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max-cut and Containment Relations in Graphs

  • Marcin Kamiński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

We study max-cut in classes of graphs defined by forbidding a single graph as a subgraph, induced subgraph, or minor. For the first two containment relations, we prove dichotomy theorems. For the minor order, we show how to solve max-cut in polynomial time for the class obtained by forbidding a graph with crossing number at most one (this generalizes a known result for K 5-minor-free graphs) and identify an open problem which is the missing case for a dichotomy theorem.

Keywords

max-cut subgraph induced subgraph minor 

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References

  1. 1.
    Vondrak, J., Galluccio, A., Loebl, M.: Optimization via enumeration: a new algorithm for the max cut problem. Mathematical Programming 90(2), 273–290 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekseev, V.E.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discrete Applied Mathematics 132(1-3), 17–26 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alekseev, V.E., Boliac, R., Korobitsyn, D.V., Lozin, V.V.: Np-hard graph problems and boundary classes of graphs. Theor. Comput. Sci. 389(1-2), 219–236 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alekseev, V.E., Korobitsyn, D.V., Lozin, V.V.: Boundary classes of graphs for the dominating set problem. Discrete Mathematics 285(1-3), 1–6 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arbib, C.: A polynomial characterization of some graph partitioning problems. Inf. Process. Lett. 26(5), 223–230 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barahona, F.: The Max-Cut problem on graphs not contractible to K. Oper. Res. Lett. 2(3), 107–111 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bienstock, D., Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a forest. J. Comb. Theory, Ser. B 52(2), 274–283 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bodlaender, H.L., Jansen, K.: On the complexity of the maximum cut problem. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds.) STACS 1994. LNCS, vol. 775, pp. 769–780. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  9. 9.
    Bodlaender, H.L., Jansen, K.: On the complexity of the maximum cut problem. Nordic J. of Computing 7(1), 14–31 (2000)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Demaine, E.D., Hajiaghayi, M.T., Kawarabayashi, K.: Algorithmic graph minor theory: Decomposition, approximation, and coloring. In: FOCS, pp. 637–646. IEEE Computer Society, Los Alamitos (2005)Google Scholar
  11. 11.
    Demaine, E.D., Hajiaghayi, M.T., Nishimura, N., Ragde, P., Thilikos, D.M.: Approximation algorithms for classes of graphs excluding single-crossing graphs as minors. J. Comput. Syst. Sci. 69(2), 166–195 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Demaine, E.D., Hajiaghayi, M.T., Thilikos, D.M.: 1.5-approximation for treewidth of graphs excluding a graph with one crossing as a minor. In: Jansen, K., Leonardi, S., Vazirani, V.V. (eds.) APPROX 2002. LNCS, vol. 2462, pp. 67–80. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    Demaine, E.D., Hajiaghayi, M.T., Thilikos, D.M.: Exponential speedup of fixed-parameter algorithms for classes of graphs excluding single-crossing graphs as minors. Algorithmica 41(4), 245–267 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Díaz, J., Kamiński, M.: Max-Cut and Max-Bisection are NP-hard on unit disk graphs. Theoretical Computer Science 377, 271–276 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Diestel, R.: Graph Theory, Electronic edn. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Földes, S., Hammer, P.L.: Split graphs. Congress. Numer., 311–315 (1978)Google Scholar
  17. 17.
    Grötschel, M., Nemhauser, G.L.: A polynomial algorithm for the Max-Cut problem on graphs without long odd cycles. Math. Prog. 29, 28–40 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grötschel, M., Pulleyblank, W.R.: Weakly bipartite graphs and the Max-Cut problem. Oper. Res. Lett. 1, 23–27 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guenin, B.: A characterization of weakly bipartite graphs. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 9–22. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  20. 20.
    Guenin, B.: A characterization of weakly bipartite graphs. Electronic Notes in Discrete Mathematics 5, 149–151 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guenin, B.: A characterization of weakly bipartite graphs. J. Comb. Theory, Ser. B 83(1), 112–168 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Guruswami, V.: Maximum cut on line and total graphs. Discrete Applied Mathematics 92(2-3), 217–221 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput. 4(3), 221–225 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–104. Plenum Press, New York (1972)CrossRefGoogle Scholar
  25. 25.
    Král, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., Van Bang, L. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  26. 26.
    Lozin, V.V., Mosca, R.: Maximum independent sets in subclasses of p5-free graphs. Inf. Process. Lett. 109(6), 319–324 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Orlova, G., Dorfman, Y.: Finding the maximum cut in a graph. Tekhnicheskaya Kibernetika (Engineering Cybernetics) 10, 502–506 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory, Ser. B 35(1), 39–61 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B 41, 92–114 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Robertson, N., Seymour, P.D.: Excluding a graph with one crossing. In: Robertson, N., Seymour, P.D. (eds.) Graph Structure Theory. Contemporary Mathematics, vol. 147, pp. 669–676. American Mathematical Society, Providence (1991)CrossRefGoogle Scholar
  31. 31.
    Schrijver, A.: A short proof of Guenin’s characterization of weakly bipartite graphs. J. Comb. Theory, Ser. B 85(2), 255–260 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wagner, K.: Über eine Eigenshaft der ebenen Komplexe. Math. Ann. 114, 570–590 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yannakakis, M.: Node- and edge-deletion NP-complete problems. In: STOC, pp. 253–264. ACM, New York (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marcin Kamiński
    • 1
  1. 1.Département d’InformatiqueUniversité Libre de BruxellesBelgium

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