Tight bounds for linkages in planar graphs

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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 prove this result for planar graphs achieving g(k) = 2 O(k). Our bound is radically better than the bounds known for general graphs. Moreover, our proof is new and self-contained, and it strongly exploits the combinatorial properties of planar graphs. We also prove that our result is optimal, in the sense that the function g(k) cannot become better than exponential. Our results suggest that any algorithm for the Disjoint-Paths Problem that runs in time better than will probably require drastically different ideas from those in the irrelevant vertex technique.

Original languageEnglish (US)
Title of host publicationAutomata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings
Number of pages12
EditionPART 1
StatePublished - 2011
Event38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland
Duration: Jul 4 2011Jul 8 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume6755 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other38th International Colloquium on Automata, Languages and Programming, ICALP 2011

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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