 
Summary: A lower bound for the treewidth of planar
graphs with vital linkages
Isolde Adler, Philipp Klaus Krause
November 9, 2010
Abstract
The disjoint paths problem asks, given an graph G and k + 1 pairs
of terminals (s0, t0), . . . , (sk, tk), whether there are k + 1 pairwise disjoint
paths P0, . . . , Pk, such that Pi connects si to ti. Robertson and Seymour
have proven that the problem can be solved in polynomial time if k is
fixed. Nevertheless, the constants involved are huge, and the algorithm
is far from implementable. The algorithm uses a bound on the tree
width of graphs with vital linkages, and deletion of irrelevant vertices.
We give single exponential lower bounds both for the treewidth of planar
graphs with vital linkages, and for the size of the grid necessary for finding
irrelevant vertices.
1 Introduction
The disjoint paths problem is the following problem.
Input: Graph G, terminals (s0, t0), . . . , (sk, tk) V (G)2(k+1)
Question: Are there k + 1 pairwise vertex disjoint paths
P0, . . . , Pk in G such that Pi has endpoints si and ti?
