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On finding the best and worst orientations for the metric dimension

Julio Araujo 1, 2 Julien Bensmail 3 Victor Campos 1 Frédéric Havet 3 Ana Karolinna Maia de Oliveira 1, 2 Nicolas Nisse 3 Ana Silva 1
2 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
3 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : 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)).
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Submitted on : Tuesday, August 25, 2020 - 11:24:46 AM
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Julio Araujo, Julien Bensmail, Victor Campos, Frédéric Havet, Ana Karolinna Maia de Oliveira, et al.. On finding the best and worst orientations for the metric dimension. [Research Report] Inria. 2020. ⟨hal-02921466⟩

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