Convex Drawings of Plane Graphs of Minimum Outer Apices

  • Kazuyuki Miura
  • Machiko Azuma
  • Takao Nishizeki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)

Abstract

In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we show that the minimum number of apices of an outer convex polygon necessary for G to have a convex drawing is, in effect, equal to the number of leaves in a triconnected component decomposition tree of a new graph constructed from G, and that a convex drawing of G having the minimum number of apices can be found in linear time.

References

  1. 1.
    Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear algorithms for convex drawings of planar graphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Progress in Graph Theory, pp. 153–173. Academic Press, London (1984)Google Scholar
  2. 2.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. International Journal of Computational Geometry and Applications 7, 211–223 (1997)CrossRefMathSciNetGoogle Scholar
  3. 3.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, NJ (1999)MATHGoogle Scholar
  5. 5.
    Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Compt. 2(3), 135–138 (1973)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Miura, K., Azuma, M., Nishizeki, T.: Canonical decomposition, realizer, Schnyder labeling and orderly spanning trees of plane graphs. International Journal of Fundations of Computer Science 16(1), 117–141 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific, Singapore (2004)MATHGoogle Scholar
  8. 8.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st Annual ACM-SIAM Symp. on Discrete Algorithms, San Francisco, pp. 138–147 (1990)Google Scholar
  9. 9.
    Thomassen, C.: Plane representations of graphs. In: Bondy, J.A., Murty, U.S.R. (eds.) Progress in Graph Theory, pp. 43–69. Academic Press, London (1984)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Kazuyuki Miura
    • 1
  • Machiko Azuma
    • 2
  • Takao Nishizeki
    • 2
  1. 1.Faculty of Symbiotic Systems ScienceFukushima UniversityFukushimaJapan
  2. 2.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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