A Note on Minimum-Area Straight-Line Drawings of Planar Graphs

  • Fabrizio Frati
  • Maurizio Patrignani
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)


Despite a long research effort, finding the minimum area for straight-line grid drawings of planar graphs is still an elusive goal. A long-standing lower bound on the area requirement for straight-line drawings of plane graphs was established in 1984 by Dolev, Leighton, and Trickey, who exhibited a family of graphs, known as nested triangles graphs, for which (2n/3 − 1) ×(2n/3 − 1) area is necessary. We show that nested triangles graphs can be drawn in 2n 2/9 + O(n) area when the outer face is not given, improving a previous n 2/3 area upper bound. Further, we show that n 2/9 + Ω(n) area is necessary for any planar straight-line drawing of a nested triangles graph. Finally, we deepen our insight into the 4/9n 2 − 4/3n + 1 lower bound by Dolev, Leighton, and Trickey, which is conjectured to be tight, showing a family of plane graphs requiring more area.


Plane Graph Minimum Area Outer Face Planar Embedding Planar Drawing 
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.


  1. 1.
    Bonichon, N., Felsner, S., Mosbah, M.: Convex drawings of 3-connected plane graphs. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 60–70. Springer, Heidelberg (2005)Google Scholar
  2. 2.
    Brandenburg, F.J., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Liotta, G., Mutzel, P.: Selected open problems in graph drawing. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 515–539. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Chrobak, M., Nakano, S.-I.: Minimum-width grid drawings of plane graphs. Comp. Geom. (11), 29–54 (1998)Google Scholar
  4. 4.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dolev, D., Leighton, F.T., Trickey, H.: Planar embedding of planar graphs. Advances in Computing Research - vol. 2. VLSI Theory (1984)Google Scholar
  6. 6.
    Miura, K., Nakano, S., Nishizeki, T.: Grid drawings of 4-connected plane graphs. Discr. Comp. Geom. (26), 73–87 (2001)Google Scholar
  7. 7.
    de Mendez, P.O.: Some problems: Grid drawings of planar graphs,
  8. 8.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. SODA 1990, pp. 138–148 (1990)Google Scholar
  9. 9.
    Zhang, H., He, X.: Compact visibility representation and straight-line grid embedding of plane graphs. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 493–504. Springer, Heidelberg (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Fabrizio Frati
    • 1
  • Maurizio Patrignani
    • 1
  1. 1.Dipartimento di Informatica e AutomazioneUniversità di Roma Tre 

Personalised recommendations