Skip to main content
SpringerLink
Log in

The Crossing Number of Graphs: Theory and Computation

  • Chapter
Efficient Algorithms

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5760))

Abstract

This survey concentrates on selected theoretical and computational aspects of the crossing number of graphs. Starting with its introduction by Turán, we will discuss known results for complete and complete bipartite graphs. Then we will focus on some historical confusion on the crossing number that has been brought up by Pach and Tóth as well as Székely. A connection to computational geometry is made in the section on the geometric version, namely the rectilinear crossing number. We will also mention some applications of the crossing number to geometrical problems. This review ends with recent results on approximation and exact computations.

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
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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. Aichholzer, O., Aurenhammer, F., Krasser, H.: On the crossing number of complete graphs. In: Symposium on Computational Geometry, pp. 19–24 (2002)

    Google Scholar 

  2. Bhatt, S.N., Leighton, F.T.: A framework for solving vlsi layout problems. Journal of Computer and System Sciences 28, 300–343 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bienstock, D.: Some provably hard crossing number problems. Discrete & Computational Geometry 6, 443–459 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bienstock, D., Dean, N.: New results on rectilinear crossing numbers and plane embeddings. J. Graph Theory 16(5), 389–398 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J. Graph Theory 17(3), 333–348 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buchheim, C., Chimani, M., Ebner, D., Gutwenger, C., Jünger, M., Klau, G.W., Mutzel, P., Weiskircher, R.: A branch-and-cut approach to the crossing number problem. Discrete Optimization, Special Issue in memory of George B. Dantzig 5(2), 373–388 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Cabello, S., Mohar, B.: Crossing and weighted crossing number of near-planar graphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 38–49. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Chimani, M.: Exact Crossing Minimization. PhD thesis, Fakultät für Informatik, Technische Universität Dortmund (2008)

    Google Scholar 

  9. Chimani, M., Gutwenger, C., Mutzel, P., Wolf, C.: Inserting a vertex into a planar graph. In: SODA, pp. 375–383. SIAM, Philadelphia (2009)

    Google Scholar 

  10. Chimani, M., Hliněný, P., Mutzel, P.: Vertex insertion approximates the crossing number for apex graphs (submitted, 2009)

    Google Scholar 

  11. Chimani, M., Mutzel, P., Bomze, I.: A new approach to exact crossing minimization. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 284–296. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. de Klerk, E., Pasechnik, D.V., Schrijver, A.: Reduction of symmetric semidefinite programs using the regular *-representation. Math. Program. 109(2-3), 613–624 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dey, T.K.: Improved bounds for planar k -sets and related problems. Discrete & Computational Geometry 19(3), 373–382 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Elekes, G.: On the number of sums and products. Acta Arithm. 81(4), 365–367 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Even, G., Guha, S., Schieber, B.: Improved approximations of crossings in graph drawings and VLSI layout areas. SIAM Journal on Computing 32(1), 231–252 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41(4), 289–308 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guy, R.K.: The decline and fall of Zarankiewicz’s theorem. In: Proof techniques in Graph Theory, pp. 63–69. Academic Press, London (1969)

    Google Scholar 

  18. Guy, R.K.: Crossing numbers of graphs. In: Proc. Graph Theory and Applications, pp. 111–124. LNM (1972)

    Google Scholar 

  19. Chojnacki, C., Hanani, H.: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundam. Math. (1934)

    Google Scholar 

  20. Hliněný, P., Salazar, G.: Crossing and weighted crossing number of near-planar graphs. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 162–173. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  21. Johnson, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kolman, P., Matousek, J.: Crossing number, pair-crossing number, and expansion. J. Combin. Theory Ser. B 92, 99–113 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pach, J., Tóth, G.: Which crossing number is it anyway? Journal of Combinatorial Theory, Series B 80, 225–246 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pan, S., Richter, R.B.: The crossing number of K \(_{\mbox{11}}\) is 100. Journal of Graph Theory 56(2), 128–134 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings on surfaces. Electronic Notes in Discrete Mathematics 29, 85–90 (2007)

    Article  MATH  Google Scholar 

  26. Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Odd crossing number and crossing number are not the same. Discrete & Computational Geometry 39(1-3), 442–454 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. The rectilinear crossing number project, http://dist.ist.tugraz.at/cape5/index.php

  28. Scheinerman, E.R., Wilf, H.: The rectilinear crossing number of a complete graph and sylvester’s “four point problem” of geometric probability. American Mathematical Monthly 101, 939–943 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Spencer, J., Szemerédi, E., Trotter, W.T.: Unit distances in the euclidean plane. Graph Theory and Combinatorics, 293–308 (1984)

    Google Scholar 

  30. Székely, L.A.: A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Mathematics 276(1-3), 331–352 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Turán, P.: A note of welcome. Journal of Graph Theory 1, 7–9 (1977)

    Article  Google Scholar 

  32. Tutte, W.T.: Toward a theory of crossing numbers. Journal of Combinatorial Theory (1970)

    Google Scholar 

  33. Vrt’o, I.: Bibliography on crossing numbers, ftp://ftp.ifi.savba.sk/pub/imrich/crobib.pdf

  34. Zarankiewicz, K.: On a problem of P. Turan concerning graphs. Fund. Math. (1954)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Informatik LS11, Algorithm Engineering, Technische Universität Dortmund, Otto-Hahn-Str. 14, 44227, Dortmund, Germany

    Petra Mutzel

Authors
  1. Petra Mutzel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität Freiburg, Georges-Köhler-Allee 79, 79110, Freiburg, Germany

    Susanne Albers

  2. Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany

    Helmut Alt

  3. Fachbereich IV - Informatik, Universität Trier, 54286, Trier, Germany

    Stefan Näher

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Mutzel, P. (2009). The Crossing Number of Graphs: Theory and Computation. In: Albers, S., Alt, H., Näher, S. (eds) Efficient Algorithms. Lecture Notes in Computer Science, vol 5760. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03456-5_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03456-5_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03455-8

  • Online ISBN: 978-3-642-03456-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation