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Embeddability of Arrangements of Pseudocircles and Graphs on Surfaces

  • Branko Grünbaum Memorial Issue
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Abstract

A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most four are embeddable into the sphere, and asked if an analogous result holds for embeddability into orientable surfaces of higher genus. We answer this question positively: An arrangement of pseudocircles is embeddable into an orientable surface of genus g if and only if all of its subarrangements of size at most \(4g+4\) are. Moreover, this bound is tight. We actually have similar results for a much general notion of arrangement, which we call an arrangement of graphs.

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Acknowledgements

We thank two anonymous referees for many helpful suggestions and corrections to an earlier version of this paper. The first author was supported by Grant ANR-17-CE40-0033 of the French National Research Agency ANR (SoS project). The second author is currently supported by a Fulbright Visiting Scholar Grant at UC Davis. The second and fourth authors were supported by Conacyt under Grant 222667, and by FRC-UASLP. The third author was supported by Conacyt under Grant 166306 and by PAPIIT-UNAM IA102118.

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Authors and Affiliations

  1. Université Paris-Est, LIGM, CNRS, ENPC, ESIEE Paris, UPEM, Marne-la-Vallée, France

    Éric Colin de Verdière

  2. Instituto de Física, UASLP, 78000, San Luis Potosí, Mexico

    Carolina Medina

  3. Centro de Ciencias Matemáticas, UNAM Campus Morelia, 58190, Morelia, Michoacán, Mexico

    Edgardo Roldán-Pensado

  4. Instituto de Física, UASLP, 78000, San Luis Potosí, Mexico

    Gelasio Salazar

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  1. Éric Colin de Verdière
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  2. Carolina Medina
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  3. Edgardo Roldán-Pensado
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  4. Gelasio Salazar
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Correspondence to Gelasio Salazar.

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Colin de Verdière, É., Medina, C., Roldán-Pensado, E. et al. Embeddability of Arrangements of Pseudocircles and Graphs on Surfaces. Discrete Comput Geom 64, 386–395 (2020). https://doi.org/10.1007/s00454-019-00126-6

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  • DOI: https://doi.org/10.1007/s00454-019-00126-6

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