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Stable Divisorial Gonality is in NP

  • Hans L. Bodlaender
  • Marieke van der WegenEmail author
  • Tom C. van der Zanden
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11376)

Abstract

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G.

In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt, we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the number of subdivisions needed for minimum stable divisorial gonality of a graph with n vertices is bounded by \(2^{p(n)}\) for a polynomial p.

Notes

Acknowledgements

We thank Gunther Cornelissen and Nils Donselaar for helpful discussions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
    • 2
  • Marieke van der Wegen
    • 1
    Email author
  • Tom C. van der Zanden
    • 1
  1. 1.Department of Information and Computing SciencesUniversiteit UtrechtUtrechtThe Netherlands
  2. 2.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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