Skip to main content

Algorithms for the Hypergraph and the Minor Crossing Number Problems

  • Conference paper
Algorithms and Computation (ISAAC 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4835))

Included in the following conference series:

Abstract

We consider the problems of hypergraph and minor crossing minimization, and point out a relation between these two problems which has not been exploited before. We present some complexity results regarding the corresponding edge and node insertion problems. Based on these results, we give the first embedding-based heuristics to tackle both problems and present a short experimental study. Furthermore, we give the first exact ILP formulation for both problems.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. Di Battista, G., Garg, A., Liotta, G., Tamassia, R., Tassinari, E., Vargiu, F.: An experimental comparison of four graph drawing algorithms. Comput. Geom. Theory Appl. 7(5-6), 303–325 (1997)

    Google Scholar 

  2. Bokal, D., Czabarkab, É., Székely, L.A., Vrt’o, I.: Graph minors and the crossing number of graphs. In: Proc. of 6th Czech-Slovak Int. Symp. on Comb. Graph Theory, Alg. and Appl. ENDM, vol. 28, pp. 169–175 (2007)

    Google Scholar 

  3. Bokal, D., Fijavz, G., Mohar, B.: The minor crossing number. SIAM J. Discrete Math. 20, 344–356 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. 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. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 37–48. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Eschbach, T., Günther, W., Becker, B.: Orthogonal hypergraph drawing for improved visibility. J. Graph Algorithms Appl. 10(2), 141–157 (2006)

    MathSciNet  Google Scholar 

  6. Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32, 826–834 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gutwenger, C., Mutzel, P.: An experimental study of crossing minimization heuristics. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 13–24. Springer, Heidelberg (2004)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Hliněný, P.: Crossing number is hard for cubic graphs. J. Combin. Theory Ser. B 96, 455–471 (2006)

    Article  MathSciNet  Google Scholar 

  10. Johnson, D.S., Pollak, H.O.: Hypergraph planarity and the complexity of drawing venn diagrams. J. Graph Theory 11(3), 309–325 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Klemetti, H., Lapinleimu, I., Mäkinen, E., Sieranta, M.: A programming project: Trimming the spring algorithm for drawing hypergraphs. SIGCSE Bull. 27(3), 34–38 (1995)

    Article  Google Scholar 

  12. Mäkinen, E.: How to draw a hypergraph. Int. J. Comput. Math. 34, 177–185 (1990)

    Article  Google Scholar 

  13. OGDF – Open Graph Drawing Framework (2007), See http://ls11-www.cs.uni-dortmund.de/ogdf

  14. Sander, G.: Layout of directed hypergraphs with orthogonal hyperedges. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 381–386. Springer, Heidelberg (2004)

    Google Scholar 

  15. Vrt’o, I.: Crossing numbers of graphs: A bibliography (2007), See ftp://ftp.ifi.savba.sk/pub/imrich/crobib.pdf

Download references

Author information

Authors and Affiliations

Authors

Editor information

Takeshi Tokuyama

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chimani, M., Gutwenger, C. (2007). Algorithms for the Hypergraph and the Minor Crossing Number Problems. In: Tokuyama, T. (eds) Algorithms and Computation. ISAAC 2007. Lecture Notes in Computer Science, vol 4835. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77120-3_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-77120-3_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-77118-0

  • Online ISBN: 978-3-540-77120-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics