Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results

Kuziak, Dorota; Yero, Ismael G.

doi:10.48550/arxiv.2107.04877

Cited by 10 publications

(13 citation statements)

References 148 publications

(380 reference statements)

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“…Even though it is newer than the vertex metric dimension, the edge metric dimension also attracted interest [12][13][14][15][16][17][18][19][20]. A nice survey of the topic of the metric dimension is given in [21].…”

Section: Introductionmentioning

confidence: 99%

Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

2022

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The vertex (respectively edge) metric dimension of a graph G is the size of a smallest vertex set in G, which distinguishes all pairs of vertices (respectively edges) in G, and it is denoted by dim(G) (respectively edim(G)). The upper bounds dim(G)≤2c(G)−1 and edim(G)≤2c(G)−1, where c(G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover, all leafless cacti that attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves, and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper, we focus on Θ-graphs, as the most simple 2-connected graphs distinct from the cycle, and show that the the upper bound 2c(G)−1 holds for both metric dimensions of Θ-graphs; we characterize all Θ-graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G)−1 in the class of leafless graphs besides already known extremal cacti and extremal Θ-graphs mentioned here.

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Section: Introductionmentioning

confidence: 99%

Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

2022

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“…This motivated the introduction of a new variant of metric dimension, namely the edge metric dimension. Even though it is newer than the vertex metric dimension, the edge metric dimension also atracted interest [6,8,12,14,17,25,26,27].A nice survey of the topic of metric dimension is given in [15].…”

Section: Introductionmentioning

confidence: 99%

Remarks on the vertex and the edge metric dimension of 2-connected graphs

Knor¹,

Sedlar²,

Škrekovski³

2022

Preprint

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The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) ≤ 2c(G) − 1 and edim(G) ≤ 2c(G) − 1, where c(G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover all leafless cacti which attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper we focus on Θ-graphs, as the most simple 2-connected graphs distinct from cycle, and show that the the upper bound 2c(G) − 1 holds for both metric dimensions of Θ-graphs and we characterize all Θ-graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G) − 1 in the class of leafless graphs besides already known extremal cacti and extremal Θ-graphs mentioned here.

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“…Fault-tolerant resolving sets were introduced in [14] and have been studied in [2,6,15,19,21,23,24,27]. For more applications and properties on metric dimension and its variants, the reader is addressed to the surveys [20,26] and the references herein.…”

Section: Introductionmentioning

confidence: 99%

“…The k-metric dimension and k-resolving sets of a graph and, concretely, of some product graphs have been studied in [2,[9][10][11][12]18,22,28]. In the survey [20], an extensive summary of known results and applications of the k-metric dimension is given.…”

Section: Introductionmentioning

confidence: 99%

Resolving sets tolerant to failures in three-dimensional grids

2022

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An ordered set S of vertices of a graph G is a resolving set for 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. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any k vertices from the set. This is equivalent to finding $$(k+1)$$ ( k + 1 ) -resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differs in at least $$k+1$$ k + 1 coordinates. This problem is also related with the study of the $$(k+1)$$ ( k + 1 ) -metric dimension of a graph, defined as the minimum cardinality of a $$(k+1)$$ ( k + 1 ) -resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of $$k\ge 1$$ k ≥ 1 for which there exists a $$(k+1)$$ ( k + 1 ) -resolving set and construct such a resolving set of minimum cardinality in almost all cases.

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Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results

Cited by 10 publications

References 148 publications

Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

Remarks on the vertex and the edge metric dimension of 2-connected graphs

Resolving sets tolerant to failures in three-dimensional grids

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