Algorithmica

, Volume 77, Issue 2, pp 309–335 | Cite as

Certifying 3-Edge-Connectivity

Article

Abstract

We present a certifying algorithm that tests graphs for 3-edge-connectivity; the algorithm works in linear time. If the input graph is not 3-edge-connected, the algorithm returns a 2-edge-cut. If it is 3-edge-connected, it returns a construction sequence that constructs the input graph from the graph with two vertices and three parallel edges using only operations that (obviously) preserve 3-edge-connectivity. Additionally, we show how to compute and certify the 3-edge-connected components and a cactus representation of the 2-cuts in linear time. For 3-vertex-connectivity, we show how to compute the 3-vertex-connected components of a 2-connected graph.

Keywords

Certifying algorithm Edge connectivity Construction sequence 

References

  1. 1.
    Alkassar, E., Böhme, S., Mehlhorn, K., Rizkallah, Ch.: A framework for the verification of certifying computations. J. Autom. Reason. 52(3), 241–273 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Corcoran, J.N., Schneider, U., Schüttler, H.-B.: Perfect stochastic summation in high order feynman graph expansions. Int. J. Mod. Phys. C 17(11), 1527–1549 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dehne, F., Langston, M., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: implementations and experiments. In: Parameterized and Exact Computation, pp. 13–24 (2006)Google Scholar
  5. 5.
    Dinits, E.A., Karzanov, A.V., Lomonosov, M.V.: On the structure of a family of minimal weighted cuts in graphs. In: Studies in Discrete Mathematics (in Russian), pp. 290–306 (1976)Google Scholar
  6. 6.
    Fleiner, T., Frank, A.: A quick proof for the cactus representation of mincuts. Technical Report QP-2009-03, Egerváry Research Group, Budapest (2009)Google Scholar
  7. 7.
    Gabow, H.N.: Path-based depth-first search for strong and biconnected components. Inf. Process. Lett. 74(3–4), 107–114 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Galil, Z., Italiano, G.F.: Reducing edge connectivity to vertex connectivity. SIGACT News 22(1), 57–61 (1991)CrossRefGoogle Scholar
  9. 9.
    Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Proceedings of the 8th International Symposium on Graph Drawing (GD’00), pp. 77–90 (2001)Google Scholar
  10. 10.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM J. Comput. 2(3), 135–158 (1973)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Karger, D.R.: Minimum cuts in near-linear time. J. ACM 47(1), 46–76 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Linial, N., Lovász, L., Wigderson, A.: Rubber bands, convex embeddings and graph connectivity. Combinatorica 8(1), 91–102 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lovász, L.: Computing ears and branchings in parallel. In: Proceedings of the 26th Annual Symposium on Foundations of Computer Science (FOCS’85) (1985)Google Scholar
  15. 15.
    Mader, W.: A reduction method for edge-connectivity in graphs. In: Bollobás, B. (ed.) Advances in Graph Theory, vol. 3 of Annals of Discrete Mathematics, pp. 145–164 (1978)Google Scholar
  16. 16.
    McConnell, R.M., Mehlhorn, K., Näher, S., Schweitzer, P.: Certifying algorithms. Comput. Sci. Rev. 5(2), 119–161 (2011)CrossRefMATHGoogle Scholar
  17. 17.
    Mehlhorn, K.: Nearly optimal binary search trees. Acta Inform. 5, 287–295 (1975)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mehlhorn, K., Näher, S., Uhrig, C.: The LEDA Platform of Combinatorial and Geometric Computing. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  19. 19.
    Nagamochi, H., Ibaraki, T.: A linear time algorithm for computing 3-edge-connected components in a multigraph. Jpn. J. Ind. Appl. Math. 9, 163–180 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nagamochi, H., Ibaraki, T.: Algorithmic Aspects of Graph Connectivity (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (2008)CrossRefMATHGoogle Scholar
  21. 21.
    Neumann, A.: Implementation of Schmidt’s algorithm for certifying triconnectivity testing. Master’s thesis, Universität des Saarlandes and Graduate School of CS, Germany (2011)Google Scholar
  22. 22.
    Noschinski, L., Rizkallah, C., Mehlhorn, K.: Verification of certifying computations through Autocorres and Simpl. In: NASA Formal Methods, vol. 8430 of LNCS, pp. 46–61 (2014)Google Scholar
  23. 23.
    Olariu, S., Zomaya, A.Y.: A time- and cost-optimal algorithm for interlocking sets-with applications. IEEE Trans. Parallel Distrib. Syst. 7(10), 1009–1025 (1996)CrossRefGoogle Scholar
  24. 24.
    Ramachandran, V.: Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. In: Synthesis of Parallel Algorithms, pp. 275–340 (1993)Google Scholar
  25. 25.
    Schmidt, J.M.: Contractions, removals and certifying 3-connectivity in linear time. Tech. Report B 10-04, Freie Universität Berlin, Germany (2010)Google Scholar
  26. 26.
    Schmidt, J.M.: Contractions, removals and certifying 3-connectivity in linear time. SIAM J. Comput. 42(2), 494–535 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Schmidt, J.M.: A simple test on 2-vertex- and 2-edge-connectivity. Inf. Process. Lett. 113(7), 241–244 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Taoka, S., Watanabe, T., Onaga, K.: A linear time algorithm for computing all 3-edge-connected components of a multigraph. IEICE Trans. Fundam. E75(3), 410–424 (1992)Google Scholar
  29. 29.
    Tsin, Y.H.: A simple 3-edge-connected component algorithm. Theory Comput. Syst. 40(2), 125–142 (2007)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tsin, Y.H.: Yet another optimal algorithm for 3-edge-connectivity. J. Discrete Algorithms 7(1), 130–146 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Vo, K.-P.: Finding triconnected components of graphs. Linear Multilinear Algebra 13, 143–165 (1983)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Vo, K.-P.: Segment graphs, depth-first cycle bases, 3-connectivity, and planarity of graphs. Linear Multilinear Algebra 13, 119–141 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Kurt Mehlhorn
    • 1
  • Adrian Neumann
    • 1
  • Jens M. Schmidt
    • 1
  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany

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