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Finding Disjoint Paths in Split Graphs

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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(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.

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Notes

  1. We briefly sketch a reduction from Edge-Disjoint Paths on general graphs to Edge-Disjoint Paths on line graphs. Replace every edge uv of the original graph G by a three-vertex path u,x u v , v (where x u v is a new vertex), and take the line graph of the resulting graph. Then each edge uv of G corresponds to an edge of the line graph (namely, the edge between the vertices corresponding to edges u x u v and x u v v). This creates a correspondence between paths in G and paths in the line graph, and completes the proof.

  2. We mention, however, that our polynomial kernel for Edge-Disjoint Paths, presented in Section 4.2, can be easily modified to take this restriction into account. It suffices to insist in Lemma 8 that, instead of the degree, the number of non-terminal neighbors is at least the number of terminals on it. The rest of the proof goes through (mutatis mutandis). Also, our NP-completeness result is not influenced by this issue.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Pinar Heggernes, Pim van ’t Hof & Reza Saei

  2. MPI für Informatik, Campus E1 4, 66123, Saarbrücken, Germany

    Erik Jan van Leeuwen

Authors
  1. Pinar Heggernes
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  2. Pim van ’t Hof
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  3. Erik Jan van Leeuwen
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  4. Reza Saei
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Correspondence to Erik Jan van Leeuwen.

Additional information

This research is supported by the Research Council of Norway. A preliminary version of this work was presented at SOFSEM 2014 [13].

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Heggernes, P., van ’t Hof, P., van Leeuwen, E.J. et al. Finding Disjoint Paths in Split Graphs. Theory Comput Syst 57, 140–159 (2015). https://doi.org/10.1007/s00224-014-9580-6

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