General vertex disjoint paths in series-parallel graphs

Abstract

Let G = ( V, E ) be an undirected graph and let ( s i , t i ), 1 ≤ i ≤ k be k pairs of vertices in G . The vertex disjoint paths problem is to find k paths P 1 ,…, P k such that P i connects s i and t i and any two of these paths may intersect only at a common endpoint. This problem is NP-complete even for planar graphs. Robertson and Seymour proved that when k is a fixed integer this problem becomes polynomial. We present a linear time algorithm for solving the decision version of the general problem when the input graph is a series-parallel graph.