# Finding edge-disjoint paths in partial *k*-trees

## Abstract

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 it has not been known whether the edge-disjoint paths problem can be solved in polynomial time for partial *k*-trees unless *p*=*O*(1). 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*.

## Key words

edge-disjoint paths partial*k*-tree bounded tree-width polynomial-time algorithm edge-coloring

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