Short Topological Decompositions of Non-Orientable Surfaces

Authors Niloufar Fuladi, Alfredo Hubard, Arnaud de Mesmay

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Author Details

Niloufar Fuladi
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France
Alfredo Hubard
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France
Arnaud de Mesmay
  • LIGM, CNRS, Univ. Gustave Eiffel, ESIEE Paris, F-77454 Marne-la-Vallée, France


We are grateful to Marcus Schaefer and Daniel Štefankovič for providing us the full version of [{Marcus} {Schaefer} and {Daniel} {Štefankovič}, 2022], to Francis Lazarus for insightful discussions, and to the anonymous reviewers for very helpful comments.

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Niloufar Fuladi, Alfredo Hubard, and Arnaud de Mesmay. Short Topological Decompositions of Non-Orientable Surfaces. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We investigate short topological decompositions of non-orientable surfaces and provide algorithms to compute them. Our main result is a polynomial-time algorithm that for any graph embedded in a non-orientable surface computes a canonical non-orientable system of loops so that any loop from the canonical system intersects any edge of the graph in at most 30 points. The existence of such short canonical systems of loops was well known in the orientable case and an open problem in the non-orientable case. Our proof techniques combine recent work of Schaefer-Štefankovič with ideas coming from computational biology, specifically from the signed reversal distance algorithm of Hannenhalli-Pevzner. This result confirms a special case of a conjecture of Negami on the joint crossing number of two embeddable graphs. We also provide a correction for an argument of Negami bounding the joint crossing number of two non-orientable graph embeddings.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Geometric topology
  • Mathematics of computing → Graphs and surfaces
  • Computational topology
  • embedded graph
  • non-orientable surface
  • joint crossing number
  • canonical system of loop
  • surface decomposition


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