Counting Hexagonal Patches and Independent Sets in Circle Graphs

  • Paul Bonsma
  • Felix Breuer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6034)


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.


graph algorithms computational complexity counting problem planar graph circle graph fullerene hexagonal patch fusene polyhex 


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  1. 1.
    Blank, S.J.: Extending immersions of the circle. PhD thesis, Brandeis University (1967)Google Scholar
  2. 2.
    Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time I: Counting hexagonal patches (2008),
  3. 3.
    Bonsma, P., Breuer, F.: Finding fullerene patches in polynomial time. In: Dong, Y., Du, D.-Z., Ibarra, O.H. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 750–759. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bouchet, A.: Reducing prime graphs and recognizing circle graphs. Combinatorica 7(3), 243–254 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes, a survey. SIAM, Philadelphia (1999)zbMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    Brinkmann, G., Delgado-Friedrichs, O., von Nathusius, U.: Numbers of faces and boundary encodings of patches. In: Graphs and discovery. DIMACS Ser. Discrete Math. Theoret. Comput. Sci, vol. 69, pp. 27–38. Amer. Math. Soc., Providence (2005)Google Scholar
  8. 8.
    Brinkmann, G., Dress, A.W.M.: A constructive enumeration of fullerenes. J. Algorithms 23(2), 345–358 (1997)CrossRefMathSciNetGoogle 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)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Diestel, R.: Graph theory, 3rd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  12. 12.
    Dutour Sikirić, M., Deza, M., Shtogrin, M.: Filling of a given boundary by p-gons and related problems. Discrete Appl. Math. 156, 1518–1535 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Endo, M., Kroto, H.W.: Formation of carbon nanofibers. J. Phys. Chem. 96, 6941–6944 (1992)CrossRefGoogle Scholar
  14. 14.
    Eppstein, D., Mumford, E.: Self-overlapping curves revisited. In: SODA 2009, pp. 160–169. SIAM, Philadelphia (2009)Google Scholar
  15. 15.
    Francis, G.K.: Extensions to the disk of properly nested plane immersions of the circle. Michigan Math. J. 17(4), 377–383 (1970)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Gabor, C.P., Supowit, K.J., Hsu, W.L.: Recognizing circle graphs in polynomial time. J. ACM 36(3), 435–473 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gavril, F.: Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks 3(3), 261–273 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Graver, J.E.: The (m,k)-patch boundary code problem. MATCH Commun. Math. Comput. Chem. 48, 189–196 (2003)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Guo, X., Hansen, P., Zheng, M.: Boundary uniqueness of fusenes. Discrete Appl. Math. 118, 209–222 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nash, N., Lelait, S., Gregg, D.: Efficiently implementing maximum independent set algorithms on circle graphs. ACM J. Exp. Algorithmics 13 (2009)Google Scholar
  21. 21.
    Seidel, R.: The nature and meaning of perturbations in geometric computing. Discrete Comput. Geom. 19(1), 1–17 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Shor, P.W., Van Wyk, C.J.: Detecting and decomposing self-overlapping curves. Comput. Geom. 2, 31–50 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Spinrad, J.P.: Recognition of circle graphs. J. Algorithms 16(2), 264–282 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Supowit, K.J.: Finding a maximum planar subset of a set of nets in a channel. IEEE T. Comput. Aid. D. 6(1), 93–94 (1987)CrossRefGoogle Scholar
  25. 25.
    Valiente, G.: A new simple algorithm for the maximum-weight independent set problem on circle graphs. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 129–137. Springer, Heidelberg (2003)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Paul Bonsma
    • 1
  • Felix Breuer
    • 2
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Institut für MathematikFreie Universität BerlinBerlinGermany

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