On the Perspectives Opened by Right Angle Crossing Drawings

  • Patrizio Angelini
  • Luca Cittadini
  • Giuseppe Di Battista
  • Walter Didimo
  • Fabrizio Frati
  • Michael Kaufmann
  • Antonios Symvonis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straight-line upward RAC drawings require exponential area. Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertex-degree at most 6 (at most 3) admits a RAC drawing with curve complexity 2 (resp. 1) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings.

References

  1. 1.
    Bárány, I., Tokushige, N.: The minimum area of convex lattice n-gons. Combinatorica 24(2), 171–185 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berge, C.: Graphs. North Holland, Amsterdam (1985)MATHGoogle Scholar
  3. 3.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)MATHGoogle Scholar
  5. 5.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science 61, 175–198 (1988)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Di Battista, G., Tamassia, R., Tollis, I.G.: Area requirement and symmetry display of planar upward drawings. Discrete & Computational Geometry 7, 381–401 (1992)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. In: WADS 2009. LNCS, vol. 5664, pp. 206–217. Springer, Heidelberg (2009)Google Scholar
  8. 8.
    Eades, P., Symvonis, A., Whitesides, S.: Three-dimensional orthogonal graph drawing algorithms. Discrete Applied Mathematics 103(1-3), 55–87 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing 31(2), 601–625 (2001)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Huang, W.: An eye tracking study into the effects of graph layout. CoRR, abs/0810.4431 (2008)Google Scholar
  11. 11.
    Huang, W., Hong, S.-H., Eades, P.: Effects of crossing angles. In: PacificVis, pp. 41–46 (2008)Google Scholar
  12. 12.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. Springer, Heidelberg (2001)MATHGoogle Scholar
  13. 13.
    Papakostas, A.: Upward planarity testing of outerplanar dags. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 298–306. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Papakostas, A., Tollis, I.G.: Algorithms for area-efficient orthogonal drawings. Computational Geometry 9(1-2), 83–110 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Petersen, J.: Die theorie der regulären graphen. Acta Mathematicae 15, 193–220 (1891)CrossRefGoogle Scholar
  16. 16.
    Purchase, H.C.: Effective information visualisation: a study of graph drawing aesthetics and algorithms. Interacting with Computers 13(2), 147–162 (2000)CrossRefGoogle Scholar
  17. 17.
    Purchase, H.C., Carrington, D.A., Allder, J.-A.: Empirical evaluation of aesthetics-based graph layout. Empirical Software Engineering 7(3), 233–255 (2002)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Schnyder, W.: Embedding planar graphs on the grid. In: SODA 1990, pp. 138–148 (1990)Google Scholar
  19. 19.
    Ware, C., Purchase, H.C., Colpoys, L., McGill, M.: Cognitive measurements of graph aesthetics. Information Visualization 1(2), 103–110 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Luca Cittadini
    • 1
  • Giuseppe Di Battista
    • 1
  • Walter Didimo
    • 2
  • Fabrizio Frati
    • 1
  • Michael Kaufmann
    • 3
  • Antonios Symvonis
    • 4
  1. 1.Dipartimento di Informatica e AutomazioneRoma Tre UniversityItaly
  2. 2.Dip. di Ingegneria Elettronica e dell’InformazionePerugia UniversityItaly
  3. 3.Wilhelm-Schickard-Institut für InformatikUniversität TübingenGermany
  4. 4.Department of MathematicsNational Technical University of AthensGreece

Personalised recommendations