Approximation complexity of Metric Dimension problem

Hauptmann, Mathias; Schmied, Richard; Viehmann, Claus

doi:10.1016/j.jda.2011.12.010

Cited by 87 publications

(77 citation statements)

References 15 publications

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“…(An impressive list of its applications can be found in [10]) It is thus clear that this dimension presents an intrinsic graph invariant. For the first time it was independently introduced in 1974 and 1975 by Harary and Melter [9] and Slater [24], respectively.…”

Section: Introductionmentioning

confidence: 99%

“…As a by-product the metric dimension of the Sierpiński graphs is determined. We point out here that in general it is very difficult to determine the exact metric dimension, see [10] and references therein for complexity issues on metric dimension. In the final section we use the derived formulas to compute the total distance of almost-extreme vertices.…”

Section: Introductionmentioning

confidence: 99%

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On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

Klavÿzar

Zemljiÿc

2013

APPL ANAL DISCRETE M

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Sierpi?ski graphs Sn p form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almost-extreme vertex of Sn p is introduced as a vertex that is either adjacent to an extreme vertex of Sn p or is incident to an edge between two subgraphs of Sn p isomorphic to Snp-1. Explicit formulas are given for the distance in Sn p between an arbitrary vertex and an almostextreme vertex. The formulas are applied to compute the total distance of almost-extreme vertices and to obtain the metric dimension of Sierpi?ski graphs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

Klavÿzar

Zemljiÿc

2013

APPL ANAL DISCRETE M

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“…Garey and Johnson proved thirty years ago that the decision version of Metric Dimension is NP-complete on general graphs [18] (another proof appears in [19]). It was shown that there exists a 2 log n-approximation algorithm on arbitrary graphs [19], which is best possible within a constant factor under reasonable complexity assumptions [2,16]. Hauptmann et al [16] show hardness of approximation on sparse graphs and on complements of sparse graphs.…”

Section: Introductionmentioning

confidence: 99%

“…It was shown that there exists a 2 log n-approximation algorithm on arbitrary graphs [19], which is best possible within a constant factor under reasonable complexity assumptions [2,16]. Hauptmann et al [16] show hardness of approximation on sparse graphs and on complements of sparse graphs. On the positive side, fifteen years ago, Khuller et al [19] gave a linear-time algorithm to compute the metric dimension of a tree, as well as characterizations for graphs with metric dimension 1 and 2.…”

Section: Introductionmentioning

confidence: 99%

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On the Complexity of Metric Dimension

Dı́az¹,

Pottonen²,

Serna³

et al. 2012

Algorithms – ESA 2012

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Abstract. The metric dimension of a graph G is the size of a smallest subset L ⊆ V (G) such that for any x, y ∈ V (G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NP-hard for general graphs, to be polynomialtime solvable on trees, and to have a log n-approximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on bounded-degree planar graphs is NP-complete. Then, we give a polynomial-time algorithm for determining the metric dimension of outerplanar graphs.

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The fault‐tolerant beacon set of hexagonal Möbius ladder network

Nadeem

Azeem

2023

Math Methods in App Sciences

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In localization, some specific nodes (beacon set) are selected to locate all nodes of a network, and if an arbitrary node stops working and still selected nodes remain in the beacon set, then the chosen nodes are called fault-tolerant beacon set.Due to the variety of metric dimension applications in different areas of sciences, many generalizations were proposed, fault-tolerant metric dimension is one of them. A resolving (beacon) set B 𝑓 is fault tolerant, if B 𝑓 ∖𝜈 for each 𝜈 ∈ B 𝑓 is also a resolving set; it is also known as a fault-tolerant beacon set; the minimum cardinality of such a beacon set is known as the fault-tolerant metric dimension of a graph G. In this paper, we find the fault-tolerant beacon set of hexagonal Möbius ladder network H(𝛼, 𝛽) and proved that all the different variations of 𝛼 and 𝛽 in H(𝛼, 𝛽) has constant fault-tolerant metric dimension.

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Approximation complexity of Metric Dimension problem

Cited by 87 publications

References 15 publications

On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

On the Complexity of Metric Dimension

The fault‐tolerant beacon set of hexagonal Möbius ladder network

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