General vertex disjoint paths in series-parallel graphs

AbstractLet G = (V, E) be an undirected graph and let (si,ti), 1 ≤ i ≤ k be k pairs of vertices in G. The vertex disjoint paths problem is to find k paths P1,…,Pk such that Pi connects si and ti 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