On Finding the Best and Worst Orientations for the Metric Dimension - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Algorithmica Année : 2023

On Finding the Best and Worst Orientations for the Metric Dimension

Résumé

The (directed) metric dimension of a digraph D, denoted by − − → MD(D), is the size of a smallest subset S of vertices such that every two vertices of D are distinguished via their distances from the vertices in S. In this paper, we investigate the graph parameters BOMD(G) and WOMD(G) which are respectively the smallest and largest metric dimension over all orientations of G. We show that those parameters are related to several classical notions of graph theory and investigate the complexity of determining those parameters. We show that BOMD(G) = 1 if and only if G is hypotraceable (that is has a path spanning all vertices but one), and deduce that deciding whether BOMD(G) ≤ k is NP-complete for every positive integer k. We also show that WOMD(G) ≥ α(G) − 1, where α(G) is the stability number of G. We then deduce that for every fixed positive integer k, we can decide in polynomial time whether WOMD(G) ≤ k. The most significant results deal with oriented forests. We provide a linear-time algorithm to compute the metric dimension of an oriented forest and a linear-time algorithm that, given a forest F , computes an orientation D − with smallest metric dimension (i.e. such that − − → MD(D −) = BOMD(F)) and an orientation D + with largest metric dimension (i.e. such that − − → MD(D +) = WOMD(F)).
Fichier principal
Vignette du fichier
metric_dimension.pdf (590.97 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04271379 , version 1 (06-11-2023)

Licence

Paternité

Identifiants

Citer

Julio Araujo, Julien Bensmail, Victor Campos, Frédéric Havet, A. Karolinna Maia, et al.. On Finding the Best and Worst Orientations for the Metric Dimension. Algorithmica, 2023, 85 (10), pp.2962-3002. ⟨10.1007/s00453-023-01132-0⟩. ⟨hal-04271379⟩
16 Consultations
9 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More