4-Labelings and Grid Embeddings of Plane Quadrangulations

  • Lali Barrière
  • Clemens Huemer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

Finding aesthetic drawings of planar graphs is a main issue in graph drawing. Of special interest are rectangle of influence drawings.The graphs considered here are quadrangulations, that is, planar graphs all whose faces have degree four.We show that each quadrangulation on n vertices has a closed rectangle of influence drawing on the (n - 2) × (n - 2) grid. Biedl, Bretscher and Meijer [2] proved that every planar graph on n vertices without separating triangle has a closed rectangle of influence drawing on the (n - 1) × (n - 1) grid.Our method, which is completely different from that of [2], is in analogy to Schnyder’s algorithm for embedding triangulations on an integer grid [9] and gives a simple algorithm.

References

  1. 1.
    Biedl, T., Brandenburg, F.: Drawing planar bipartite graphs with small area. In: Proceedings of the 17th Canadian Conference on Computational Geometry, Windsor, Canada, pp. 105–108 (2005)Google Scholar
  2. 2.
    Biedl, T., Bretscher, A., Meijer, H.: Rectangle of Influence Drawings of Graphs without Filled 3-Cycles. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 359–368. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Felsner, S.: Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes. Order 18, 19–37 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Felsner, S., Fusy, E., Noy, M., Orden, D.: Bijections for Baxter Families and Related Objects (2008), http://arxiv.org/abs/0803.1546
  5. 5.
    Felsner, S., Huemer, C., Kappes, S., Orden, D.: Binary Labelings for Plane Quadrangulations and their Relatives (2008), http://arxiv.org/abs/math/0612021v3
  6. 6.
    de Fraysseix, H., de Mendez, P.O., Pach, J.: A left-first search algorithm for planar graphs. Discrete and Computational Geometry 13, 459–468 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fusy, E.: Straight-line drawing of quadrangulations. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 234–239. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Huemer, C., Kappes, S.: A binary labelling for plane Laman graphs and quadrangulations. In: Proceedings of the 22nd European Workshop on Computational Geometry, Delphi, Greece, pp. 83–86 (2006)Google Scholar
  9. 9.
    Schnyder, W.: Embedding Planar Graphs on the Grid. In: Proceedings of the first annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148 (1990)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Lali Barrière
    • 1
  • Clemens Huemer
    • 1
  1. 1.Departament de Matemàtica Aplicada IVUniversitat Politècnica de Catalunya 

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