound was known. In particular, Khuller et al [17] and Chartrand et al [5] independently proved that every graph in G β,D has at most D β + β vertices. This bound is tight only for D ≤ 3 or β = 1.…”

Section: Introductionmentioning

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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