NC algorithms for computing the number of perfect matchings in K3,3-free graphs and related problems

  • Vijay V. Vazirani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 318)


We show that the problem of computing the number of perfect matchings in K3,3-free graphs is in NC. This stands in striking contrast with the #P-completeness of counting the number of perfect matchings in arbitrary graphs. As corollaries we obtain NC algorithms for checking if a given K3,3-free graph has a perfect matching and if it has an EXACT MATCHING. Our result also opens up the possibility of obtaining an NC algorithm for finding a perfect matching in K3,3-free graphs.


Planar Graph Perfect Match Parallel Algorithm Polynomial Time Algorithm Decomposition Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [As]
    T. Asano. An Approach to the Subgraph Homeomorphism Problem. Theor. Comput. Sci. 38 (1985), 249–267.Google Scholar
  2. [Be]
    C. Berge. Graphs and Hypergraphs, North Holland Publishing Co., Amsterdam, 1973.Google Scholar
  3. [BCP]
    A. Borodin, S.A. Cook, and N. Pippinger. Parallel Computation for Well-endowed Rings and Space Bounded Probabilistic Machines. Information and Control 58, 1–3 (1983), 113–136.Google Scholar
  4. [Cs]
    L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM J. Computing 5 (1976), 618–623.Google Scholar
  5. [GK]
    D. Grigoriev and M. Karpinski. The Matching Problem for Bipartite Graphs with Polynomially Bounded Permanents is in NC. to appear.Google Scholar
  6. [Ha]
    D.W. Hall. A note on primitive skew curves. Bull. Amer. Math. Soc., 49 (1943), 935–937.Google Scholar
  7. [HT]
    J.E. Hopcroft and R.E. Tarjan. Dividing a Graph into Triconnected Components. SIAM J. Computing 2,3 (Sept. 1973), 135–151.Google Scholar
  8. [Jo]
    D.S. Johnson. The NP-Completeness Column: An Ongoing Guide. J. of Algorithms (June 1987).Google Scholar
  9. [JS]
    J. JaJa and J. Simon. Parallel Algorithms in Graph Theory: Planarity Testing. SIAM J. Comput. 11 (1982), 314–328.Google Scholar
  10. [KaRa]
    A. Kanevsky and V. Ramachandran. Improved Algorithms for Graph Four-Connectivity. Proceedings of FOCS Conference (1987).Google Scholar
  11. [KUW]
    R. M. Karp, E. Upfal, and A. Wigderson. Constructing a Maximum Matching is in Random NC. Combinatorica 6, 1 (1986), 35–48.Google Scholar
  12. [Ka]
    P. W. Kasteleyn. Graph Theory and Crystal Physics. Graph Theory and Theoretical Physics, Ed.: F. Harary, Academic Press, NY (1967), 43–110.Google Scholar
  13. [Kl]
    S. Khuller. Extending Planar Graph Algorithms to K 3,3-free Graphs. Technical Report No. 88-902, Department of Computer Science, Cornell University (1988).Google Scholar
  14. [KR]
    P.N. Klein and J.H. Reif. An Efficient Parallel Algorithm for Planarity. Proceedings of FOCS Conference 1986, 465–477.Google Scholar
  15. [KVV]
    D. Kozen, U.V. Vazirani, and V.V. Vazirani. NC Algorithms for Comparability Graphs, Interval Graphs and Testing for Unique Perfect Matching. Fifth Annual Foundations of Software Technology and Theoretical Computer Science Conference (1985), to appear in Theoretical Computer Science.Google Scholar
  16. [Li]
    C. H. C. Little. An extension of Kasteleyn's method of enumerating the 1-factors of planar graphs. Combinatorial Mathematics, Proc. Second Australian Conference, Ed.: D. Holton, Lecture Notes in Math. 403, Springer-Verlag Berlin (1974), 63–72.Google Scholar
  17. [LP]
    L. Lovasz and M. Plummer. Matching Theory, Academic Press, Budapest, Hungary.Google Scholar
  18. [MR]
    G.L. Miller and V. Ramachandran. A New Graph Triconnectivity Algorithm and its Parallelization. Proceedings of STOC Conference (1987), 335–344.Google Scholar
  19. [MVV]
    K. Mulmuley, U. V. Vazirani, and V. V. Vazirani. Matching Is as Easy as Matrix Multiplication. Combinatorica 7, 1 (1987), 105–113.Google Scholar
  20. [Pa]
    V. Pan. Fast and Efficient Algorithms for the Exact Inversion of Matrices. Fifth Annual Foundations of Software Technology and Theoretical Computer Science Conference, (1985).Google Scholar
  21. [PY]
    C.H. Papadimitriou and M. Yannakakis. The Complexity of Restricted Spanning Tree Problems. JACM 33, 1 (Aug 1986), 75–87Google Scholar
  22. [Tu1]
    W.T. Tutte. The Factorization of Linear Graphs. J. London Math. Soc. 22 (1947), 107–111.Google Scholar
  23. [Tu2]
    W.T. Tutte. Connectivity in Graphs, University of Toronto Press. 1966 Google Scholar
  24. [Va]
    L. G. Valiant. The Complexity of Computing the Permanent. Theoretical Computer Science 8 (1979), 189–201.Google Scholar
  25. [VY]
    V.V. Vazirani and M. Yannakakis. Pfaffian Orientations, 0/1 Permanents, and Even Cycles in Directed Graphs. Proceedings of ICALP Conference (1988), to appear in Discrete Applied Mathematics.Google Scholar
  26. [Wa]
    K. Wagner. ‘Bemerkung zu Hadwigers Vermutung. Math. Ann. 141 (1960), 433–451.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Vijay V. Vazirani
    • 1
  1. 1.Computer Science DepartmentCornell UniversityUSA

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