Finding Fullerene Patches in Polynomial Time

  • Paul Bonsma
  • Felix Breuer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)


We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2-connected plane graph G in which inner faces have length 5 or 6, non-boundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is called the boundary code of G. We show that the question whether a given sequence S is a boundary code of some fullerene patch can be answered in polynomial time when such patches have at most five 5-faces. We conjecture that our algorithm gives the correct answer for any number of 5-faces, and sketch how to extend the algorithm to the problem of counting the number of different patches with a given boundary code.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bonsma, P., Breuer, F.: Counting hexagonal patches and independent sets in circle graphs (2009),
  2. 2.
    Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time (2009),
  3. 3.
    Bornhöft, J., Brinkmann, G., Greinus, J.: Pentagon–hexagon-patches with short boundaries. European J. Combin. 24(5), 517–529 (2003)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brinkmann, G., Coppens, B.: An efficient algorithm for the generation of planar polycyclic hydrocarbons with a given boundary. MATCH Commun. Math. Comput. Chem. (2009)Google Scholar
  5. 5.
    Brinkmann, G., Dress, A.W.M.: A constructive enumeration of fullerenes. J. Algorithms 23(2), 345–358 (1997)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Brinkmann, G., Fowler, P.W.: A catalogue of growth transformations of fullerene polyhedra. J. Chem. Inf. Comput. Sci. 43, 1837–1843 (2003)Google Scholar
  7. 7.
    Brinkmann, G., Nathusius, U.v., Palser, A.H.R.: A constructive enumeration of nanotube caps. Discrete Appl. Math. 116(1-2), 55–71 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    Diestel, R.: Graph theory, 3rd edn. Springer, Berlin (2005)MATHGoogle Scholar
  10. 10.
    Endo, M., Kroto, H.W.: Formation of carbon nanofibers. J. Phys. Chem. 96, 6941–6944 (1992)CrossRefGoogle Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
  12. 12.
    Graver, J.E.: The (m,k)-patch boundary code problem. MATCH Commun. Math. Comput. Chem. 48, 189–196 (2003)MATHMathSciNetGoogle Scholar
  13. 13.
    Guo, X., Hansen, P., Zheng, M.: Boundary uniqueness of fusenes. Discrete Appl. Math. 118, 209–222 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Paul Bonsma
    • 1
  • Felix Breuer
    • 2
  1. 1.Computer Science DepartmentHumboldt Universität zu BerlinBerlin
  2. 2.Mathematics DepartmentFreie Universität BerlinBerlin

Personalised recommendations