Finding Edge-Disjoint Paths in Partial k -Trees

Abstract

For a given graph G and p pairs (s i ,t i ) , $1\leq i\leq p$ , of vertices in G , the edge-disjoint paths problem is to find p pairwise edge-disjoint paths P i , $1\leq i\leq p$ , 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 $p\geq \log n$ .