Vertex Disjoint Paths on Clique-Width Bounded Graphs

Extended Abstract
  • Frank Gurski
  • Egon Wanke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)


We show that l vertex disjoint paths between l pairs of vertices can be found in linear time for co-graphs but is NP-complete for graphs of NLC-width at most 4 and clique-width at most 7. This is the first inartificial graph problem known to be NP-complete on graphs of bounded clique-width but solvable in linear time on co-graphs and graphs of bounded tree-width.


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  1. [AGKR02]
    Alstrup, S., Gavoille, C., Kaplan, H., Rauhe, T.: Nearest common ancestors: A survey and a new distributed algorithm. In: Proceedings of the Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 258–264. ACM, New York (2002)Google Scholar
  2. [BDLM02]
    Brandstädt, A., Dragan, F.F., Le, H.-O., Mosca, R.: New graph classes of bounded clique width. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 57–67. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. [Bod90]
    Bodlaender, H.L.: Polynomial algorithms for chromatic index and graph isomorphism on partial k-trees. Journal of Algorithms 11(4), 631–643 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [CHL+00]
    Corneil, D.G., Habib, M., Lanlignel, J.M., Reed, B., Rotics, U.: Polynomial time recognition of clique-width at most three graphs. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776. Springer, Heidelberg (2000)Google Scholar
  5. [CMR00]
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems 33(2), 125–150 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [CO00]
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Applied Mathematics 101, 77–114 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [CPS85]
    Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14(4), 926–934 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [CR01]
    Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 78–90. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. [EGW01]
    Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 117–128. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. [EGW03]
    Espelage, W., Gurski, F., Wanke, E.: Deciding clique-width for graphs of bounded tree-width. Journal of Graph Algorithms and Applications - Special Issue of JGAA on WADS 2001 7(2), 141–180 (2003)zbMATHMathSciNetGoogle Scholar
  11. [EIS76]
    Even, S., Itai, A., Shamir, A.: On the complexity of timetable and multicommodity flow problems. SIAM Journal on Computing 5, 691–703 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [FM91]
    Feder, T., Motwani, R.: Clique partitions, graph compression and speeding up algorithms. In: Proceedings of the Annual ACM Symposium on Theory of Computing, pp. 123–133. ACM, New York (1991)Google Scholar
  13. [GR99]
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 135–147. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. [GW00]
    Gurski, F., Wanke, E.: The tree-width of clique-width bounded graphs without K n,n. In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 196–205. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  15. [Joh98]
    Johansson, Ö.: Clique-decomposition, NLC-decomposition, and modular decomposition - relationships and results for random graphs. Congressus Numerantium 132, 39–60 (1998)zbMATHMathSciNetGoogle Scholar
  16. [Joh00]
    Johansson, Ö.: NLC2-decomposition in polynomial time. International Journal of Foundations of Computer Science 11(3), 373–395 (2000)CrossRefMathSciNetGoogle Scholar
  17. [KR01]
    Kobler, D., Rotics, U.: Polynomial algorithms for partitioning problems on graphs with fixed clique-width. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 468–476. ACM-SIAM (2001)Google Scholar
  18. [MP95]
    Middendorf, M., Pfeiffer, F.: On the complexity of the disjoint paths problems. Combinatorica 35(1), 97–107 (1995)MathSciNetGoogle Scholar
  19. [NVZ01]
    Nishizeki, T., Vygen, J., Zhou, X.: The edge-disjoint paths problem is NP-complete for series-parallel graphs. Discrete Applied Mathematics 115, 177–186 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [RS86]
    Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree width. Journal of Algorithms 7, 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [RS95]
    Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B 63(1), 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  22. [Sch94]
    Scheffler, P.: A practical linear time algorithm for vertex disjoint paths in graphs with bounded treewidth. Technical Report 396, Dept. of Mathematics, Technische Universität Berlin (1994)Google Scholar
  23. [Tod03]
    Todinca, I.: Coloring powers of graphs of bounded clique-width. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 370–382. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  24. [Wan94]
    Wanke, E.: k-NLC graphs and polynomial algorithms. Discr. Applied Mathematics 54, 251–266 (1994); revised version: zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Frank Gurski
    • 1
  • Egon Wanke
    • 1
  1. 1.Department of Computer ScienceDüsseldorfGermany

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