The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-metric dimension of the lexicographic product of graphs

Estrada‐Moreno, Alejandro; Yero, Ismael G.; Rodríguez-Velázquez, Juan A.

doi:10.1016/j.disc.2015.12.024

Cited by 35 publications

(29 citation statements)

References 8 publications

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“…Example 5.4. The Petersen graph, which is illustrated in Figure 1, has dimension sequence (3,4,7,8,9,10, +∞, . .…”

Section: Some Examplesmentioning

confidence: 99%

“…The vertex set V of a graph G supports a natural graph metric d, where d(u, v) is the smallest number of edges that can be used to join u to v. Some basic results on the k-metric dimension of a graph have recently been obtained in [7][8][9][10][11]. Moreover, it was shown in [20] that the problem of computing the k-metric dimension of a graph is NP-complete.…”

Section: The Metric Dimensions Of Graphsmentioning

confidence: 99%

“…The theory was developed further in 2013 for general metric spaces [1]. More recently, the theory of metric dimension has been generalised, again in the context of graph theory, to the notion of a k-metric dimension, where k is any positive integer, and where the case k = 1 corresponds to the original theory [7][8][9][10][11]. Here we develop the idea of the k-metric dimension both in graph theory and in metric spaces.…”

Section: Introductionmentioning

confidence: 99%

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On the k-metric dimension of metric spaces

Beardon

Rodríguez-Velázquez²

2018

Ars Math. Contemp.

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“…Example 5.4. The Petersen graph, which is illustrated in Figure 1, has dimension sequence (3,4,7,8,9,10, +∞, . .…”

Section: Some Examplesmentioning

confidence: 99%

Section: The Metric Dimensions Of Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the k-metric dimension of metric spaces

Beardon

Rodríguez-Velázquez²

2018

Ars Math. Contemp.

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“…Resolving sets have applications in robot navigation [14] and network discovery and verification [1], for example. For some recent developments, see [6,12,13].…”

Section: Introductionmentioning

confidence: 99%

On the metric dimensions for sets of vertices

Hakanen¹,

Junnila²,

Laihonen³

et al. 2020

Discuss. Math. Graph Theory

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Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, { }-resolving sets were recently introduced. In this paper, we present new results regarding the { }-resolving sets of a graph. In addition to proving general results, we consider {2}-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept ofsolid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for-solid-resolving sets and show how-solid-and { }-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the-solid-and { }-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

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“…The concepts of resolving set of a graph was first introduced by Slater [1] in 1975 and independently by Harary and Melter [2] in 1976. The metric dimension of a graph has been widely studied and a large number of related concepts have been extended (see [3][4][5][6][7][8][9][10][11]). As a parameter of a graph, it has been applied to lots of practical problems, such as robot navigation [12], connected joins in graphs and combinatorial optimization [13], and pharmaceutical chemistry [14].…”

Section: Introductionmentioning

confidence: 99%

Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

2019

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Let G = ( V ( G ) , E ( G ) ) be a connected graph. An ordered set W ⊂ V ( G ) is a resolving set for G if every vertex of G is uniquely determined by its vector of distances to the vertices in W. The metric dimension of G is the minimum cardinality of a resolving set. In this paper, we characterize the graphs of metric dimension n − 3 by constructing a special distance matrix, called metric matrix. The metric matrix makes it so a class of graph and its twin graph are bijective and the class of graph is obtained from its twin graph, so it provides a basis for the extension of graphs with respect to metric dimension. Further, the metric matrix gives a new idea of the characterization of extremal graphs based on metric dimension.

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The k-metric dimension of the lexicographic product of graphs

Cited by 35 publications

References 8 publications

On the k-metric dimension of metric spaces

On the k-metric dimension of metric spaces

On the metric dimensions for sets of vertices

Characterization of n-Vertex Graphs of Metric Dimension n − 3 by Metric Matrix

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