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Recognizing Map Graphs of Bounded Treewidth

Authors Patrizio Angelini , Michael A. Bekos , Giordano Da Lozzo , Martin Gronemann , Fabrizio Montecchiani , Alessandra Tappini

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Author Details

Patrizio Angelini
  • Department of Mathematics, Natural, and Applied Sciences, John Cabot University, Rome, Italy
Michael A. Bekos
  • Department of Mathematics, University of Ioannina, Greece
Giordano Da Lozzo
  • Department of Engineering, Roma Tre University, Rome, Italy
Martin Gronemann
  • Algorithms and Complexity Group, Technische Universität Wien, Austria
Fabrizio Montecchiani
  • Department of Engineering, University of Perugia, Italy
Alessandra Tappini
  • Department of Engineering, University of Perugia, Italy

Funding

  • Bekos, Michael A.: Partially supported by DFG grant KA812/18-1.
  • Da Lozzo, Giordano: Partially supported by MSCA-RISE project "CONNECT", N^∘ 734922, and by MIUR, grant 20174LF3T8 "AHeAD: efficient Algorithms for HArnessing networked Data".
  • Montecchiani, Fabrizio: Partially supported by MIUR, grant 20174LF3T8 "AHeAD: efficient Algorithms for HArnessing networked Data", and by Dipartimento di Ingegneria, University of Perugia, grants RICBA20ED and RICBA21LG.
  • Tappini, Alessandra: Partially supported by MIUR, grant 20174LF3T8 "AHeAD: efficient Algorithms for HArnessing networked Data", and by Dipartimento di Ingegneria, University of Perugia, grants RICBA20ED and RICBA21LG.

Acknowledgements

We thank the anonymous reviewers of a previous version of this paper for pointing out that the map recognition problem admits an MSO₂ formulation.

Cite AsGet BibTex

Patrizio Angelini, Michael A. Bekos, Giordano Da Lozzo, Martin Gronemann, Fabrizio Montecchiani, and Alessandra Tappini. Recognizing Map Graphs of Bounded Treewidth. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SWAT.2022.8

Abstract

A map graph is one admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. The algorithm has time complexity that is linear in the size of the graph and, if the input is a yes-instance, it reports a certificate in the form of a so-called witness. Furthermore, this result is developed within a more general algorithmic framework that allows to test, for any k, if the input graph admits a k-map (where at most k nations meet at a common point) or a hole-free k-map (where each point is covered by at least one nation). We point out that, although bounding the treewidth of the input graph also bounds the size of its largest clique, the latter alone does not seem to be a strong enough structural limitation to obtain an efficient time complexity. In fact, while the largest clique in a k-map graph is ⌊ 3k/2 ⌋, the recognition of k-map graphs is still open for any fixed k ≥ 5.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Mathematics of computing → Graph algorithms
Keywords
  • Map graphs
  • Recognition
  • Parameterized complexity

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