Theory of Computing Systems

, Volume 57, Issue 1, pp 140–159 | Cite as

Finding Disjoint Paths in Split Graphs

  • Pinar Heggernes
  • Pim van ’t Hof
  • Erik Jan van Leeuwen
  • Reza Saei
Article

Abstract

The well-known Disjoint Paths problem takes as input a graph G and a set of k pairs of terminals in G, and the task is to decide whether there exists a collection of k pairwise vertex-disjoint paths in G such that the vertices in each terminal pair are connected to each other by one of the paths. This problem is known to be NP-complete, even when restricted to planar graphs or interval graphs. Moreover, although the problem is fixed-parameter tractable when parameterized by k due to a celebrated result by Robertson and Seymour, it is known not to admit a polynomial kernel unless NP ⊆ coNP/poly. We prove that Disjoint Paths remains NP-complete on split graphs, and show that the problem admits a kernel with O(k2) vertices when restricted to this graph class. We furthermore prove that, on split graphs, the edge-disjoint variant of the problem is also NP-complete and admits a kernel with O(k3) vertices. To the best of our knowledge, our kernelization results are the first non-trivial kernelization results for these problems on graph classes.

Keywords

Disjoint paths Computational complexity Parameterized complexity Polynomial kernel Split graphs 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Pinar Heggernes
    • 1
  • Pim van ’t Hof
    • 1
  • Erik Jan van Leeuwen
    • 2
  • Reza Saei
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.MPI für InformatikSaarbrückenGermany

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