Constant Time Generation of Biconnected Rooted Plane Graphs

  • Bingbing Zhuang
  • Hiroshi Nagamochi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6213)


A plane graph is a drawing of a planar graph in the plane such that no two edges cross each other. In a rooted plane graph, an outer (directed) edge is designated as the root. For a given positive integer n ≥ 1, we give an O(1)-time delay algorithm that enumerates all plane graphs with exactly n vertices using O(n) space. Our algorithm can generates only plane graphs such that the size of each inner face is bounded from above by a prescribed integer g ≥ 3 in the same time and space complexity.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beyer, T., Hedetniemi, S.M.: Constant time generation of rooted trees. SIAM Journal on Computing 9, 706–712 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Fujiwara, H., Wang, J., Zhao, L., Nagamochi, H., Akutsu, T.: Enumerating tree-like chemical graphs with given path frequency. Journal of Chemical Information and Modeling 48, 1345–1357 (2008)CrossRefGoogle Scholar
  3. 3.
    Goldberg, L.A.: Efficient Algorithms for Listing Combinatorial Structures. Cambridge University Press, New York (1993)CrossRefzbMATHGoogle Scholar
  4. 4.
    Hall, L.H., Dailey, E.S.: Design of molecules from quantitative structure-activity relationship models. 3. role of higher order path counts: path 3. J. Chem. Inf. Comp. Sci. 33, 598–603 (1993)CrossRefGoogle Scholar
  5. 5.
    Hopcroft, J.E., Wong, J.K.: Linear time algorithm for isomorphism of planar graphs. In: STOC 1974, pp. 172–184 (1974)Google Scholar
  6. 6.
    Horváth, T., Ramon, J., Wrobel, S.: Frequent subgraph mining in outerplanar graphs. In: Proc. 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 197–206 (2006)Google Scholar
  7. 7.
    Imada, T., Ota, S., Nagamochi, H., Akutsu, T.: Enumerating stereoisomers of tree structured molecules using dynamic programming. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 14–23. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Ishida, Y., Zhao, L., Nagamochi, H., Akutsu, T.: Improved algorithm for enumerating tree-like chemical graphs. In: The 19th International Conference on Genome Informatics, Gold Coast, Australia, December 1- 3 (2008); Genome Informatics 21, 53-64 (2008) Google Scholar
  9. 9.
    Kreher, D.L., Stinson, D.R.: Combinatorial Algorithms. CRC Press, Boca Raton (1998)zbMATHGoogle Scholar
  10. 10.
    Li, Z., Nakano, S.: Efficient generation of plane triangulations without repetitions. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 433–443. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Li, G., Ruskey, F.: The advantage of forward thinking in generating rooted and free trees. In: Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 939–940 (1999)Google Scholar
  12. 12.
    Mauser, H., Stahl, M.: Chemical fragment spaces for de novo design. J. Chem. Inf. Comp. Sci. 47, 318–324 (2007)CrossRefGoogle Scholar
  13. 13.
    McKay, B.D.: Isomorph-free exhaustive generation. J. of Algorithms 26, 306–324 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nakano, S.: Efficient generation of plane trees. Information Processing Letters 84, 167–172 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nakano, S.: Efficient generation of triconnected plane triangulations. Computational Geometry Theory and Applications 27(2), 109–122 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nakano, S., Uno, T.: Efficient generation of rooted trees, NII Technical Report, NII-2003-005 (2003)Google Scholar
  17. 17.
    Nakano, S., Uno, T.: Generating colored trees. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 249–260. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Read, R.C.: How to avoid isomorphism search when cataloguing combinatorial configurations. Annals of Discrete Mathematics 2, 107–120 (1978)CrossRefzbMATHGoogle Scholar
  19. 19.
    Wilf, H.S.: Combinatorial Algorithms: An Update. SIAM, Philadelphia (1989)CrossRefzbMATHGoogle Scholar
  20. 20.
    Wright, R.A., Richmond, B., Odlyzko, A., McKay, B.D.: Constant time generation of free trees. SIAM J. Comput. 15, 540–548 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yamanaka, K., Nakano, S.: Listing all plane graphs. In: Nakano, S.-i., Rahman, M. S. (eds.) WALCOM 2008. LNCS, vol. 4921, pp. 210–221. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Zhuang, B., Nagamochi, H.: Enumerating rooted biconnected planar graphs with internally triangulated faces, Dept. of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Technical Report 2009-018 (2009),
  23. 23.
    Zhuang, B., Nagamochi, H.: Enumerating biconnected rooted plane graphs, Dept. of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Technical Report 2010-001 (2010),

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Bingbing Zhuang
    • 1
  • Hiroshi Nagamochi
    • 1
  1. 1.Graduate School of InformaticsKyoto UniversityJapan

Personalised recommendations