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Short Topological Decompositions of Non-orientable Surfaces

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Abstract

In this article, 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 on 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. The existence of short canonical non-orientable systems of loops 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. Finally, we provide a generalization of O(g)-universal shortest path metrics to non-orientable surfaces.

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Notes

  1. Throughout the article, we decompose surface-embedded graphs by cutting them along embedded graphs which are transverse to the original graph, and count the number of intersections. This is equivalent to the primal setting studied in, e.g., Lazarus et al. [20] via graph duality.

  2. This statement is slightly stronger than Conjecture 1.1 since it enforces a control on the number of crossings between each pair of edges instead of the total number of crossings, but it is equally open.

  3. A handle is obtained by removing a small disk and gluing a punctured torus along its boundary to the boundary circle of the resulting hole (see the left picture in Fig. 3).

  4. A cross-cap is obtained by removing a small disk from the sphere and gluing in a Möbius band along its boundary to the boundary circle of the resulting hole (see the right picture in Fig. 3).

  5. This terminology is slightly non-standard, we follow Schaefer and Štefankovič [27] who attribute it to B. Mohar.

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Acknowledgements

We are grateful to Marcus Schaefer and Daniel Štefankovič for providing us the full version of [27], to Francis Lazarus for insightful discussions, and to the anonymous reviewers for very helpful comments.

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  1. Univ Gustave Eiffel, CNRS, LIGM, 77454, Marne-la-Vallée, France

    Niloufar Fuladi, Alfredo Hubard & Arnaud de Mesmay

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  1. Niloufar Fuladi
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  2. Alfredo Hubard
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  3. Arnaud de Mesmay
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Correspondence to Alfredo Hubard.

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This article is dedicated to the memory of Eli Goodman, whose allowable sequences inspired some of this work.

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This work was partially supported by the ANR project SoS (ANR-17-CE40-0033). A. Hubard was also supported by the ANR projet CAAPS (ANR-17-CE40-0018).

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Fuladi, N., Hubard, A. & de Mesmay, A. Short Topological Decompositions of Non-orientable Surfaces. Discrete Comput Geom (2023). https://doi.org/10.1007/s00454-023-00580-3

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