Finding Disjoint Paths in Split Graphs

  • Pinar Heggernes
  • Pim van ’t Hof
  • Erik Jan van Leeuwen
  • Reza Saei
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8327)


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 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(k 2) 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(k 3) vertices. To the best of our knowledge, our kernelization results are the first non- trivial kernelization results for these problems on graph classes.


Planar Graph Polynomial Kernel Disjoint Path Chordal Graph Reduction Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing Switzerland 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 BergenNorway
  2. 2.MPI für InformatikSaarbrückenGermany

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