Representing and enumerating edge connectivity cuts in RNC

  • Dalit Naor
  • Vijay V. Vazirani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 519)


An undirected edge-weighted graph can have at most (\(\left( {\begin{array}{*{20}c}n \\2 \\\end{array} } \right)\)) edge connectivity cuts. A succinct and algorithmically useful representation for this set of cuts was given by [DKL], and an efficient sequential algorithm for obtaining it was given by [KT]. In this paper, we present a fast parallel algorithm for obtaining this representation; our algorithm is an RNC algorithm in case the weights are given in unary. We also observe that for a unary weighted graph, the problems of counting and enumerating the connectivity cuts are in RNC.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ADK]
    G.M. Adelson-Velskii, E.A. Dinits, A.V. Karzanov, Flow Algorithms [In Russian], Nauka, Moscow, 1976.Google Scholar
  2. [B]
    R.E. Bixby, The Minimum Number of Edges and Vertices in a Graph with Edge Connectivity n and m n-Bonds, Networks, Vol. 5, pp. 253–298, 1975.Google Scholar
  3. [BP]
    M.O. Ball, J.S. Provan, Calculating Bounds on Reachability in Computer Networks, Networks, Vol. 18, pp. 1–12, 1988.Google Scholar
  4. [DKL]
    E.A. Dinits, A.V. Karzanov, M.V. Lomosonov, On the Structure of a Family of Minimal Weighted Cuts in a Graph, Studies in Discrete Optimization [In Russian], A.A. Fridman (Ed), Nauka, Moscow, pp. 290–306, 1976.Google Scholar
  5. [EH]
    A.H. Esfahanian, S.L. Hakimi, On Computing the Connectivities of Graphs and Digraphs, Networks, Vol. 14 (1984), pp. 355–366.Google Scholar
  6. [ET]
    S. Even, R.E. Tarjan, Network Flow and Testing Graph Connectivity, Siam J. Computing, Vol. 4, No. 4, pp. 507–518, 1975.Google Scholar
  7. [Ga]
    H. Gabow, A Matroid Approach to Finding Edge Connectivity and Packing Arborescences, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, New Orleans, pp. 112–122, 1991.Google Scholar
  8. [GN]
    D. Gusfield, D. Naor, Extracting Maximal Information about Sets of Minimum Cuts, Tech. Report CSE-88-14, UC Davis.Google Scholar
  9. [K1]
    A. Kanevsky, Graphs with Odd and Even Edge Connectivity are Inherently Different, Tech. Report TAMU-89-10, June 1989.Google Scholar
  10. [K2]
    A. Kanevsky, On the Number of Minimum Size Separating Vertex Sets in a Graph and How to Find All of Them, Proc. of the 1st Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, January 1990.Google Scholar
  11. [KT]
    A.V. Karzanov, E.A. Timofeev, Efficient Algorithm for Finding all Minimal Edge Cuts of a Nonoriented Graph, Cybernetics, (1986) pp. 156–162, Translated from Kibernetika, No. 2, pp. 8–12, March–April 1986.Google Scholar
  12. [KUW]
    R.M. Karp, E. Upfal, A. Wigderson, Constructing a Perfect Matching is in Random NC, Combinatorica, 6(1) 1986, pp. 35–48.Google Scholar
  13. [M]
    D. Matula, Determining Edge Connectivity in O(nm), Proc. of the 28th Annual IEEE Symposium on Foundations of Computer Science, Los-Angeles, pp. 249–251, 1987.Google Scholar
  14. [MVV]
    K. Mulmuley, U.V. Vazirani, V.V. Vazirani, Matching is As Easy As Matrix Inversion, Combinatorica, 7(1) 1987, pp. 105–113.Google Scholar
  15. [NGM]
    D. Naor, D. Gusfield, C. Martel, A Fast Algorithm for Optimally Increasing the Edge-Connectivity, Proc. of the 31th Annual IEEE Symposium on Foundations of Computer Science, St. Louis, pp. 698–707, 1990.Google Scholar
  16. [PQ1]
    J.C. Picard, M. Queyranne, On the Structure of All Minimum Cuts in a Network and Applications, Mathematical Programming Study 13 (1980), 8–16.Google Scholar
  17. [PQ2]
    J.C. Picard, M. Queyranne, Selected Applications of Minimum Cuts in Networks, INFOR — Can. J. Oper. Res. Inf. Process. 20 (1982), 394–422.Google Scholar
  18. [Po]
    V.D. Podderyugin, An Algorithm for Finding the Edge Connectivity of Graphs, Vopr. Kibern., no. 2, 136 (1973).Google Scholar
  19. [PS]
    J.S. Provan, D.R. Shier, A Paradigm for Listing (s,t)-Cuts in Graphs, Technical Report UNC/OR TR91-3, Department of Operations Research, University of North Carolina at Chapel Hill, February 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Dalit Naor
    • 1
  • Vijay V. Vazirani
    • 2
  1. 1.Division of Computer ScienceUniversity of California at DavisDavis
  2. 2.Dept. of Computer Science and Engg.Indian Institute of TechnologyNew DelhiIndia

Personalised recommendations