Irrelevant vertices for the planar Disjoint Paths Problem

Isolde Adler, Stavros G. Kolliopoulos, Philipp Klaus Krause, Daniel Lokshtanov, Saket Saurabh, Dimitrios M. Thilikos

Research output: Contribution to journalArticlepeer-review


The DISJOINT PATHS PROBLEM asks, given a graph G and a set of pairs of terminals (s1,t1),…,(sk,tk), whether there is a collection of k pairwise vertex-disjoint paths linking si and ti, for i=1,…,k. In their f(k)⋅n3 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(k3/2⋅2k). Our bound is radically better than the bounds known for general graphs.

Original languageEnglish (US)
Pages (from-to)815-843
Number of pages29
JournalJournal of Combinatorial Theory. Series B
StatePublished - Jan 1 2017


  • Disjoint Paths Problem
  • Graph Minors
  • Treewidth

ASJC Scopus subject areas

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


Dive into the research topics of 'Irrelevant vertices for the planar Disjoint Paths Problem'. Together they form a unique fingerprint.

Cite this