Improved Algorithms for the 2-Vertex Disjoint Paths Problem
Given distinct vertices s1,s2,t1, and t2 the 2-vertex-disjoint paths problem (2-VDPP) consists in determining two vertex-disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such paths exist.
We show that by using some kind of sparsification technique the previously best known time bound of O(n + mα(m,n)) can be reduced to O(m + nα(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.
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