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)

Abstract

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.

References

  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)MATHCrossRefMathSciNetGoogle 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, http://www.ehess.fr/centres/cams/person/pom/langen/openpb.html
  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