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Noncrossing Longest Paths and Cycles

Authors Greg Aloupis, Ahmad Biniaz, Prosenjit Bose , Jean-Lou De Carufel, David Eppstein, Anil Maheshwari , Saeed Odak, Michiel Smid, Csaba D. Tóth , Pavel Valtr

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Longest Paths
  • Longest Cycles
  • Noncrossing Paths
  • Noncrossing Cycles

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Abstract

Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest.
Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. 
In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.

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Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, David Eppstein, Anil Maheshwari, Saeed Odak, Michiel Smid, Csaba D. Tóth, and Pavel Valtr. Noncrossing Longest Paths and Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.36

Author Details

Greg Aloupis
  • Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA
Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada
Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
David Eppstein
  • Computer Science Department, University of California, Irvine, CA, USA
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Saeed Odak
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada
Csaba D. Tóth
  • Department of Mathematics, California State , University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, USA
Pavel Valtr
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

Funding

  • Biniaz, Ahmad: Research supported by NSERC.
  • Bose, Prosenjit: Research supported by NSERC.
  • De Carufel, Jean-Lou: Research supported by NSERC.
  • Eppstein, David: Research supported by NSF grant CCF-2212129.
  • Maheshwari, Anil: Research supported by NSERC.
  • Odak, Saeed: Research supported by NSERC.
  • Smid, Michiel: Research supported by NSERC.
  • Tóth, Csaba D.: Research supported in part by the NSF award DMS-2154347.
  • Valtr, Pavel: Research supported by Czech Science Foundation grant GAČR 23-04949X.

Acknowledgements

This work was initiated at the 10th Annual Workshop on Geometry and Graphs, held at Bellairs Research Institute in Barbados in February 2023. We thank the organizers and the participants.

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