Theory of Computing Systems

, Volume 39, Issue 1, pp 51–78

Solving the 2-Disjoint Paths Problem in Nearly Linear Time


DOI: 10.1007/s00224-005-1256-9

Cite this article as:
Tholey, T. Theory Comput Syst (2006) 39: 51. doi:10.1007/s00224-005-1256-9


Given four distinct vertices s1,s2,t1, and t2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p1 from s1 to t1 and p2 from s2 to t2, if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified to solve the 2-vertex-disjoint paths problem in only O(n + mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and where 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 2-vertex- and the 2-edge-disjoint paths problem on dags.

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Institut fur Informatik, Universitat Augsburg, D-86135 AugsburgGermany