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Acute Tours in the Plane

Author Ahmad Biniaz

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Author Details

Ahmad Biniaz
  • School of Computer Science, University of Windsor, Canada

Funding

Supported by NSERC.

Acknowledgements

I am very grateful to the anonymous reviewer who meticulously verified our proof, and provided valuable feedback that reduced the number of subcases to two (which was three in our original proof) and improved the bound on n to 20 (which was 36 originally).

Cite AsGet BibTex

Ahmad Biniaz. Acute Tours in the Plane. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 16:1-16:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.16

Abstract

We confirm the following conjecture of Fekete and Woeginger from 1997: for any sufficiently large even number n, every set of n points in the plane can be connected by a spanning tour (Hamiltonian cycle) consisting of straight-line edges such that the angle between any two consecutive edges is at most π/2. Our proof is constructive and suggests a simple O(nlog n)-time algorithm for finding such a tour. The previous best-known upper bound on the angle is 2π/3, and it is due to Dumitrescu, Pach and Tóth (2009).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • planar points
  • acute tour
  • Hamiltonian cycle
  • equitable partition

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