Generating Labeled Planar Graphs Uniformly at Random

  • Manuel Bodirsky
  • Clemens Gröpl
  • Mihyun Kang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2719)


We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1-, 2-, and 3- connected components. For 3-connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.


Planar Graph Recurrence Formula Outer Face Tuning Ratio Counting Formula 
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.


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  1. 1.
    C. Banderier, P. Flajolet, G. Schaeffer, and M. Soria. Planar maps and Airy phenomena. In ICALP’00, number 1853 in LNCS, pages 388–402, 2000.Google Scholar
  2. 2.
    C. Banderier, P. Flajolet, G. Schaeffer, and M. Soria. Random maps, coalescing saddles, singularity analysis, and Airy phenomena. Random Structures and Algorithms, 19:194–246, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Bender, Z. Gao, and N. Wormald. The number of labeled 2-connected planar graphs. Preprint, 2000.Google Scholar
  4. 4.
    M. Bodirsky and M. Kang. Generating random outerplanar graphs. Presented at ALICE 03, 2003. Journal version submitted.Google Scholar
  5. 5.
    N. Bonichon, C. Gavoille, and N. Hanusse. An information-theoretic upper bound of planar graphs using triangulation. In In 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2003.Google Scholar
  6. 6.
    A. Denise, M. Vasconcellos, and D. Welsh. The random planar graph. Congressus Numerantium, 113:61–79, 1996.zbMATHMathSciNetGoogle Scholar
  7. 7.
    R. Diestel. Graph Theory. Springer-Verlag, New York, 1997.zbMATHGoogle Scholar
  8. 8.
    P. Duchon, P. Flajolet, G. Louchard, and G. Schaeffer. Random sampling from Boltzmann principles. In ICALP’ 02, LNCS, pages 501–513, 2002.Google Scholar
  9. 9.
    S. Gerke and C. McDiarmid. On the number of edges in random planar graphs. Submitted.Google Scholar
  10. 10.
    The GNU multiple precision arithmetic library, version 4.1.2. Scholar
  11. 11.
    I. Köthnig. Personal communication. Humboldt-Universität zu Berlin, 2002.Google Scholar
  12. 12.
    C. McDiarmid, A. Steger, and D. J. Welsh. Random planar graphs. Preprint, 2001.Google Scholar
  13. 13.
    R. Mullin and P. Schellenberg. The enumeration of c-nets via quadrangulations. Journal of Combinatorial Theory, 4:259–276, 1968.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D. Osthus, H. J. Prömel, and A. Taraz. On random planar graphs, the number of planar graphs and their triangulations. Jombinatorial Theory, Series B, to appear.Google Scholar
  15. 15.
    G. Schaeffer. Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis, Université Bordeaux I, 1998.Google Scholar
  16. 16.
    G. Schaeffer. Random sampling of large planar maps and convex polyhedra. In Proc. of the thirty-first annual ACM symposium on theory of computing (STOC’99), pages 760–769, Atlanta, Georgia, May 1999. ACM press.Google Scholar
  17. 17.
    G. Schaeffer. Personal communication, 2002.Google Scholar
  18. 18.
    N. J. A. Sloane. The on-line encyclopedia of integer sequences., 2002.Google Scholar
  19. 19.
    B. A. Trakhtenbrot. Towards a theory of non-repeating contact schemes. Trudi Mat. Inst. Akad. Nauk SSSR, 51:226–269, 1958. [In Russian].Google Scholar
  20. 20.
    W. Tutte. A census of planar maps. Canad. J. Math., 15:249–271, 1963.zbMATHMathSciNetGoogle Scholar
  21. 21.
    T. Walsh. Counting labelled three-connected and homeomorphically irreducible two-connected graphs. J. Combin. Theory, 32:1–11, 1982.zbMATHGoogle Scholar
  22. 22.
    T. Walsh. Counting nonisomorphic three-connected planar maps. J. Combin. Theory, 32:33–44, 1982.zbMATHCrossRefGoogle Scholar
  23. 23.
    T. Walsh. Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs. J. Combin. Theory, 32:12–32, 1982.zbMATHCrossRefGoogle Scholar
  24. 24.
    T. Walsh and V. A. Liskovets. Ten steps to counting planar graphs. In Eighteenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing, Congr. Numer., volume 60, pages 269–277, 1987.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Manuel Bodirsky
    • 1
  • Clemens Gröpl
    • 2
  • Mihyun Kang
    • 1
  1. 1.Humboldt-Universität zu BerlinGermany
  2. 2.Freie Universität BerlinGermany

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