Advertisement

On the Complexity of Computing the k-restricted Edge-connectivity of a Graph

  • Luis Pedro Montejano
  • Ignasi SauEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

The k -restricted edge-connectivity of a graph G, denoted by \(\lambda _k(G)\), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing \(\lambda _k(G)\). Very recently, in the parameterized complexity community the notion of good edge separation of a graph has been defined, which happens to be essentially the same as the k-restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.

Keywords

Graph cut k-restricted edge-connectivity Good edge separation Parameterized complexity FPT-algorithm polynomial kernel 

Supplementary material

References

  1. 1.
    Balbuena, C., Carmona, A., Fàbrega, J., Fiol, M.A.: Extraconnectivity of graphs with large minimum degree and girth. Discrete Math. 167, 85–100 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Berman, P., Karpinski, M.: Approximation hardness of bounded degree MIN-CSP and MIN-BISECTION. Electron. Colloquium Comput. Complex. 8(26) (2001)Google Scholar
  3. 3.
    Berman, P., Karpinski, M.: Approximation hardness of bounded degree MIN-CSP and MIN-BISECTION. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 623–632. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discrete Math. 28(1), 277–305 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonsma, P., Ueffing, N., Volkmann, L.: Edge-cuts leaving components of order at least three. Discrete Math. 256(1), 431–439 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chitnis, R.H., Cygan, M., Hajiaghayi, M., Pilipczuk, M., Pilipczuk, M.: Designing FPT algorithms for cut problems using randomized contractions. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 460–469 (2012)Google Scholar
  7. 7.
    Cygan, M., Fomin, F., Jansen, B.M., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M.: Open problems from School on Parameterized Algorithms and Complexity (2014). http://fptschool.mimuw.edu.pl/opl.pdf
  8. 8.
    Cygan, M., Kowalik, L., Pilipczuk, M.: Open problems from Update Meeting on Graph Separation Problems (2013). http://worker2013.mimuw.edu.pl/slides/update-opl.pdf
  9. 9.
    Cygan, M., Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Minimum bisection is fixed parameter tractable. In: Proceedings of the 46th ACM Symposium on Theory of Computing (STOC), pp. 323–332 (2014)Google Scholar
  10. 10.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  11. 11.
    Downey, R.G., Estivill-Castro, V., Fellows, M.R., Prieto, E., Rosamond, F.A.: Cutting up is hard to do: the parameterized complexity of \(k\)-cut and related problems. Electron. Notes Theor. Comput. Sci. 78, 209–222 (2003)CrossRefzbMATHGoogle Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dyer, M.E., Frieze, A.M.: On the complexity of partitioning graphs into connected subgraphs. Discrete Appl. Math. 10, 139–153 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dyer, M.E., Frieze, A.M.: Planar 3DM is NP-complete. J. Algorithms 7(2), 174–184 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Esfahanian, A.-H., Hakimi, S.L.: On computing a conditional edge-connectivity of a graph. Inf. Process. Lett. 27(4), 195–199 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fàbrega, J., Fiol, M.A.: Extraconnectivity of graphs with large girth. Discrete Math. 127, 163–170 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  18. 18.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., New York (1979)zbMATHGoogle Scholar
  19. 19.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Holtkamp, A.: Connectivity in Graphs and Digraphs. Maximizing vertex-, edge- and arc-connectivity with an emphasis on local connectivity properties. Ph.D. thesis, RWTH Aachen University (2013)Google Scholar
  21. 21.
    Holtkamp, A., Meierling, D., Montejano, L.P.: \(k\)-restricted edge-connectivity in triangle-free graphs. Discrete Appl. Math. 160(9), 1345–1355 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kawarabayashi, K., Thorup, M.: The minimum \(k\)-way cut of bounded size is fixed-parameter tractable. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), pp. 160–169 (2011)Google Scholar
  23. 23.
    Kim, E.J., Oum, S., Paul, C., Sau, I., Thilikos, D.M.: The List Allocation Problem and Some of its Applications in Parameterized Algorithms. Manuscript submitted for publication (2015). http://www.lirmm.fr/~sau/Pubs/LA.pdf
  24. 24.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Marx, D., Pilipczuk, M.: Everything you always wanted to know about the parameterized complexity of subgraph isomorphism (but were afraid to ask). In: Proceedings of the 31st International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 542–553 (2014)Google Scholar
  26. 26.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and near-optimal derandomization. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), pp. 182–191 (1995)Google Scholar
  27. 27.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  28. 28.
    Stoer, M., Wagner, F.: A simple min-cut algorithm. J. ACM 44(4), 585–591 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    van Bevern, R., Feldmann, A.E., Sorge, M., Suchý, O.: On the parameterized complexity of computing graph bisections. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 76–87. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  30. 30.
    Yuan, J., Liu, A.: Sufficient conditions for \(\lambda _k\)-optimality in triangle-free graphs. Discrete Math. 310, 981–987 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang, Z., Yuan, J.: A proof of an inequality concerning k-restricted edge-connectivity. Discrete Math. 304, 128–134 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Montpellier 2MontpellierFrance
  2. 2.AlGCo project team, CNRS, LIRMMMontpellierFrance

Personalised recommendations