Summary: A lower bound for the tree-width of planar
graphs with vital linkages
Isolde Adler, Philipp Klaus Krause
November 9, 2010
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 tree-width of planar
graphs with vital linkages, and for the size of the grid necessary for finding
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?