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DOI: 10.4230/LIPIcs.SoCG.2022.41

Short Topological Decompositions of Non-Orientable Surfaces

Authors: Niloufar Fuladi, Alfredo Hubard, and Arnaud de Mesmay

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract

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.

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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)

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@InProceedings{fuladi_et_al:LIPIcs.SoCG.2022.41,
  author =	{Fuladi, Niloufar and Hubard, Alfredo and de Mesmay, Arnaud},
  title =	{{Short Topological Decompositions of Non-Orientable Surfaces}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{41:1--41:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.41},
  URN =		{urn:nbn:de:0030-drops-160492},
  doi =		{10.4230/LIPIcs.SoCG.2022.41},
  annote =	{Keywords: Computational topology, embedded graph, non-orientable surface, joint crossing number, canonical system of loop, surface decomposition}
}

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Short Topological Decompositions of Non-Orientable Surfaces

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