Theory of Computing Systems

, Volume 59, Issue 3, pp 416–439 | Cite as

Counting the Number of Perfect Matchings in K5-Free Graphs

Article

Abstract

Counting the number of perfect matchings in graphs is a computationally hard problem. However, in the case of planar graphs, and even for K3,3-free graphs, the number of perfect matchings can be computed efficiently. The technique to achieve this is to compute a Pfaffian orientation of a graph. In the case of K5-free graphs, this technique will not work because some K5-free graphs do not have a Pfaffian orientation. We circumvent this problem and show that the number of perfect matchings in K5-free graphs can be computed in polynomial time. We also parallelize the sequential algorithm and show that the problem is in TC2. We remark that our results generalize to graphs without singly-crossing minor.

Keywords

Perfect matching Counting Complexity 

References

  1. 1.
    Barahona, F.: Balancing signed toroidal graphs in polynomial time. Technical report, University of Chile (1983)Google Scholar
  2. 2.
    Di Battista, G., Tamassia, R.: Incremental planarity testing. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 436–441 (1989)Google Scholar
  3. 3.
    Di Battista, G., Tamassia, R.: On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15(4), 302–318 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cayley, A.: Sur les déterminants gauches. J. Pure Appl. Math. 38, 93–96 (1847)MathSciNetGoogle Scholar
  5. 5.
    Curticapean, R.: Counting perfect matchings in graphs that exclude a single-crossing minor. arXiv:1406.4056 (2014)
  6. 6.
    Demaine, E.D., Hajiaghayi, M., Nishimura, N., Ragde, P., Thilikos, D.M.: Approximation algorithms for classes of graphs excluding single-crossing graphs as minors. J. Comput. Syst. Sci. 69(2), 166–195 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Datta, S., Nimbhorkar, P., Thierauf, T., Wagner, F.: Isomorphism for K 3,3-free and K 5-free graphs is in log-space. In: Proceedings of the 29th Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 145–156 (2009)Google Scholar
  8. 8.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Elberfeld, M., Jakoby, A., Tantau, T.: Logspace versions of the theorems of Bodlaender and Courcelle. In: 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 143–152 (2010)Google Scholar
  10. 10.
    Galbiati, G., Maffioli, F.: On the computation of pfaffians. Discrete Appl. Math. 51(3), 269–275 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gibbons, A., Rytter, W.: Efficient Parallel Algorithms. Cambridge University Press, Cambridge (1988)MATHGoogle Scholar
  12. 12.
    Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)MATHGoogle Scholar
  13. 13.
    Hopcroft, J.E., Tarjan, R.E.: A \(V\log V\) algorithm for isomorphism of triconnected planar graphs. J. Comput. Syst. Sci. 7 (3), 323–331 (1973)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kasteleyn, P.W.: Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics, pp 43–110. Academic, New York (1967)Google Scholar
  15. 15.
    Kulkarni, R., Mahajan, M., Varadarajan, K.: Some perfect matchings and perfect half-integral matchings in NC. Chic. J. Theor. Comput. Sci. 2008(4), 1–26 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kuratowski, K.: Sur le probléme des courbes gauches en topologie. Fundam. Math. 15, 271–283 (1930)MATHGoogle Scholar
  17. 17.
    Little, C.H.C.: An extension of Kasteleyn’s method of enumerating the 1-factors of planar graphs. In: Holton, D.A. (ed.) Combinatorial Mathematics, volume 403 of Lecture Notes in Mathematics, pp 63–72. Springer, Berlin Heidelberg (1974)Google Scholar
  18. 18.
    Miller, G.L., Ramachandran, V.: A new graph triconnectivity algorithm and its parallelization. Combinatorica 12, 53–76 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mahajan, M., Subramanya, P.R., Vinay, V.: The combinatorial approach yields an NC algorithm for computing Pfaffians. Discrete Appl. Math. 143(1–3), 1–16 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Robertson, N., Seymour, P.: Excluding a graph with one crossing. In: Graph Structure Theory, pp. 669–675. American Mathematical Society (1993)Google Scholar
  21. 21.
    Tutte, W.T.: Connectivity in Graphs. University of Toronto Press, Toronto (1966)MATHGoogle Scholar
  22. 22.
    Thierauf, T., Wagner, F.: Reachability in K 3,3-free graphs and K 5-free graphs is in unambiguous log-space. Chic. J. Theor. Comput. Sci., To appear (2014)Google Scholar
  23. 23.
    Valiant, L.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Valiant, L.: Holographic algorithms. SIAM J. Comput. 37(5), 1565–1594 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Vazirani, V.: NC algorithms for computing the number of perfect matchings in K 3,3-free graphs and related problems. Inf. Comput. 80(2), 152–164 (1989)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Vollmer, H.: Introduction to Circuit Complexity. Springer, Berlin (1999)CrossRefMATHGoogle Scholar
  27. 27.
    Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114 (1), 570–590 (1937)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Ulm UniversityUlmGermany
  2. 2.Aalen UniversityAalenGermany

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