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A faster edge splitting algorithm in multigraphs and its application to the edge-connectivity augmentation problem

  • Hiroshi Nagamochi
  • Toshihide Ibaraki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 920)

Abstract

This paper first shows that, given a multigraph G and a vertex s with even degree, all edges incident to s can be split off (i.e., if G is k-edge-connected, then the resulting multigraph is also k-edge-connected) in O(mn2+n2 log n) time, where n and m are the numbers of vertices and edges in G, respectively. This algorithm is unique in the sense that it does not rely on the maximum flow computations. Based on this, we then show that, given a positive integer k, the problem of making a multigraph Gk-edge-connected by adding the smallest number of new edges can be solved in O(m+minen2+n3 log n, kn3) time, where e (≤n2) is the number of pairs of vertices between which G has an edge.

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References

  1. 1.
    A. A. Benczúr, Augmenting undirected connectivity in Õ(n 3) time, Proceedings 26th ACM Symposium on Theory of Computing, 1994, pp. 658–667.Google Scholar
  2. 2.
    G.-R. Cai and Y.-G. Sun, The minimum augmentation of any graph to k-edge-connected graph, Networks, Vol. 19, 1989, pp. 151–172.Google Scholar
  3. 3.
    A. Frank, Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Mathematics, Vol. 5, 1992, pp. 25–53.Google Scholar
  4. 4.
    A. Frank, T. Ibaraki and H. Nagamochi, On sparse subgraphs preserving connectivity properties, J. Graph Theory, Vol. 17, 1993, pp. 275–281.Google Scholar
  5. 5.
    H.N. Gabow, Applications of a poset representation to edge connectivity and graph rigidity, Proc. 32nd IEEE Symp. Found. Comp. Sci., San Juan, Puerto Rico (1991), pp. 812–821.Google Scholar
  6. 6.
    H.N. Gabow, Efficient splitting off algorithms for graphs, Proceedings 26th ACM Symposium on Theory of Computing, 1994, pp. 696–705.Google Scholar
  7. 7.
    L. Lovász, Combinatorial Problems and Exercises, North-Holland 1979.Google Scholar
  8. 8.
    H. Nagamochi and T. Ibaraki, Computing edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Mathematics, Vol. 5, 1992, pp. 54–66.Google Scholar
  9. 9.
    H. Nagamochi, K. Nishimura and T. Ibaraki, Computing all small cuts in undirected networks, Lectures Notes in Computer Science 834, Springer-Verlag, Ding-Zhu Du and Xiang-Sun Zhang (Eds.), Algorithms and Computation, 5th International Symposium, ISAAC'94 Beijin, Aug. 1994, pp. 190–198.Google Scholar
  10. 10.
    D. Naor, D. Gusfield and C. Martel, A fast algorithm for optimally increasing the edge connectivity, Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science, 1990, pp. 698–707.Google Scholar
  11. 11.
    T. Watanabe and A. Nakarnura, Edge-connectivity augmentation problems, Comp. System Sci., Vol. 35, 1987, pp. 96–144.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Hiroshi Nagamochi
    • 1
  • Toshihide Ibaraki
    • 1
  1. 1.Kyoto UniversityKyotoJapan

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