Finding edge-disjoint paths in partial k-trees

For a given graph G and p pairs (s_i, t_i), 1 ≤ i ≤ p, of vertices in G, the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P_i, 1 ≤ i ≤ p, connecting s_i and t_i. Many combinatorial problems can be efficiently solved for partial k-trees (graphs of treewidth bounded by a fixed integer k), but the edge-disjoint paths problem is NP-complete even for partial 3-trees. This paper gives two algorithms for the edge-disjoint paths problem on partial k-trees. The first one solves the problem for any partial k-tree G and runs in polynomial time if p = O(log n) and in linear time if p = O(1), where n is the number of vertices in G. The second one solves the problem under some restriction on the location of terminal pairs even if p ≥ log n.