Drawing high degree graphs with low bend numbers

  • Ulrich Fößmeier
  • Michael Kaufmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)

Abstract

We consider the problem of drawing plane graphs with an arbitrarily high vertex degree orthogonally into the plane such that the number of bends on the edges should be minimized. It has been known how to achieve the bend minimum without any restriction of the size of the vertices. Naturally, the vertices should be represented by uniformly small squares. In addition we might require that each face should be represented by a non-empty region. This would allow a labeling of the faces. We present an efficient algorithm which provably achieves the bend minimum following these constraints. Omitting the latter requirement we conjecture that the problem becomes NP-hard. For that case we give advices for good approximations. We demonstrate the effectiveness of our approaches giving some interesting examples.

References

  1. 1.
    Batini, C., E. Nardelli, and R. Tamassia, A Layout Algorithm for Data-Flow Diagrams, IEEE Trans. on Software Engineering, Vol. SE-12 (4), pp. 538–546, 1986.Google Scholar
  2. 2.
    Batini, C., M. Talamo, and R. Tamassia, Computer Aided Layout Of Entity-Relationship Diagrams, The Journal of Systems and Software, Vol. 4, pp. 163–173, 1984.CrossRefGoogle Scholar
  3. 3.
    H.K.B. Beck, H.-P. Galil, R. Henkel, and E. Sedlmayr: Chemistry in circumstellar shells, I. Chromospheric radiation fields and dust formation in optimcally thin shells of M. giants, Astron. Astrophys. 265 (1992) 626–642.Google Scholar
  4. 4.
    Di Battista G., P. Eades, R. Tamassia and I.G. Tollis, Algorithms for Automatic Graph Drawing: An Annotated Bibliography, Tech.Rep., Dept.of Comp.Sc., Brown Univ., 1993.Google Scholar
  5. 5.
    Di Battista, E. Pietrosanti, R. Tamassia and I.G. Tollis, Automatic Layout of PERT Diagrams with XPERT, Proc. IEEE Workshop on Visual Lang. (VL'89), 171–176, 1989.Google Scholar
  6. 6.
    Di Battista, G., L. Vismara, Angles of Planar Triangular Graphs, Proc. of the 25th ACM Symposium on the Theory of Computing, San Diego, California, 1993.Google Scholar
  7. 7.
    Fößmeier, U., and M. Kaufmann, Drawing High Degree Graphs with Low Bend Numbers, Technical Report WSI-95-21, Univ. Tubingen 1995.Google Scholar
  8. 8.
    Garg, A. and R. Tamassia, On the Computational Complexity of Upward and Rectilinear Planarity Testing, Proc. of GD '94, Princeton, 1994.Google Scholar
  9. 9.
    Himsolt, M., Konzeption und Implementierung von Grapheneditoren, Doctoral Dissertation, Passau 1993.Google Scholar
  10. 10.
    Lengauer, Th., Combinatorial Algorithms for Integrated Circuit Layout, Teubner/Wiley & Sons, Stuttgart/Chichester, 1990.Google Scholar
  11. 11.
    Mutzel, P., The Maximum Planar Subgraph Problem, Doctoral Dissertation, Köln 1994.Google Scholar
  12. 12.
    Reiner, D., et al., A Database Designer's Workbench in Entity-Relationship Approach, ed. S. Spaccapietra, pp. 347–360, North-Holland, 1987.Google Scholar
  13. 13.
    Rosenstiehl, P., and R.E. Tarjan, Rectilinear planar layouts and bipolar orientations of planar graphs, Discrete and Comp. Geometry 1 (1986), pp. 343–353.CrossRefGoogle Scholar
  14. 14.
    Protsko, L.B., P.G. Sorenson, J.P. Tremblay, and D.A. Schaefer, Towards the Automatic Generation of Software Diagrams, IEEE Trans. on Software Engineering, Vol. SE-17 (1), pp. 10–21, 1991.CrossRefGoogle Scholar
  15. 15.
    Storer, J.A., The node cost measure for embedding graphs in the planar grid, Proc. 12th ACM Symposium on the Theory of Computing, 1980, pp. 201–210.Google Scholar
  16. 16.
    Tamassia, R., On Embedding a Graph in the Grid with the Minimum Number of Bends, SIAM Journal of Computing, vol. 16, 3, 421–444, 1987.CrossRefGoogle Scholar
  17. 17.
    Tamassia, R., and I.G. Tollis, A unified approach to visibility representations of planar graphs, Discr. and Comp. Geometry 1 (1986), pp. 321–341.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Ulrich Fößmeier
    • 1
  • Michael Kaufmann
    • 1
  1. 1.Wilhelm-Schickard-InstitutUniversität TübingenTübingenGermany

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