, Volume 63, Issue 3, pp 645–671 | Cite as

Counting Hexagonal Patches and Independent Sets in Circle Graphs

  • Paul Bonsma
  • Felix Breuer


A hexagonal patch is a plane graph in which inner faces have length 6, inner vertices have degree 3, and boundary vertices have degree 2 or 3. We consider the following counting problem: given a sequence of twos and threes, how many hexagonal patches exist with this degree sequence along the outer face? This problem is motivated by the enumeration of benzenoid hydrocarbons and fullerenes in computational chemistry. We give the first polynomial time algorithm for this problem. We show that it can be reduced to counting maximum independent sets in circle graphs, and give a simple and fast algorithm for this problem. It is also shown how to subsequently generate hexagonal patches.


Counting problem Planar graph Circle graph Fullerene Hexagonal patch Fusene Polyhex 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blank, S.J.: Extending immersions of the circle. Ph.D. thesis, Brandeis University, Waltham, MA, USA (1967) Google Scholar
  2. 2.
    Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time I: counting hexagonal patches. arXiv:0808.3881v1 (2008)
  3. 3.
    Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time. In: Proceedings of the 20th International Symposium on Algorithm and Computation (ISAAC 2009). Lecture Notes in Computer Science, vol. 5878, pp. 750–759. Springer, Berlin (2009) Google Scholar
  4. 4.
    Bornhöft, J., Brinkmann, G., Greinus, J.: Pentagon–hexagon-patches with short boundaries. Eur. J. Comb. 24(5), 517–529 (2003) zbMATHCrossRefGoogle Scholar
  5. 5.
    Bouchet, A.: Reducing prime graphs and recognizing circle graphs. Combinatorica 7(3), 243–254 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (1999) zbMATHCrossRefGoogle Scholar
  7. 7.
    Brinkmann, G., Coppens, B.: An efficient algorithm for the generation of planar polycyclic hydrocarbons with a given boundary. MATCH Commun. Math. Comput. Chem. 62(1), 209–220 (2009) MathSciNetzbMATHGoogle Scholar
  8. 8.
    Brinkmann, G., Dress, A.W.M.: A constructive enumeration of fullerenes. J. Algorithms 23(2), 345–358 (1997) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Brinkmann, G., Nathusius, U. v., Palser, A.H.R.: A constructive enumeration of nanotube caps. Discrete Appl. Math. 116(1–2), 55–71 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brinkmann, G., Delgado-Friedrichs, O., von Nathusius, U.: Numbers of faces and boundary encodings of patches. In: Graphs and Discovery, Proceedings of the DIMACS Workshop on Computer Generated Conjectures from Graph Theoretic and Chemical Databases. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 69, pp. 27–38. AMS, Providence (2005) Google Scholar
  11. 11.
    Brinkmann, G., Graver, J.E., Justus, C.: Numbers of faces in disordered patches. J. Math. Chem. 45(2), 263–278 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Deza, M., Fowler, P.W., Grishukhin, V.: Allowed boundary sequences for fused polycyclic patches and related algorithmic problems. J. Chem. Inf. Comput. Sci. 41, 300–308 (2001) CrossRefGoogle Scholar
  13. 13.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2005) zbMATHGoogle Scholar
  14. 14.
    Dutour Sikirić, M., Deza, M., Shtogrin, M.: Filling of a given boundary by p-gons and related problems. Discrete Appl. Math. 156(9), 1518–1535 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Endo, M., Kroto, H.W.: Formation of carbon nanofibers. J. Phys. Chem. 96, 6941–6944 (1992) CrossRefGoogle Scholar
  16. 16.
    Eppstein, D., Mumford, E.: Self-overlapping curves revisited. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 160–169. SIAM, Philadelphia (2009) Google Scholar
  17. 17.
    Francis, G.K.: Extensions to the disk of properly nested plane immersions of the circle. Mich. Math. J. 17(4), 377–383 (1970) MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gabor, C.P., Supowit, K.J., Hsu, W.L.: Recognizing circle graphs in polynomial time. J. ACM 36(3), 435–473 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Gavril, F.: Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks 3(3), 261–273 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Graver, J.E.: The (m,k)-patch boundary code problem. MATCH Commun. Math. Comput. Chem. 48, 189–196 (2003) MathSciNetzbMATHGoogle Scholar
  21. 21.
    Guo, X., Hansen, P., Zheng, M.: Boundary uniqueness of fusenes. Discrete Appl. Math. 118(3), 209–222 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. John Hopkins University Press, Baltimore (2001) zbMATHGoogle Scholar
  23. 23.
    Nash, N., Lelait, S., Gregg, D.: Efficiently implementing maximum independent set algorithms on circle graphs. ACM J. Exp. Algorithmics 13, 1.9 (2008) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Seidel, R.: The nature and meaning of perturbations in geometric computing. Discrete Comput. Geom. 19(1), 1–17 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Shor, P.W., Van Wyk, C.J.: Detecting and decomposing self-overlapping curves. Comput. Geom. 2(1), 31–50 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Spinrad, J.P.: Recognition of circle graphs. J. Algorithms 16(2), 264–282 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Supowit, K.J.: Finding a maximum planar subset of a set of nets in a channel. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 6(1), 93–94 (1987) CrossRefGoogle Scholar
  28. 28.
    Valiente, G.: A new simple algorithm for the maximum-weight independent set problem on circle graphs. In: Proceedings of the 14th International Symposium on Algorithm and Computation (ISAAC 2003). Lecture Notes in Computer Science, vol. 2906, pp. 129–137. Springer, Berlin (2003) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany

Personalised recommendations