## Abstract

The DISJOINT PATHS PROBLEM asks, given a graph G and a set of pairs of terminals (s_{1},t_{1}),…,(s_{k},t_{k}), whether there is a collection of k pairwise vertex-disjoint paths linking s_{i} and t_{i}, for i=1,…,k. In their f(k)⋅n^{3} algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose – very technical – proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k)=O(k^{3/2}⋅2^{k}). Our bound is radically better than the bounds known for general graphs.

Original language | English (US) |
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Pages (from-to) | 815-843 |

Number of pages | 29 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 122 |

DOIs | |

State | Published - Jan 1 2017 |

## Keywords

- Disjoint Paths Problem
- Graph Minors
- Treewidth

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics