On bend-minimum orthogonal upward drawing of directed planar graphs

  • Ulrich Fößmeier
  • Michael Kaufmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 894)


In last year's graph drawing workshop GD'93 we considered a restricted version of the problem of minimization of bends in orthogonal upward drawings. Inserting the severe restriction that each node should have an incoming edge from below and an outgoing edge upwards (if such edges exist), we were able to get optimal bounds on the number of bends in linear time. In this paper now, we release this restriction completely. The problem becomes much harder. Starting from a fixed planar topological embedding we are able to reduce the problem to a min-cut problem and present three algorithms: a) Find an orthogonal upward drawing without any bend, if such a drawing exists (in linear time), b) find a bend-minimum solution, if the undirected version of the graph requires no bends (in time O(n2·logn), n being the number of vertices of the graph), c) apply our technique to the general case; here we could not prove the optimality up to now. But the achieved number of bends does not exceed the optimum by more than the optimal number of bends in Tamassia's undirected case, i.e. our algorithm needs at most twice the number of bends as necessary in this case.


  1. 1.
    Bertolazzi, P., R.F. Cohen, G. Di Battista, R. Tamassia and I.G. Tollis, How to Draw a Series-Parallel Digraph, Proc. 3rd Scandinavian Workshop on Algorithm Theory, LNCS 621, (1992), pp. 272–283.Google Scholar
  2. 2.
    Di Battista, G., G. Liotta and F. Vargiu, Spirality of Orthogonal Representations and Optimal Drawings of Series-Parallel Graphs and 3-planar Graphs, Proc. 3nd Workshop on Algorithms and Data Structures, (1993), Lecture Notes in Comp. Science 709, pp. 151–162.Google Scholar
  3. 3.
    Di Battista, G., W.P. Liu, and I. Rival, Bipartite Graphs, Upward Drawings and Planarity, Information Processing Letters. vol. 36, (1990), pp. 317–322.Google Scholar
  4. 4.
    Di Battista, G. and R. Tamassia, Algorithms for Plane Representations of Acyclic Digraphs, Theoretical Computer Science Vol.61, (1988), pp. 175–198.Google Scholar
  5. 5.
    Di Battista, G., R. Tamassia and I.G. Tollis, Area Requirement and Symmetry Display in Drawing Graphs, Discrete and Comp. Geometry 7 (1992), pp. 381–401.Google Scholar
  6. 6.
    Eades, P., B. McKay and N. Wormald, On an Edge Crossing Problem, Proc. 9th Australian Computer Science Conf., (1986), pp. 327–334.Google Scholar
  7. 7.
    Fößmeier, U. and M. Kaufmann, An Approach for Bend-Minimal Upward Drawing, Workshop of GD'93, Paris (1993), pp. 27–29.Google Scholar
  8. 8.
    Garg, A. and R. Tamassia, On the Computational Complexity of Upward and Rectilinear Planarity Testing, Report CS-94-10, Comp. Sci. Dep., Brown Univ., Providence (1994), this proceedings.Google Scholar
  9. 9.
    Kelly, I. and I. Rival, Planar Lattices, Canadian J. Mathematics, Vol. 27, (1975), pp. 636–66.Google Scholar
  10. 10.
    Storer, J.A., On Minimal Node-cost Planar Embeddings, Networks 14 (1984), pp. 181–212.Google Scholar
  11. 11.
    Sugiyama, K., S. Tagawa and M. Toda, Methods for Visual Understanding of Hierarchical Systems, IEEE Trans. on Systems, Man and Cybernetics, Vol, SMC-11, (1981), pp. 109–125.Google Scholar
  12. 12.
    Tamassia, R., On Embedding a Graph in the Grid with the Minimum Number of Bends, SIAM J. Comput. 16 (1987), pp. 421–444.Google Scholar
  13. 13.
    Tamassia, R. and I.G. Tollis, A Unified Approach to Visibility Representations of Planar Graphs, Discrete and Comput. Geometry 1 (1986), pp. 321–341.Google Scholar
  14. 14.
    Tamassia, R. and I.G. Tollis, Efficient Embedding of Planar Graphs in Linear Time, Proc. IEEE Int. Symp. on Circuits and Systems, Philadelphia, (1987), pp. 495–498.Google Scholar
  15. 15.
    Tarjan, R.E., Data Structures and Network Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, (1983), Chapter 8.Google Scholar
  16. 16.
    Thomassen, C., Planar Acyclic Oriented Graphs, Order, Vol. 5, (1989), pp. 349–361.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

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

Personalised recommendations