Morphing Planar Graph Drawings Efficiently

  • Patrizio Angelini
  • Fabrizio Frati
  • Maurizio Patrignani
  • Vincenzo Roselli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


A morph between two straight-line planar drawings of the same graph is a continuous transformation from the first to the second drawing such that planarity is preserved at all times. Each step of the morph moves each vertex at constant speed along a straight line. Although the existence of a morph between any two drawings was established several decades ago, only recently it has been proved that a polynomial number of steps suffices to morph any two planar straight-line drawings. Namely, at SODA 2013, Alamdari et al. [1] proved that any two planar straight-line drawings of a planar graph can be morphed in O(n 4) steps, while O(n 2) steps suffice if we restrict to maximal planar graphs.

In this paper, we improve upon such results, by showing an algorithm to morph any two planar straight-line drawings of a planar graph in O(n 2) steps; further, we show that a morph with O(n) steps exists between any two planar straight-line drawings of a series-parallel graph.


  1. 1.
    Alamdari, S., Angelini, P., Chan, T.M., Di Battista, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V., Singla, S., Wilkinson, B.T.: Morphing planar graph drawings with a polynomial number of steps. In: SODA 2013, pp. 1656–1667 (2013)Google Scholar
  2. 2.
    Angelini, P., Cortese, P.F., Di Battista, G., Patrignani, M.: Topological morphing of planar graphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 145–156. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Angelini, P., Didimo, W., Kobourov, S., Mchedlidze, T., Roselli, V., Symvonis, A., Wismath, S.: Monotone drawings of graphs with fixed embedding. Algorithmica, 1–25 (2013)Google Scholar
  4. 4.
    Angelini, P., Frati, F., Patrignani, M., Roselli, V.: Morphing planar graph drawings efficiently. CoRR cs.CG (2013),
  5. 5.
    Angelini, P., Colasante, E., Di Battista, G., Frati, F., Patrignani, M.: Monotone drawings of graphs. J. of Graph Algorithms and Appl. 16(1), 5–35 (2012); In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 13–24. Springer, Heidelberg (2011)Google Scholar
  6. 6.
    Biedl, T.C., Lubiw, A., Spriggs, M.J.: Morphing planar graphs while preserving edge directions. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 13–24. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Cairns, S.S.: Deformations of plane rectilinear complexes. American Math. Monthly 51, 247–252 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete & Computational Geometry 6(5), 485–524 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged. Sect. Sci. Math. 11, 229–233 (1948)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Grunbaum, B., Shephard, G.: The geometry of planar graphs. Cambridge University Press (1981),
  11. 11.
    Lubiw, A., Petrick, M.: Morphing planar graph drawings with bent edges. Electronic Notes in Discrete Mathematics 31, 45–48 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lubiw, A., Petrick, M., Spriggs, M.: Morphing orthogonal planar graph drawings. In: SODA 2006, pp. 222–230. ACM (2006)Google Scholar
  13. 13.
    Thomassen, C.: Deformations of plane graphs. J. Comb. Th., Series B 34, 244–257 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Fabrizio Frati
    • 2
  • Maurizio Patrignani
    • 1
  • Vincenzo Roselli
    • 1
  1. 1.Engineering DepartmentRoma Tre UniversityItaly
  2. 2.School of Information TechnologiesThe University of SydneyAustralia

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