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On the Approximation Hardness of Geodetic Set and Its Variants

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Computing and Combinatorics (COCOON 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 13025))

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Abstract

Given a graph, a geodetic set (resp. edge geodetic set) is a subset of its vertices such that every vertex (resp. edge) of the graph is on a shortest path between two vertices of the subset. A strong geodetic set is a subset S of vertices and a choice of a shortest path for every pair of vertices of S such that every vertex is on one of these shortest paths. The geodetic number (resp. edge geodetic number) of a graph is the minimum size of a geodetic set (resp. edge geodetic set) and the strong geodetic number is the minimum size of a strong geodetic set. We first prove that, given a subset of vertices, it is \(\mathcal {NP}\)-hard to determine whether it is a strong geodesic set. Therefore, it seems natural to study the problem of maximizing the number of covered vertices by a choice of a shortest path for every pair of a provided subset of vertices. We provide a tight 2-approximation algorithm to solve this problem. Then, we show that there is no polynomial-time approximation algorithm for edge geodetic number and strong geodetic number on subcubic bipartite graphs with arbitrarily high girth. We also prove that geodetic number and edge geodetic number are both LOG-\(\mathcal {APX}\)-hard, even on subcubic bipartite graphs with arbitrarily high girth. Finally, we disprove a conjecture of Iršiš and Konvalinka by proving that the strong geodetic number can be computed in polynomial time in complete multipartite graphs.

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Authors and Affiliations

  1. LIRMM, Univ Montpellier, CNRS, Montpellier, France

    Tom Davot

  2. Aix-Marseille Université, Marseille, France

    Lucas Isenmann

  3. INSA Centre Val de Loire, Univ. Orléans, LIFO EA 4022, Orléans, France

    Jocelyn Thiebaut

Authors
  1. Tom Davot
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  2. Lucas Isenmann
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  3. Jocelyn Thiebaut
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Correspondence to Tom Davot .

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Editors and Affiliations

  1. National Cheng Kung University, Tainan, Taiwan

    Chi-Yeh Chen

  2. National Tsing Hua University, Hsinchu, Taiwan

    Wing-Kai Hon

  3. National Taipei University of Business, Taoyuan, Taiwan

    Ling-Ju Hung

  4. National Taitung University, Taitung, Taiwan

    Chia-Wei Lee

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Davot, T., Isenmann, L., Thiebaut, J. (2021). On the Approximation Hardness of Geodetic Set and Its Variants. In: Chen, CY., Hon, WK., Hung, LJ., Lee, CW. (eds) Computing and Combinatorics. COCOON 2021. Lecture Notes in Computer Science(), vol 13025. Springer, Cham. https://doi.org/10.1007/978-3-030-89543-3_7

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  • DOI: https://doi.org/10.1007/978-3-030-89543-3_7

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  • Print ISBN: 978-3-030-89542-6

  • Online ISBN: 978-3-030-89543-3

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