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 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.
This research is supported by the Research Council of Norway.
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Heggernes, P., van ’t Hof, P., van Leeuwen, E.J., Saei, R. (2014). Finding Disjoint Paths in Split Graphs. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds) SOFSEM 2014: Theory and Practice of Computer Science. SOFSEM 2014. Lecture Notes in Computer Science, vol 8327. Springer, Cham. https://doi.org/10.1007/978-3-319-04298-5_28
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DOI: https://doi.org/10.1007/978-3-319-04298-5_28
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