Advertisement

Bitonic st-orderings for Upward Planar Graphs

  • Martin GronemannEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

Canonical orderings serve as the basis for many incremental planar drawing algorithms. All these techniques, however, have in common that they are limited to undirected graphs. While st-orderings do extend to directed graphs, especially planar st-graphs, they do not offer the same properties as canonical orderings. In this work we extend the so called bitonic st-orderings to directed graphs. We fully characterize planar st-graphs that admit such an ordering and provide a linear-time algorithm for recognition and ordering. If for a graph no bitonic st-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. With this new technique we are able to draw every upward planar graph on n vertices by using at most one bend per edge, at most \(n - 3\) bends in total and within quadratic area.

Keywords

Directed Graph Planar Graph Span Subgraph Canonical Ordering Quadratic Area 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abbasi, S., Healy, P., Rextin, A.: Improving the running time of embedded upward planarity testing. Inf. Process. Lett. 110(7), 274–278 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 6(12), 476–497 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biedl, T.C., Derka, M.: The (3, 1)-ordering for 4-connected planar triangulations. CoRR abs/1511.00873 (2015)Google Scholar
  4. 4.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)zbMATHGoogle Scholar
  5. 5.
    Di Battista, G., Frati, F.: A survey on small-area planar graph drawing. CoRR abs/1410.1006 (2014)Google Scholar
  6. 6.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theoret. Comput. Sci. 61(2–3), 175–198 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Di Battista, G., Tamassia, R., Tollis, I.: Area requirement and symmetry display of planar upward drawings. Discrete Comput. Geom. 7(1), 381–401 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Didimo, W., Giordano, F., Liotta, G.: Upward spirality and upward planarity testing. SIAM J. Discrete Math. 23(4), 1842–1899 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: Bipolar orientations revisited. Discrete Appl. Math. 56(2–3), 157–179 (1995). 5th Franco-Japanese DaysMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 286–297. Springer, Heidelberg (1995). doi: 10.1007/3-540-58950-3_384 CrossRefGoogle Scholar
  12. 12.
    Gronemann, M.: Bitonic st-orderings for Upward Planar Graphs. arXiv e-prints, August 2016. http://arxiv.org/abs/1608.08578v1
  13. 13.
    Gronemann, M.: Bitonic st-orderings of biconnected planar graphs. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 162–173. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-45803-7_14 Google Scholar
  14. 14.
    Gronemann, M.: Algorithms for incremental planar graph drawing and two-page book embeddings. Ph.D. thesis, University of Cologne (2015)Google Scholar
  15. 15.
    Harel, D., Sardas, M.: An algorithm for straight-line drawing of planar graphs. Algorithmica 20(2), 119–135 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hutton, M.D., Lubiw, A.: Upward planarity testing of single-source acyclic digraphs. SIAM J. Comput. 25(2), 291–311 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoret. Comput. Sci. 172(1), 175–193 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Papakostas, A.: Upward planarity testing of outerplanar dags (extended abstract). In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 298–306. Springer, Heidelberg (1995). doi: 10.1007/3-540-58950-3_385 CrossRefGoogle Scholar
  20. 20.
    Samee, M.A.H., Rahman, M.S.: Upward planar drawings of series-parallel digraphs with maximum degree three. In: WALCOM 2012, pp. 28–45. Bangladesh Academy of Sciences (BAS) (2007)Google Scholar
  21. 21.
    Schmidt, J.M.: The mondshein sequence. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 967–978. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43948-7_80 Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.University of CologneCologneGermany

Personalised recommendations