Advertisement

An SPQR-Tree-Like Embedding Representation for Upward Planarity

  • Guido BrücknerEmail author
  • Markus Himmel
  • Ignaz Rutter
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

The SPQR-tree is a data structure that compactly represents all planar embeddings of a biconnected planar graph. It plays a key role in constrained planarity testing.

We develop a similar data structure, called the UP-tree, that compactly represents all upward planar embeddings of a biconnected single-source directed graph. We demonstrate the usefulness of the UP-tree by solving the upward planar embedding extension problem for biconnected single-source directed graphs.

References

  1. 1.
    Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32:1–32:42 (2015).  https://doi.org/10.1145/2629341
  2. 2.
    Angelini, P., Di Battista, G., Patrignani, M.: Finding a minimum-depth embedding of a planar graph in \(o(n^4)\) time. Algorithmica 60(4), 890–937 (2011).  https://doi.org/10.1007/s00453-009-9380-6MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertolazzi, P., Di Battista, G., Mannino, C., Tamassia, R.: Optimal upward planarity testing of single-source digraphs. SIAM J. Comput. 27(1), 132–169 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, Discrete Mathematics and its Applications, pp. 349–373. CRC Press (2014)Google Scholar
  6. 6.
    Bläsius, T., Lehmann, S., Rutter, I.: Orthogonal graph drawing with inflexible edges. Comput. Geom. 55, 26–40 (2016).  https://doi.org/10.1016/j.comgeo.2016.03.001MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bläsius, T., Rutter, I., Wagner, D.: Optimal orthogonal graph drawing with convex bend costs. ACM Trans. Algorithms 12(3), 33:1–33:32 (2016).  https://doi.org/10.1145/2838736MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bläsius, T., Karrer, A., Rutter, I.: Simultaneous embedding: edge orderings, relative positions, cutvertices. Algorithmica 80(4), 1214–1277 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brückner, G., Himmel, M., Rutter, I.: An SPQR-tree-like embedding representation for upward planarity. CoRR abs/1908.00352v1 (2019). https://arxiv.org/abs/1908.00352v1
  10. 10.
    Brückner, G., Rutter, I.: Partial and constrained level planarity. In: Klein, P.N. (ed.) Proceedings of 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 2000–2011. SIAM (2017)Google Scholar
  11. 11.
    Brückner, G., Rutter, I., Stumpf, P.: Level planarity: transitivity vs. even crossings. In: Biedl, T., Kerren, A. (eds.) GD 2018. LNCS, vol. 11282, pp. 39–52. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-04414-5_3CrossRefGoogle Scholar
  12. 12.
    Da Lozzo, G., Di Battista, G., Frati, F.: Extending upward planar graph drawings. CoRR abs/1902.06575 (2019)Google Scholar
  13. 13.
    Da Lozzo, G., Jelínek, V., Kratochvíl, J., Rutter, I.: Planar embeddings with small and uniform faces. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 633–645. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13075-0_50CrossRefzbMATHGoogle Scholar
  14. 14.
    Da Lozzo, G., Rutter, I.: Approximation algorithms for facial cycles in planar embeddings. In: Hsu, W.L., Lee, D.T., Liao, C.S. (eds.) Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC 2018). LIPIcs, vol. 123, pp. 41:1–41:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018).  https://doi.org/10.4230/LIPIcs.ISAAC.2018.41
  15. 15.
    Di Battista, G., Tamassia, R.: Incremental planarity testing. In: Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pp. 436–441, October 1989.  https://doi.org/10.1109/SFCS.1989.63515
  16. 16.
    Di Battista, G., Tamassia, R.: On-line graph algorithms with SPQR-trees. In: Paterson, M.S. (ed.) ICALP 1990. LNCS, vol. 443, pp. 598–611. Springer, Heidelberg (1990).  https://doi.org/10.1007/BFb0032061CrossRefGoogle Scholar
  17. 17.
    Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996).  https://doi.org/10.1007/BF01961541MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Didimo, W., Liotta, G., Patrignani, M.: Bend-minimum orthogonal drawings in quadratic time. In: Biedl, T., Kerren, A. (eds.) GD 2018. LNCS, vol. 11282, pp. 481–494. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-04414-5_34CrossRefGoogle Scholar
  19. 19.
    Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995).  https://doi.org/10.1007/3-540-60313-1_145CrossRefGoogle Scholar
  20. 20.
    Fulek, R., Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Hanani-Tutte, monotone drawings, and level-planarity. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 263–287. Springer, New York (2013).  https://doi.org/10.1007/978-1-4614-0110-0_14CrossRefGoogle Scholar
  21. 21.
    Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30(2), 209–221 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-44541-2_8CrossRefGoogle Scholar
  23. 23.
    Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Comput. 2(3), 135–158 (1973)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hutton, M.D., Lubiw, A.: Upward planar drawing of single-source acyclic digraphs. SIAM J. Comput. 25(2), 291–311 (1996)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom.: Theory Appl. 46(4), 466–492 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Jünger, M., Leipert, S.: Level planar embedding in linear time. In: Kratochvíyl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 72–81. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-46648-7_7CrossRefGoogle Scholar
  27. 27.
    Mac Lane, S.: A structural characterization of planar combinatorial graphs. Duke Math. J. 3(3), 460–472 (1937).  https://doi.org/10.1215/S0012-7094-37-00336-3MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Platt, C.R.: Planar lattices and planar graphs. J. Comb. Theory Ser. B 21(1), 30–39 (1976)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Randerath, B., Speckenmeyer, E., Boros, E., Hammer, P., Kogan, A., Makino, K., Simeone, B., Cepek, O.: A satisfiability formulation of problems on level graphs. Electron. Notes Discret. Math. 9, 269–277 (2001). lICS 2001 Workshop on Theory and Applications of Satisfiability Testing (SAT 2001)Google Scholar
  30. 30.
    Tutte, W.T.: Connectivity in Graphs. University of Toronto Press, Toronto (1966)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.University of PassauPassauGermany

Personalised recommendations