Advertisement

Graphs and Combinatorics

, Volume 22, Issue 2, pp 185–202 | Cite as

Planar Graphs, via Well-Orderly Maps and Trees

  • Nicolas BonichonEmail author
  • Cyril Gavoille
  • Nicolas Hanusse
  • Dominique Poulalhon
  • Gilles Schaeffer
Article

Abstract

The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2 αn + O (log n ), where α≈4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log2p(n)≤4.91n.

The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges.

Finally we obtain as an outcome of our combinatorial analysis an explicit linear-time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge.

Keywords

Planar graph Triangulation Realizer Well-orderly 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bonichon, N., Gavoille, C., Hanusse, N.: An information upper bound of planar graphs using triangulation. Research Report RR-1279-02, LaBRI, University of Bordeaux, 351, cours de la Libération, 33405 Talence Cedex, France, September 2002Google Scholar
  2. 2.
    Bonichon, N., Gavoille, C., Hanusse, N.: An information-theoretic upper bound of planar graphs using triangulation. In: 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), volume 2607 of Lecture Notes in Computer Science, pages 499–510. Springer, February 2003Google Scholar
  3. 3.
    Bodirsky, M., Gröpl, C., Kang, M.: Generating labeled planar graphs uniformly at random. In: 30th International Colloquium on Automata, Languages and Programming (ICALP), volume 2719 of LNCS, pages 1095–1107, 2003Google Scholar
  4. 4.
    Bender, E.A., Gao, Z., Wormald, N.C.: The number of labeled 2-connected planar graphs. The Electronic Journal of Combinatorics 9(1), R43 (2002)Google Scholar
  5. 5.
    Bonichon, N., Le Saëc, B., Mosbah, M.: Optimal area algorithm for planar polyline drawings. In: 28th International Workshop, Graph - Theoretic Concepts in Computer Science (WG), volume 2573 of LNCS, pages 35–46. Springer, 2002Google Scholar
  6. 6.
    Bonichon, N., Le Saëc, B., Mosbah, M.: Wagner's theorem on realizers. In: 29th International Colloquium on Automata, Languages and Programming (ICALP). volume 2380 of LNCS, pages 1043–1053. Springer, 2002Google Scholar
  7. 7.
    Bonichon, N.: A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths. In: Formal Power Series & Algebraic Combinatorics (FPSAC). July 2002Google Scholar
  8. 8.
    Chih-Nan Chuang, R., Garg, A., He X., Kao M.-Y., Lu H.-I.: Compact encodings of planar graphs via canonical orderings and multiple parentheses. In: 25th International Colloquium on Automata, Languages and Programming (ICALP), volume 1443 of LNCS, pages 118–129. Springer, July 1998Google Scholar
  9. 9.
    Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications to graph encoding and graph drawing. In: 12th Symposium on Discrete Algorithms (SODA), pages 506–515. ACM-SIAM, January 2001Google Scholar
  10. 10.
    Denise, A., Vasconcellos, M., Welsh, D. J. A.: The random planar graph. Congressus Numerantium 113, 61–79 (1996)Google Scholar
  11. 11.
    Frederickson, G. N., Janardan, R.: Efficient message routing in planar networks. SIAM Journal on Computing, 18(4), 843–857, August (1989)Google Scholar
  12. 12.
    Flajolet, P., Sedgewick,R.: Analytic combinatorics. Future book available online at the URL http://algo.inria.fr/flajolet/Publications/books.html
  13. 13.
    Gavoille, C., Hanusse, N.: Compact routing tables for graphs of bounded genus. In: 26th International Colloquium on Automata, Languages and Programming (ICALP), volume 1644 of LNCS, pages 351–360. Springer, July 1999Google Scholar
  14. 14.
    Goulden, I. P., Jackson, D. M.: Combinatorial Enumeration. John Wiley & Sons, 1983Google Scholar
  15. 15.
    Gerke, S., McDiarmid, C. J. H.: On the number of edges in random planar graphs. Combinatorics, Probability & Computing, 2002 (to appear)Google Scholar
  16. 16.
    Giménez, O., Noy, M.: Asymptotic enumeration and limit laws of planar graphs. preprint arXiv:Math.CO/051269Google Scholar
  17. 17.
    Khodakovsky, A., Alliez, P., Desbrun, M. Schröder, P.: Near-optimal connectivity encoding of 2-manifold polygon meshes. Graphical Models, 2002. To appear in a special issueGoogle Scholar
  18. 18.
    King, D., Rossignac, J.: Guaranteed 3.67V bit encoding of planar triangle graphs. In: 11th Canadian Conference on Computational Geometry. pp 146–149, August 1999Google Scholar
  19. 19.
    Keeler, K. Westbrook, J.: Short encodings of planar graphs and maps. Discrete Applied Mathematics 58, 239–252 (1995)Google Scholar
  20. 20.
    Lu, H.-I.: Improved compact routing tables for planar networks via orderly spanning trees. In: 8th Annual International Computing & Combinatorics Conference (COCOON), volume 2387 of LNCS, pages 57–66. Springer, August 2002Google Scholar
  21. 21.
    Liskovets, V. A., Walsh, T. R.: Ten steps to counting planar graphs. Congressus Numerantium 60, 269–277 (1987)Google Scholar
  22. 22.
    Munro, J. I., Raman, V.: Succinct representation of balanced parentheses, static trees and planar graphs. In: 38th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 118–126. IEEE Computer Society Press, October 1997Google Scholar
  23. 23.
    McDiarmid, C., Steger, A., Welsh, D. J. A.: Random planar graphs. J. Comb. Theory Ser. B 93(2), 187–205 (2005)Google Scholar
  24. 24.
    Osthus, D., Prömel, H. J., Taraz, A.: On random planar graphs, the number of planar graphs and their triangulations. Journal of Combinatorial Theory, Series B 88, 119–134 (2003)Google Scholar
  25. 25.
    Poulalhon, D., Schaeffer, G.: Optimal coding and sampling of triangulations. In: 30th International Colloquium on Automata, Languages and Programming (ICALP), volume 2719 of LNCS, pages 1080–1094. Springer, July 2003Google Scholar
  26. 26.
    Rossignac, J.: Edgebreaker: Connectivity compression for triangle meshes. IEEE Transactions on Visualization and Computer Graphics 5(1), 47–61 (1999)Google Scholar
  27. 27.
    Schnyder, W.: Embedding planar graphs on the grid. In: 1st Symposium on Discrete Algorithms (SODA), pp 138–148. ACM-SIAM, 1990Google Scholar
  28. 28.
    Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. In 42th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society Press, October 2001.Google Scholar
  29. 29.
    Turán, G.: Succinct representations of graphs. Discrete Applied Mathematics 8, 289–294 (1984)Google Scholar
  30. 30.
    Tutte, W. T.: A census of planar triangulations. Canadian Journal of Mathematics 14, 21–38 (1962)Google Scholar
  31. 31.
    Wright, E. M.: Graphs on unlabelled nodes with a given number of edges. Acta Math. 126, 1–9 (1971)Google Scholar
  32. 32.
    Yannakakis, M.: Embedding planar graphs in four pages. Journal of Computer and System Sciences 38, 36–67 (1989)Google Scholar
  33. 33.
    Zhang, H., He, X.: Compact visibility representation and straight-line grid embedding of plane graphs. In: Workshop on Algorithms and Data Structures (WADS), volume 2748 of LNCS, pages 493–504. Springer, July 2003Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nicolas Bonichon
    • 1
    Email author
  • Cyril Gavoille
    • 1
  • Nicolas Hanusse
    • 1
  • Dominique Poulalhon
    • 2
  • Gilles Schaeffer
    • 3
  1. 1.Laboratoire Bordelais de Recherche en InformatiqueUniversité Bordeaux IFrance
  2. 2.Laboratoire d'Informatique AlgorithmiqueFondements et Applications (LIAFA) case 7014Paris Cedex 05France
  3. 3.Laboratoire d'Informatique de l'École Polytechnique (LIX) École polytechniquePalaiseau CedexFrance

Personalised recommendations