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Computing Shortest Closed Curves on Non-Orientable Surfaces

Authors Denys Bulavka, Éric Colin de Verdière, Niloufar Fuladi

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graphs and surfaces
Keywords
  • Surface
  • Graph
  • Algorithm
  • Non-orientable surface

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Abstract

We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a self-homeomorphism of the surface, and computing shortest such curves has been vastly studied, for non-orientable ones the classification of non-separating simple closed curves up to ambient homeomorphism is subtler, depending on whether the curve is one-sided or two-sided, and whether it is orienting or not (whether it cuts the surface into an orientable one).
We prove that computing a shortest orienting (weakly) simple closed curve on a non-orientable combinatorial surface is NP-hard but fixed-parameter tractable in the genus of the surface. In contrast, we can compute a shortest non-separating non-orienting (weakly) simple closed curve with given sidedness in g^{O(1)} ⋅ n log n time, where g is the genus and n the size of the surface.
For these algorithms, we develop tools that can be of independent interest, to compute a variation on canonical systems of loops for non-orientable surfaces based on the computation of an orienting curve, and some covering spaces that are essentially quotients of homology covers.

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Denys Bulavka, Éric Colin de Verdière, and Niloufar Fuladi. Computing Shortest Closed Curves on Non-Orientable Surfaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 28:1-28:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.28

Author Details

Denys Bulavka
  • Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel
Éric Colin de Verdière
  • LIGM, CNRS, Univ. Gustave Eiffel, F-77454 Marne-la-Vallée, France
Niloufar Fuladi
  • LORIA, CNRS, INRIA, Université de Lorraine, F-54000 Nancy, France

Funding

  • Bulavka, Denys: Work done while this author was at Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic. This work was initiated while this author was visiting LIGM, funded by the French National Research Agency grant ANR-17-CE40-0033 (SoS), by the grant no. 21-32817S of the Czech Science Foundation (GAČR) and by Charles University project PRIMUS/21/SCI/014.
  • Colin de Verdière, Éric: The work of this author is partly funded by the French National Research Agency grants ANR-17-CE40-0033 (SoS) and ANR-19-CE40-0014 (MIN-MAX).
  • Fuladi, Niloufar: Work done while this author was at LIGM, CNRS, Univ. Gustave Eiffel, F-77454 Marne-la-Vallée, France. The work of this author is partly funded by the French National Research Agency grants ANR-17-CE40-0033 (SoS) and ANR-19-CE40-0014 (MIN-MAX).

Acknowledgements

We would like to thank Arnaud de Mesmay for stimulating discussions, and the anonymous reviewers for their useful comments.

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