On the metric dimensions for sets of vertices

Hakanen, Anni; Junnila, Ville; Laihonen, Tero; Puertas, María Luz

doi:10.7151/dmgt.2367

Cited by 4 publications

(2 citation statements)

References 13 publications

(30 reference statements)

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“…However, they solved the problem in linear time for the particular case of trees. Recently, Hakanen et al [6] obtained some results on this topic.…”

Section: Introductionmentioning

confidence: 99%

Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C(n : 1, 2)

Saha¹,

Das²,

Tiwary

et al. 2023

Mathematics

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Let G=(V(G),E(G)) be a simple connected unweighted graph. A set R⊂V(G) is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set Sx,y={w∈R:d(w,x)≠d(w,y)} is at least k for every pair of distinct vertices x,y of G. A k-level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k. In this article, we calculate and determine the k-level metric dimension for the circulant graph C(n:1,2) for all possible values of k and n. The optimal fault-tolerant resolving sets with k tolerance are also delineated.

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“…However, they solved the problem in linear time for the particular case of trees. Recently, Hakanen et al [6] obtained some results on this topic.…”

Section: Introductionmentioning

confidence: 99%

Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C(n : 1, 2)

Saha¹,

Das²,

Tiwary

et al. 2023

Mathematics

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“…In [8] and [9], forced vertices of ℓ-solid-resolving sets for all ℓ, and {ℓ}-resolving sets for ℓ ≥ 2 were characterised. These characterisations used the local properties of the vertices and their neighbourhoods.…”

Section: Introductionmentioning

confidence: 99%

On Vertices Contained in All or in No Metric Basis

Hakanen¹,

Junnila²,

Laihonen³

et al. 2021

Preprint

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A set R ⊆ V (G) is a resolving set of a graph G if for all distinct vertices v, u ∈ V (G) there exists an element r ∈ R such that d(r, v) = d(r, u). The metric dimension dim(G) of the graph G is the minimum cardinality of a resolving set of G. A resolving set with cardinality dim(G) is called a metric basis of G. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.

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