Improved Algorithms for the 2-Vertex Disjoint Paths Problem

Given distinct vertices s_1, s_2, t_1, and t_2 the 2-vertex-disjoint paths problem 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. As a first result we show that by using some kind of sparsification technique the previously best known time bound of O(n + m alpha(m, n)) can be reduced to O(m + n alpha(n, n)), where alpha denotes the inverse of the Ackermann function. Moreover, we extend the very practical and simple algorithm of Hagerup for solving the 2-vertex-disjoint-paths problem on 3-connected planar graphs to a practical linear time algorithm for the 2-VDPP on general planar graphs thereby avoiding the computation of planar embeddings or triconnected components