The Degenerate Crossing Number and Higher-Genus Embeddings

  • Marcus SchaeferEmail author
  • Daniel Štefankovič
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


If a graph embeds in a surface with k crosscaps, does it always have an embedding in the same surface in which every edge passes through each crosscap at most once? This well-known open problem can be restated using crossing numbers: the degenerate crossing number, dcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps in which every edge passes through each crosscap at most once. The genus crossing number, gcr(G), of G equals the smallest number k so that G has an embedding in a surface with k crosscaps. The question then becomes whether dcr(G) = gcr(G), and it is in this form that it was first asked by Mohar.

We show that dcr(G) \(\le \) 6 gcr(G), and dcr(G) = gcr(G) as long as dcr(G) \(\le \) 3. We can separate dcr and gcr for (single-vertex) graphs with embedding schemes, but it is not clear whether the separating example can be extended into separations on simple graphs. Finally, we show that if a graph can be embedded in a surface with crosscaps, then it has an embedding in that surface in which every edge passes through each crosscap at most twice. This implies that dcr is \(\mathrm {\mathbf {NP}}\)-complete.


Degenerate crossing number Non-orientable genus Genus crossing number 



We would like to thank Bojan Mohar for suggesting the question, and giving us detailed feedback on earlier drafts of this paper. We are also grateful for helpful comments by the anonymous reviewers.


  1. 1.
    Ackerman, E., Pinchasi, R.: On the degenerate crossing number. Discrete Comput. Geom. 49(3), 695–702 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dey, T.K., Schipper, H.: A new technique to compute polygonal schema for \(2\)-manifolds with application to null-homotopy detection. Discrete Comput. Geom. 14(1), 93–110 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Planar drawings of higher-genus graphs. J. Graph Algorithms Appl. 15(1), 7–32 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erickson, J., Har-Peled, S.: Optimally cutting a surface into a disk. Discrete Comput. Geom. 31(1), 37–59 (2004). ACM Symposium on Computational Geometry, Barcelona 2002MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Harborth, H.: Drawings of graphs and multiple crossings. In: Alavi, Y., Chartrand, G., Lick, D.R., Wall, C.E., Lesniak, L. (eds.) Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984), pp. 413–421. Wiley-Interscience Publication, New York (1985)Google Scholar
  7. 7.
    Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an orientable triangulated surface. In: Proceedings of the Seventeenth Annual Symposium on Computational Geometry (SCG-01), pp. 80–89. ACM Press, New York, 3–5 2001Google Scholar
  8. 8.
    Mohar, B.: The genus crossing number. Ars Math. Contemp. 2(2), 157–162 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD (2001)zbMATHGoogle Scholar
  10. 10.
    Pach, J., Tóth, G.: Degenerate crossing numbers. Discrete Comput. Geom. 41(3), 376–384 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings on surfaces. European J. Combin. 30(7), 1704–1717 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schaefer, M.: The graph crossing number and its variants: a survey. Electron. J. Comb. 20, 1–90 (2013). Dynamic Survey, #DS21MathSciNetGoogle Scholar
  13. 13.
    Thomassen, C.: The genus problem for cubic graphs. J. Comb. Theory Ser. B 69(1), 52–58 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.DePaul UniversityChicagoUSA
  2. 2.University of RochesterRochesterUSA

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