Stable divisorial gonality is in NP

Hans L. Bodlaender; Marieke van der Wegen; Tom C. van der Zanden

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arXiv · 1808.06921

Stable divisorial gonality is in NP

Hans L. Bodlaender·Marieke van der Wegen·Tom C. van der Zanden

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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 consist 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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Hans L. Bodlaender, Marieke van der Wegen, Tom C. van der Zanden. 2018-08-21. Stable divisorial gonality is in NP. https://arxiv.org/abs/1808.06921

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