Improved Algorithms for the 2-Vertex Disjoint Paths Problem

  • Torsten Tholey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5404)


Given distinct vertices s 1,s 2,t 1, and t 2 the 2-vertex-disjoint paths problem (2-VDPP) consists in determining two vertex-disjoint paths p 1, from s 1 to t 1, and p 2, from s 2 to t 2, if such paths exist.

We show that by using some kind of sparsification technique the previously best known time bound of O(n + (m,n)) can be reduced to O(m + (n,n)), where α denotes the inverse of the Ackermann function. Moreover, we extend the very practical and simple algorithm of Hagerup for solving the 2-VDPP on 3-connected planar graphs to a simple linear time algorithm for the 2-VDPP on general planar graphs thereby avoiding the computation of planar embeddings or triconnected components.


Planar Graph Linear Time Improve Algorithm Linear Time Algorithm Original Instance 


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  1. 1.
    Aggarwal, A., Kleinberg, J., Williamson, D.P.: Node-disjoint paths on the mesh and a new trade-off in VLSI layout. SIAM J. Comput. 29, 1321–1333 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Fortune, S., Hopcroft, J., Wyllie, J.: The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121 (1980)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Hagerup, T.: A very practical algorithm for the two-paths problem in 3-connected planar graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 145–150. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Kanevsky, A., Tamassia, R., Di Battista, G., Chen, J.: On-line maintenance of the four-connected components of a graph. In: Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 793–801 (1991)Google Scholar
  5. 5.
    Khuller, S., Mitchell, S.G., Vazirani, V.V.: Processor efficient parallel algorithms for the two disjoint paths problem and for finding a Kuratowski homeomorph. SIAM J. Comput. 21, 486–506 (1992)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Lucchesi, C.L., Giglio, M.C.M.T.: On the irrelevance of edge orientations on the acyclic directed two disjoint paths problem, IC Technical Report DCC-92-03, Universidade Estadual de Campinas, Instituto de Computação (1992)Google Scholar
  7. 7.
    Nagamochi, H., Ibaraki, T.: Linear time algorithms for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583–596 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ohtsuki, T.: The two disjoint path problem and wire routing design. In: Saito, N., Nishizeki, T. (eds.) Graph Theory and Algorithms. LNCS, vol. 108, pp. 207–216. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  9. 9.
    Perković, L., Reed, B.: An improved algorithm for finding tree decompositions of small width. International Journal of Foundations of Computer Science (IJFCS) 11, 365–371 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Perl, Y., Shiloach, Y.: Finding two disjoint paths between two pairs of vertices in a graph. J. ACM 25, 1–9 (1978)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Scheffler, P.: A practical linear time algorithm for disjoint paths in graphs with bounded tree-width, Report No. 396/1994, TU Berlin, FB Mathematik (1994)Google Scholar
  13. 13.
    Schrijver, A.: Finding k disjoint paths in a directed planar graph. SIAM J. Comput. 23, 780–788 (1994)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Seymour, P.D.: Disjoint paths in graphs. Discrete Math. 29, 293–309 (1980)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Shiloach, Y.: A polynomial solution to the undirected two paths problem. J. ACM 27, 445–456 (1980)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Tholey, T.: Solving the 2-disjoint paths problem in nearly linear time. Theory Comput. Systems 39, 51–78 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Thomassen, C.: 2-linked graphs. Europ. J. Combinatorics 1, 371–378 (1980)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Woeginger, G.: A simple solution to the two paths problem in planar graphs. Inform. Process. Lett. 36, 191–192 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Torsten Tholey
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

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