Skip to main content
SpringerLink
Log in

The Crossing-Angle Resolution in Graph Drawing

  • Chapter
  • First Online:
Thirty Essays on Geometric Graph Theory

Abstract

The crossing-angle resolution of a drawing of a graph measures the smallest angle formed by any pair of crossing edges. In this chapter, we survey some of the most recent results and discuss the current research agenda on drawings of graphs with good crossing-angle resolution.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and 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
Hardcover Book
USD 109.99
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

References

  1. E. Ackerman, On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discr. Comput. Geom. 41(3), 365–375 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. E. Ackerman, R. Fulek, C.D. Tóth, On the size of graphs that admit polyline drawings with few bends and crossing angles, in Proceedings of GD 2010. LNCS, vol. 6502 (Springer, Berlin, 2010), pp. 1–12

    Google Scholar 

  3. E. Ackerman, G. Tardos, On the maximum number of edges in quasi-planar graphs. J. Comb. Theor. Ser. A 114(3), 563–571 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Angelini, L. Cittadini, G. Di Battista, W. Didimo, F. Frati, M. Kaufmann, A. Symvonis, On the perspectives opened by right angle crossing drawings, in Proceedings of GD 2009. LNCS, vol. 5849 (Springer, Berlin, 2010), pp. 21–32

    Google Scholar 

  5. P. Angelini, L. Cittadini, G. Di Battista, W. Didimo, F. Frati, M. Kaufmann, A. Symvonis, On the perspectives opened by right angle crossing drawings. J. Graph Algorithm. Appl. 15(1), 53–78 (2011) (Special issue on GD 2009)

    Google Scholar 

  6. E. Argyriou, M. Bekos, A. Symvonis, Maximizing the total resolution of graphs, in Proceedings of GD 2010. LNCS, vol. 6502 (Springer, Berlin, 2010), pp. 62–67

    Google Scholar 

  7. E.N. Argyriou, M.A. Bekos, A. Symvonis, The straight-line RAC drawing problem is NP-hard, in SOFSEM. Lecture Notes in Computer Science, vol. 6543 (Springer, Berlin, 2011), pp. 74–85

    Google Scholar 

  8. K. Arikushi, R. Fulek, B. Keszegh, F. Moric, Csaba D. Tóth, Graphs that admit right angle crossing drawings. Comput. Geom. Springer, Berlin, 45(4), 169–177 (2012)

    Google Scholar 

  9. C. Berge, Graphs (North Holland, Amsterdam, 1885)

    Google Scholar 

  10. O.V. Borodin, A.V. Kostochka, A. Raspaud, E. Sopena, Acyclic colouring of 1-planar graphs. Discr. Appl. Math. 114(1–3), 29–41 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Brightwell, E.R. Scheinerman, Representations of planar graphs. SIAM J. Discr. Math. 6(2), 214–229 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. H. de Fraysseix, J. Pach, R. Pollack, How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Di Battista, P. Eades, R. Tamassia, I.G. Tollis, Graph Drawing (Prentice Hall, Upper Saddle River, NJ, 1999)

    MATH  Google Scholar 

  14. E. Di Giacomo, W. Didimo, P. Eades, S.-H. Hong, G. Liotta, On the crossing resolution of complete geometric graphs. Technical report, DIEI RT-002-11, University of Perugia, Italy (2011). http://www.diei.unipg.it/rt/rt_frames.htm

  15. E. Di Giacomo, W. Didimo, P. Eades, S.-H. Hong, G. Liotta, Bounds on the crossing resolution of complete geometric graphs. Discr. Appl. Math., 160(1–2), 132–139 (2012). Available online, doi:10.1016/j.dam.2011.09.016

    Google Scholar 

  16. E. Di Giacomo, W. Didimo, P. Eades, G. Liotta, 2-layer right angle crossing drawings, in IWOCA 2011, Springer, Berlin, LNCS, 7056, 156–169 (2011)

    Google Scholar 

  17. E. Di Giacomo, W. Didimo, G. Liotta, H. Meijer, Area, curve complexity, and crossing resolution of non-planar graph drawings, in Proceedings of GD 2009. LNCS, vol. 5849 (Springer, Berlin, 2010), pp. 15–20

    Google Scholar 

  18. E. Di Giacomo, W. Didimo, G. Liotta, H. Meijer, Area, curve complexity, and crossing resolution of non-planar graph drawings. Theor. Comput. Syst. 49(3), 565–575 (2011)

    Article  MATH  Google Scholar 

  19. W. Didimo, P. Eades, G. Liotta, Drawing graphs with right angle crossings, in Proceedings of WADS 2009. LNCS, vol. 5664 (Springer, Berlin, 2009), pp. 206–217

    Google Scholar 

  20. W. Didimo, P. Eades, G. Liotta, A characterization of complete bipartite rac graphs. Inform. Process. Lett. 110(16), 687–691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Didimo, P. Eades, G. Liotta, Drawing graphs with right angle crossings. Theor. Comput. Sci. 412(39), 5156–5166 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. W. Didimo, G. Liotta, S.A. Romeo, Graph visualization techniques for conceptual web site traffic analysis, in PacificVis, IEEE, 2010, pp. 193–200

    Google Scholar 

  23. W. Didimo, G. Liotta, S.A. Romeo, Topology-driven force-directed algorithms, in Proceedings of GD 2010. LNCS, vol. 6502 (Springer, Berlin, 2010), pp. 165–176

    Google Scholar 

  24. V. Dujmovic, J. Gudmundsson, P. Morin, T. Wolle, Notes on large angle crossing graphs. CoRR, abs/0908.3545 (2009). http://arxiv.org/abs/0908.3545

  25. V. Dujmovic, J. Gudmundsson, P. Morin, T. Wolle, Notes on large angle crossing graphs. Chicago J. Theor. Comput. Sci., Brisbane, Australia, January 18–21 (2011)

    Google Scholar 

  26. C.A. Duncan, D. Eppstein, S.G. Kobourov, The geometric thickness of low degree graphs, in Symposium on Computational Geometry, ACM, 2004, pp. 340–346

    Google Scholar 

  27. P. Eades, G. Liotta, Right angle crossing graphs and 1-planarity, in proceedings of the 27th European Workshop on Computational Geometry (EuroCG), March 28–30, Morschach, Switzerland, pp. 155–158 (2011)

    Google Scholar 

  28. P. Eades, A. Symvonis, S. Whitesides, Three-dimensional orthogonal graph drawing algorithms. Discr. Appl. Math. 103(1–3), 55–87 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. I. Fabrici, T. Madaras, The structure of 1-planar graphs. Discr. Math. 307(7–8), 854–865 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. F. Frati, Improved lower bounds on the area requirements of series-parallel graphs, in Proceedings of GD 2010. LNCS, vol. 6502 (Springer, Berlin, 2010), pp. 220–225

    Google Scholar 

  31. A. Garg, R. Tamassia, On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. W. Huang, Using eye tracking to investigate graph layout effects, in APVIS 2007, 6th International Asia-Pacific Symposium on Visualization, Sydney, Australia, 5–7 February 2007, IEEE, pp. 97–100 (2007)

    Google Scholar 

  33. W. Huang, An eye tracking study into the effects of graph layout. CoRR, abs/0810.4431 (2008). http://arxiv.org/abs/0810.4431

  34. W. Huang, P. Eades, S.-H. Hong, C.-C. Lin, Improving force-directed graph drawings by making compromises between aesthetics, in VL/HCC, IEEE, 2010, pp. 176–183

    Google Scholar 

  35. W. Huang, S.-H. Hong, P. Eades, Effects of crossing angles, in IEEE VGTC Pacific Visualization Symposium 2008, PacificVis 2008, Kyoto, Japan, March 5–7, pp. 41–46 (2008)

    Google Scholar 

  36. M. Jünger, P. Mutzel (eds.), Graph Drawing Software (Mathematics and Visualization) (Springer-Verlag, Berlin, 2003)

    Google Scholar 

  37. M. Kaufmann, D. Wagner (eds.), Drawing Graphs, Methods and Models (the Book Grow Out of a Dagstuhl Seminar, April 1999). Lecture Notes in Computer Science, vol. 2025 (Springer-Verlag, Berlin, 2001)

    Google Scholar 

  38. V.P. Korzhik, B. Mohar, Minimal obstructions, Springer, Berlin, Lecture Notes in Computer Science, 5417, 302–312 (2008)

    Article  Google Scholar 

  39. Q. Nguyen, P. Eades, S.-H. Hong, W. Huang, Large crossing angles in circular layouts, in Proceedings of GD 2010. LNCS, vol. 6502 (Springer, Berlin, 2010), pp. 397–399

    Google Scholar 

  40. J.E. Goodman, J. O’Rourke (eds.), Geometric graph theory, in Handbook of Discrete and Computational Geometry (CRC, 2004), Chapman and Hall/CRC, New York, United States of America, pp. 219–238

    Google Scholar 

  41. J. Pach, R. Radoičić, G. Tardos, G. Tóth, Improving the crossing lemma by finding more crossings in sparse graphs. Discr. Comput. Geom. 36(4), 527–552 (2006)

    Article  MATH  Google Scholar 

  42. J. Pach, G. Tóth, Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Petersen, Die theorie der regulären graphen. Acta Math. 15, 193–220 (1891)

    Article  MathSciNet  MATH  Google Scholar 

  44. H.C. Purchase, Effective information visualisation: a study of graph drawing aesthetics and algorithms. Interact. Comput. 13(2), 147–162 (2000)

    Article  Google Scholar 

  45. H.C. Purchase, D.A. Carrington, J.-A. Allder, Empirical evaluation of aesthetics-based graph layout. Empir. Softw. Eng. 7(3), 233–255 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. K. Sugiyama, Graph Drawing and Applications for Software and Knowledge Engineers (World Scientific, Singapore, 2002)

    Book  MATH  Google Scholar 

  47. Y. Suzuki, Optimal 1-planar graphs which triangulate other surfaces. Discr. Math. 310(1), 6–11 (2010)

    Article  MATH  Google Scholar 

  48. J.J. Thomas, K.A. Cook, Illuminating the Path: The Research and Development Agenda for Visual Analytics (National Visualization and Analytics Ctr, 2005), Pacific Northwest National Laboratory, United States of America

    Google Scholar 

  49. M. van Kreveld, The quality ratio of RAC drawings and planar drawings of planar graphs, in Proceedings of GD 2010. LNCS, vol. 6502 (Springer, Berlin, 2010), pp. 371–376

    Google Scholar 

  50. C. Ware, H.C. Purchase, L. Colpoys, M. McGill, Cognitive measurements of graph aesthetics. Inform. Vis. 1(2), 103–110 (2002)

    Article  Google Scholar 

  51. D.R. Wood, Grid drawings of k-colourable graphs. Comput. Geom. 30(1), 25–28 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Our work is supported in part by MIUR of Italy under project AlgoDEEP prot. 2008TFBWL4.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Elettronica e dell’Informazione, Università degli Studi di Perugia, Perugia, Italy

    Walter Didimo & Giuseppe Liotta

Authors
  1. Walter Didimo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuseppe Liotta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Walter Didimo .

Editor information

Editors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

  2. École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    János Pach (Chair of Combinatorial Geometry) (Chair of Combinatorial Geometry)

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Didimo, W., Liotta, G. (2013). The Crossing-Angle Resolution in Graph Drawing. In: Pach, J. (eds) Thirty Essays on Geometric Graph Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0110-0_10

Download citation

Publish with us

Policies and ethics

Search

Navigation