Minimum Depth Graph Embeddings and Quality of the Drawings: An Experimental Analysis

  • Maurizio Pizzonia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)


The depth of a planar embedding of a graph is a measure of the topological nesting of the biconnected components of the graph in that embedding. Motivated by the intuition that lower depth values lead to better drawings, previous works proposed efficient algorithms for finding embeddings with minimum depth. We present an experimental study that shows the impact of embedding depth minimization on important aesthetic criteria and relates the effectiveness of this approach with measures of how much the graph resembles a tree or a biconnected graph. In our study, we use a well known test suite of graphs obtained from real-world applications and a randomly generated one with favorable biconnectivity properties. In the experiments we consider orthogonal drawings computed using the topology-shape-metrics approach.


  1. 1.
    Alberts, D., Gutwenger, C., Mutzel, P., Näher, S.: AGD-Library: A library of algorithms for graph drawing. In: Proc. Workshop on Algorithm Engineering, pp. 112–123 (1997)Google Scholar
  2. 2.
    Di Battista, G., Didimo, W., Patrignani, M., Pizzonia, M.: Orthogonal and quasi-upward drawings with vertices of prescribed size. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, p. 297. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Bertolazzi, P., Di Battista, G., Didimo, W.: Computing orthogonal drawings with the minimum number of bends. In: Rau-Chaplin, A., Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 1997. LNCS, vol. 1272, pp. 331–344. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  4. 4.
    Bertolazzi, P., Di Battista, G., Didimo, W.: Computing orthogonal drawings with the minimum number of bends. IEEETC: IEEE Transactions on Computers 49 (2000)Google Scholar
  5. 5.
    Bienstock, D., Monma, C.L.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5(1), 93–109 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brandes, U., Wagner, D.: Dynamic grid embedding with few bends and changes. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, p. 89. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Brandes, U., Wagner, D.: A Bayesian paradigm for dynamic graph layout. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 236–247. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  8. 8.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)MATHGoogle Scholar
  9. 9.
    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, 303–325 (1997)MATHGoogle Scholar
  10. 10.
    Di Battista, G., et al: Graph Drawing Toolkit. University of Rome III, Italy,
  11. 11.
    Didimo, W., Liotta, G.: Computing orthogonal drawings in a variable embedding setting. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, pp. 79–88. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Fößmeier, U., Kaufmann, M.: Drawing high degree graphs with low bend numbers. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 254–266. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gutwenger, C., Mutzel, P.: Graph embedding with minimum depth and maximum external face. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 259–272. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jünger, M., Mutzel, P.: Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica 16(1), 33–59 (1996)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Jünger, M., Leipert, S., Mutzel, P.: Pitfalls of using PQ-Trees in automatic graph drawing. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 193–204. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    Klau, G.W., Mutzel, P.: Optimal compaction of orthogonal grid drawings. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, p. 304. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  19. 19.
    Lauer, H., Ettrich, M., Soukup, K.: GraVis - system demonstration. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 344–349. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  20. 20.
    Patrignani, M.: On the complexity of orthogonal compaction. Computational Geometry: Theory and Applications 19(1), 47–67 (2001)MATHMathSciNetGoogle Scholar
  21. 21.
    Pizzonia, M.: Engineering of Graph Drawing Algorithms for Applications. PhD thesis, Dipartimento di Informatica e Sistemistica, University degli Studi “La Sapienza” di Roma (2001)Google Scholar
  22. 22.
    Pizzonia, M., Tamassia, R.: Minimum depth graph embedding. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 356–367. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  23. 23.
    Purchase, H.C., Cohen, R.F., James, M.: Validating graph drawing aesthetics. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 435–446. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  24. 24.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Maurizio Pizzonia
    • 1
  1. 1.Dipartimento di Informatica e AutomazioneUniversità Roma TreItaly

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