Solving the 2-Disjoint Paths Problem in Nearly Linear Time

  • Torsten Tholey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2996)


Given four distinct vertices s 1,s 2,t 1, and t 2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p 1 from s 1 to t 1 and p 2 from s 2 to t 2, if such paths exist. Disjoint can mean vertex- or edge-disjoint.

Both, the edge- and the vertex-disjoint version of the problem, are \(\mathcal{NP}\)-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified so as to solve the 2-(vertex-)disjoint paths problem in O(n+(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the decision versions of the 2-vertex- and the 2-edge-disjoint paths problem on dags.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Torsten Tholey
    • 1
  1. 1.Institut für InformatikJohann Wolfgang Goethe-Universität FrankfurtFrankfurt am MainGermany

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