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Parameterized Counting of Trees, Forests and Matroid Bases

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Computer Science – Theory and Applications (CSR 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10304))

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Abstract

We prove \(\mathsf {\#W[1]}\)-hardness of counting (1) trees with k edges in a given graph, (2) forests with k edges in a given graph, and (3) bases of a given matroid of rank (or nullity) k representable over an arbitrary field of characteristic two, where k is the parameter.

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Notes

  1. 1.

    That is, \(\sup \{\tau (H) \mid H \in {\mathcal {H}}\} < \infty \), where \(\tau (H)\) is the size of a minimum vertex cover of H.

  2. 2.

    A slightly weaker version of this result with a simpler proof that still suffices for our application seems to follow along the lines of Snook [14].

  3. 3.

    This terminology stems from the fact that the removal of a leaves v isolated.

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Acknowledgements

The authors wish to thank Markus Bläser, Radu Curticapean, Holger Dell and Petr Hliněný for helpful comments on this work.

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Authors and Affiliations

  1. Saarland University and Cluster of Excellence (MMCI), Saarbrücken, Germany

    Cornelius Brand & Marc Roth

Authors
  1. Cornelius Brand
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  2. Marc Roth
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Correspondence to Cornelius Brand .

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Editors and Affiliations

  1. University of Bordeaux , Talence, France

    Pascal Weil

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Brand, C., Roth, M. (2017). Parameterized Counting of Trees, Forests and Matroid Bases. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_10

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  • DOI: https://doi.org/10.1007/978-3-319-58747-9_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58746-2

  • Online ISBN: 978-3-319-58747-9

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