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# Stable Divisorial Gonality is in NP

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.

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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