Chakraborty, D. orcid.org/0000-0003-0534-6417, Das, S., Foucaud, F. et al. (3 more authors) (2020) Algorithms and complexity for geodetic sets on planar and chordal graphs. In: Leibniz International Proceedings in Informatics, LIPIcs. ISAAC 2020, 14-18 Dec 2020, Hong Kong, China (Virtual Conference). Leibniz International Proceedings in Informatics (LIPIcs), 181 . Schloss-Dagstuhl - Leibniz Zentrum für Informatik , Wadern, Merzig-Wadern, Saarland , 7:1-7:15. ISBN 978-3-95977-173-3
Abstract
We study the complexity of finding the geodetic number on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that MGS remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that MGS is fixed parameter tractable for inputs of this class when parameterized by their treewidth (which equals the clique number minus one). This implies a linear-time algorithm for k-trees, for fixed k. Then, we show that MGS is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Dibyayan Chakraborty, Sandip Das, Florent Foucaud, Harmender Gahlawat, Dimitri Lajou, and Bodhayan Roy; licensed under Creative Commons License (Creative Commons Attribution 3.0 Unported license). |
Keywords: | Geodetic set, Planar graph, Chordal graph, Interval graph, FPT algorithm |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 05 Feb 2024 13:12 |
Last Modified: | 05 Feb 2024 13:20 |
Published Version: | https://drops.dagstuhl.de/entities/document/10.423... |
Status: | Published |
Publisher: | Schloss-Dagstuhl - Leibniz Zentrum für Informatik |
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) |
Identification Number: | 10.4230/LIPIcs.ISAAC.2020.7 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:208684 |
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