Skip to main content
SpringerLink
Log in

Hanani-Tutte and Related Results

  • Chapter
Geometry — Intuitive, Discrete, and Convex

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 24))

Abstract

We investigate under what conditions crossings of adjacent edges and pairs of edges crossing an even number of times are unnecessary when drawing graphs. This leads us to explore the Hanani-Tutte theorem and its close relatives, emphasizing the intuitive geometric content of these results.

We are taking the view that crossings of adjacent edges are trivial, and easily got rid of. Bill Tutte

We interpret this sentence as a philosophical view and not a mathematical claim. László Székely

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Pankaj K. Agarwal, Eran Nevo, János Pach, Rom Pinchasi, Micha Sharir, and Shakhar Smorodinsky, Lenses in arrangements of pseudo-circles and their applications, J. ACM, 51(2):139–186 (electronic), 2004.

    Article  MathSciNet  Google Scholar 

  2. Dan Archdeacon and R. Bruce Richter, On the parity of crossing numbers, J. Graph Theory, 12(3):307–310, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  3. Sanjeev Arora, László Babai, Jacques Stern, and Z. Sweedyk, The hardness of approximate optima in lattices, codes, and systems of linear equations, J. Comput. System Sci.U, 54(2, part 2):317–331, 1997. 34th Annual Symposium on Foundations of Computer Science (Palo Alto, CA, 1993).

    Article  MATH  MathSciNet  Google Scholar 

  4. Peter Brass, William Moser, and János Pach, Research Problems in Discrete Geometry. Springer, New York, 2005.

    MATH  Google Scholar 

  5. Henning Bruhn and Reinhard Diestel, Maclane’s theorem for arbitrary surfaces, Journal of Combinatorial Theory, Series B, 99(2):275–286, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  6. Sarit Buzaglo, Rom Pinchasi, and Günter Rote, Topological hyper-graphs. Unpublished manuscript, 2007.

    Google Scholar 

  7. Grant Cairns and Yury Nikolayevsky, Bounds for generalized thrackles, Discrete Comput. Geom., 23(2):191–206, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  8. Grant Cairns and Yury Nikolayevsky, Generalized thrackle drawings of non-bipartite graphs, Discrete Comput. Geom., 41(1):119–134, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  9. Jakub Černý, Combinatorial and Computational Geometry. PhD thesis, Charles University, Prague, 2008.

    Google Scholar 

  10. Hubert de Fraysseix and Pierre Rosenstiehl, A characterization of planar graphs by Trémaux orders, Combinatorica, 5(2):127–135, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  11. Peter Eades, Qing-Wen Feng, and Xuemin Lin, Straight-line drawing algorithms for hierarchical graphs and clustered graphs. In Graph drawing, volume 1190 of Lecture Notes in Comput. Sci., pages 113–128, London, UK, 1997. Springer.

    Google Scholar 

  12. Esther Ezra and Micha Sharir, Lower envelopes of 3-intersecting surfaces in ℝ3. Unpublished manuscript, 2007.

    Google Scholar 

  13. Radoslav Fulek and János Pach, A computational approach to conway’s thrackle conjecture, CoRR, abs/1002.3904, Feb 2010.

    Google Scholar 

  14. Michael R. Garey and David S. Johnson, Crossing number is NP-complete, SIAM Journal on Algebraic and Discrete Methods, 4(3):312–316, 1983.

    Article  Google Scholar 

  15. Chaim Chojnacki (Haim Hanani), Über wesentlich unplättbare Kurven im dreidimensionalen Raume, Fundamenta Mathematicae, 23:135–142, 1934.

    Google Scholar 

  16. Daniel J. Kleitman. A note on the parity of the number of crossings of a graph, J. Combinatorial Theory Ser. B, 21(1):88–89, 1976.

    Article  MathSciNet  Google Scholar 

  17. Roy B. Levow, On Tutte’s algebraic approach to the theory of crossing numbers. In Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1972), pages 315–314, Boca Raton, Fla., 1972. Florida Atlantic Univ.

    Google Scholar 

  18. Martin Loebl and Gregor Masbaum, On the optimality of the arf invariant formula for graph polynomials, CoRR, abs/0908.2925, 2009.

    Google Scholar 

  19. Laszlo Lovász, János Pach, and Mario Szegedy, On Conway’s thrackle conjecture, Discrete Comput. Geom., 18(4):369–376, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  20. Karl Menger, Ergebnisse eines mathematischen Kolloquiums. Springer-Verlag, Vienna, 1998.

    MATH  Google Scholar 

  21. Bojan Mohar and Carsten Thomassen, Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, 2001.

    MATH  Google Scholar 

  22. Serguei Norine, Pfaffian graphs, T-joins and crossing numbers, Combinatorica, 28(1):89–98, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  23. János Pach, Rom Pinchasi, Gábor Tardos, and Géza Tóth, Geometric graphs with no self-intersecting path of length three, European J. Combin., 25(6):793–811, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  24. János Pach, Rado0161 Radoicić, Gábor Tardos, and Géza Tóth, Improving the crossing lemma by finding more crossings in sparse graphs: [extended abstract]. In SCG’ 04: Proceedings of the twentieth annual symposium on Computational geometry, pages 68–75, New York, NY, USA, 2004. ACM.

    Google Scholar 

  25. János Pach and Micha Sharir, On the boundary of the union of planar convex sets, Discrete Comput. Geom., 21(3):321–328, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  26. János Pach and Micha Sharir, Geometric incidences. In Towards a theory of geometric graphs, volume 342 of Contemp. Math., pages 185–223. Amer. Math. Soc., Providence, RI, 2004.

    Google Scholar 

  27. János Pach and Micha Sharir, Combinatorial geometry and its algorithmic applications: The Alcalá lectures, volume 152 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2009.

    Google Scholar 

  28. János Pach and Géza Tóth, Thirteen problems on crossing numbers, Geombinatorics, 9(4):194–207, 2000.

    MATH  MathSciNet  Google Scholar 

  29. János Pach and Géza Tóth, Which crossing number is it anyway? J. Combin. Theory Ser. B, 80(2):225–246, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  30. János Pach and Géza Tóth, Monotone drawings of planar graphs, J. Graph Theory, 46(1):39–47, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  31. János Pach and Géza Tóth. Disjoint edges in topological graphs, In Combinatorial geometry and graph theory, volume 3330 of Lecture Notes in Comput. Sci., pages 133–140. Springer, Berlin, 2005.

    Google Scholar 

  32. Michael J. Pelsmajer, Marcus Schaefer, and Despina Stasi, Strong Hanani-Tutte on the projective plane, SIAM Journal on Discrete Mathematics, 23(3):1317–1323, 2009.

    Article  MathSciNet  Google Scholar 

  33. Michael J. Pelsmajer, Marcus Schaefer, and Štefankovič, Crossing numbers and parameterized complexity. In Seok-Hee Hong, Takao Nishizeki, and Wu Quan, editors, Graph Drawing, volume 4875 of Lecture Notes in Computer Science, pages 31–36. Springer, 2007.

    Google Scholar 

  34. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Removing even crossings, J. Combin. Theory Ser. B, 97(4):489–500, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  35. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Odd crossing number and crossing number are not the same, Discrete Comput. Geom., 39(1):442–454, 2008.

    MATH  MathSciNet  Google Scholar 

  36. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Removing even crossings on surfaces, European J. Combin., 30(7):1704–1717, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  37. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Removing independently even crossings, SIAM Journal on Discrete Mathematics, 24(2):379–393, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  38. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič, Crossing numbers of graphs with rotation systems, Algorithmica, 60:679–702, 2011. 10.1007/s00453-009-9343-y.

    Article  MATH  MathSciNet  Google Scholar 

  39. Amitai Perlstein and Rom Pinchasi, Generalized thrackles and geometric graphs in ℝ3 with no pair of strongly avoiding edges, Graphs Combin., 24(4):373–389, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  40. Rom Pinchasi and Rado0161 Radoi010Cić, Topological graphs with no self-intersecting cycle of length 4. In Towards a theory of geometric graphs, volume 342 of Contemp. Math., pages 233–243. Amer. Math. Soc., Providence, RI, 2004.

    Google Scholar 

  41. K. S. Sarkaria, A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion, Israel J. Math., 73(1):79–89, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  42. Marcus Schaefer, Removing incident crossings. Unpublished Manuscript, 2010.

    Google Scholar 

  43. Marcus Schaefer, Eric Sedgwick, and Daniel Štefankovič, Recognizing string graphs in NP, J. Comput. System Sci., 67(2):365–380, 2003. Special issue on STOC2002 (Montreal, QC).

    Article  MATH  MathSciNet  Google Scholar 

  44. Marcus Schaefer, Eric Sedgwick, and Daniel Štefankovič, Computing Dehn twists and geometric intersection numbers in polynomial time. In Proceedings of the 20th Canadian Conference on Computational Geometry, pages 111–114, 2008.

    Google Scholar 

  45. Shakhar Smorodinsky and Micha Sharir, Selecting points that are heavily covered by pseudo-circles, spheres or rectangles, Combin. Probab. Comput., 13(3):389–411, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  46. Géza Tóth, A better bound for the pair-crossing number. Presentation at the Conference on Geoemtric Graph Theory, Lausanne, Switzerland, 2010.

    Google Scholar 

  47. Géza Tóth, Note on the pair-crossing number and the odd-crossing number, Discrete Comput. Geom., 39(4):791–799, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  48. Géza Tóth, 2010. Personal communication.

    Google Scholar 

  49. William T. Tutte, Toward a theory of crossing numbers, J. Combinatorial Theory, 8:45–53, 1970.

    Article  MATH  MathSciNet  Google Scholar 

  50. Pavel Valtr, On the pair-crossing number. In Combinatorial and Computational Geometry, volume 52 of Math. Sci. Res. Inst. Publ., pages 569–575. Cambridge University Press, Cambridge, 2005.

    Google Scholar 

  51. Hein van der Holst, Algebraic characterizations of outerplanar and planar graphs, European J. Combin., 28(8):2156–2166, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  52. Hein van der Holst, A polynomial-time algorithm to find a linkless embedding of a graph, J. Combin. Theory Ser. B, 99(2):512–530, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  53. Hein van der Holst and Rudi Pendavingh, On a graph property generalizing planarity and flatness, Combinatorica, 29(3):337–361, 2009.

    Article  MathSciNet  Google Scholar 

  54. E. R. van Kampen, Komplexe in euklidischen Räumen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 9(1):72–78, December 1933.

    Article  Google Scholar 

  55. Hassler Whitney, The self-intersections of a smooth n-manifold in 2n-space, The Annals of Mathematics, 45(2):220–246, 1944.

    Article  MATH  MathSciNet  Google Scholar 

  56. Wen Jun Wu, On the planar imbedding of linear graphs. I, J. Systems Sci. Math. Sci., 5(4):290–302, 1985.

    MATH  MathSciNet  Google Scholar 

  57. Wen Jun Wu, On the planar imbedding of linear graphs (continued), J. Systems Sci. Math. Sci., 6(1):23–35, 1986.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing, DePaul University, Chicago, Illinois, 60604, USA

    Marcus Schaefer

Authors
  1. Marcus Schaefer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Imre Bárány , Károly J. Böröczky , Gábor Fejes Tóth  & János Pach , ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2013 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Schaefer, M. (2013). Hanani-Tutte and Related Results. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds) Geometry — Intuitive, Discrete, and Convex. Bolyai Society Mathematical Studies, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41498-5_10

Download citation

Publish with us

Policies and ethics

Search

Navigation