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Tight Lower Bounds for the Number of Inclusion-Minimal st-Cuts

  • Alessio Conte
  • Roberto Grossi
  • Andrea Marino
  • Romeo Rizzi
  • Takeaki Uno
  • Luca VersariEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)

Abstract

We study the number of inclusion-minimal cuts in an undirected connected graph G, also called \(st\)-cuts, for any two distinct nodes s and t: the \(st\)-cuts are in one-to-one correspondence with the partitions \(S \cup T\) of the nodes of G such that \(S \cap T = \emptyset \), \(s \in S\), \(t \in T\), and the subgraphs induced by S and T are connected. It is easy to find an exponential upper bound to the number of \(st\)-cuts (e.g. if G is a clique) and a constant lower bound. We prove that there is a more interesting lower bound on this number, namely, \(\varOmega (m)\), for undirected m-edge graphs that are biconnected or triconnected (2- or 3-node-connected). The wheel graphs show that this lower bound is the best possible asymptotically.

Notes

Acknowledgements

This work was partially supported by JST CREST, grant number JPMJCR1401, Japan, and MIUR, Italy.

References

  1. 1.
    Abel, U., Bicker, R.: Determination of all minimal cut-sets between a vertex pair in an undirected graph. IEEE Trans. Reliab. 31(2), 167–171 (1982)CrossRefGoogle Scholar
  2. 2.
    Ball, M.O., Provan, J.S.: Calculating bounds on reachability and connectedness in stochastic networks. Networks 13(2), 253–278 (1983)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bellmore, M., Jensen, P.A.: An implicit enumeration scheme for proper cut generation. Technometrics 12(4), 775–788 (1970)CrossRefGoogle Scholar
  4. 4.
    Berge, C.: La theorie des graphes. Presses Universitaires de France, Paris (1958)zbMATHGoogle Scholar
  5. 5.
    Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bixby, R.E.: The minimum number of edges and vertices in a graph with edge connectivity n and m n-bonds. Networks 5(3), 253–298 (1975)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brecht, T.B., Colbourn, C.J.: Lower bounds on two-terminal network reliability. Discrete Appl. Math. 21(3), 185–198 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chandran, L.S., Ram, L.S.: On the number of minimum cuts in a graph. SIAM J. Discrete Math. 18(1), 177–194 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Shimon Even and Robert Endre Tarjan: Computing an st-numbering. Theor. Comput. Sci. 2(3), 339–344 (1976)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gardner, M.L.: Algorithm to aid in the design of large scale networks. Large Scale Syst. 8(2), 147–156 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Goldberg, L.A.: Efficient Algorithms for Listing Combinatorial Structures, vol. 5. Cambridge University Press, Cambridge (2009)zbMATHGoogle Scholar
  12. 12.
    Hamacher, H.W., Picard, J.-C., Queyranne, M.: On finding the K best cuts in a network. Oper. Res. Lett. 2(6), 303–305 (1984)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Harada, H., Sun, Z., Nagamochi, H.: An exact lower bound on the number of cut-sets in multigraphs. Networks 24(8), 429–443 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Harary, F.: The maximum connectivity of a graph. Proc. Nat. Acad. Sci. 48(7), 1142–1146 (1962)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jasmon, G.B., Foong, K.W.: A method for evaluating all the minimal cuts of a graph. IEEE Trans. Reliab. 36(5), 539–545 (1987)CrossRefGoogle Scholar
  16. 16.
    Katona, G.: A theorem for finite sets. In: Theory of Graphs, pp. 187–207 (1968)Google Scholar
  17. 17.
    Khachiyan, L., Boros, E., Borys, K., Elbassioni, K., Gurvich, V., Makino, K.: Generating cut conjunctions in graphs and related problems. Algorithmica 51(3), 239–263 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kruskal, J.B.: The number of simplices in a complex. Math. Optim. Tech. 10, 251–278 (1963)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Martelli, A.: A Gaussian elimination algorithm for the enumeration of cut sets in a graph. J. ACM 23(1), 58–73 (1976)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Picard, J.-C., Queyranne, M.: On the structure of all minimum cuts in a network and applications. Math. Program. 22(1), 121–121 (1982)CrossRefGoogle Scholar
  21. 21.
    Prasad, V.C., Sankar, V., Rao, K.S.P.: Generation of vertex and edge cutsets. Microelectron. Reliab. 32(9), 1291–1310 (1992)CrossRefGoogle Scholar
  22. 22.
    Provan, J.S., Ball, M.O.: Computing network reliability in time polynomial in the number of cuts. Oper. Res. 32(3), 516–526 (1984)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Provan, J.S., Shier, D.R.: A paradigm for listing (s, t)-cuts in graphs. Algorithmica 15(4), 351–372 (1996)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pierre Rosenstiehl and Robert Endre Tarjan: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shier, D.R., Whited, D.E.: Iterative algorithms for generating minimal cutsets in directed graphs. Networks 16(2), 133–147 (1986)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Tamassia, R., Tollis, I.G.: A unified approach a visibility representation of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Tsukiyama, S., Shirakawa, I., Ozaki, H., Ariyoshi, H.: An algorithm to enumerate all cutsets of a graph in linear time per cutset. J. ACM (JACM) 27(4), 619–632 (1980)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Li, Y., Taha, H.A., Landers, T.L.: A recursive approach for enumerating minimal cutsets in a network. IEEE Trans. Reliab. 43(3), 383–388 (1994)CrossRefGoogle Scholar
  29. 29.
    Yeh, L.-P., Wang, B.-F., Hsin-Hao, S.: Efficient algorithms for the problems of enumerating cuts by non-decreasing weights. Algorithmica 56(3), 297–312 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Alessio Conte
    • 1
  • Roberto Grossi
    • 2
  • Andrea Marino
    • 2
  • Romeo Rizzi
    • 3
  • Takeaki Uno
    • 1
  • Luca Versari
    • 2
    Email author
  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.Università di PisaPisaItaly
  3. 3.Università di VeronaVeronaItaly

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