Computing shortest closed curves on non-orientable surfaces - Université Gustave Eiffel
Communication Dans Un Congrès Année : 2024

Computing shortest closed curves on non-orientable surfaces

Résumé

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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Dates et versions

hal-04732932 , version 1 (11-10-2024)

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Denys Bulavka, Éric Colin de Verdière, Niloufar Fuladi. Computing shortest closed curves on non-orientable surfaces. International Symposium on Computational Geometry, Jun 2024, Athens, Greece. ⟨10.4230/LIPIcs.SoCG.2024.28⟩. ⟨hal-04732932⟩
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