A lower bound for the tree-width of planar graphs with vital linkages
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 tree-width of planar graphs with vital linkages, and for the size of the grid necessary for finding irrelevant vertices.
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