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.
H. L. Bodlaender—This work was supported by the NETWORKS project, funded by the Netherlands Organization for Scientific Research NWO under project no. 024.002.003.
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We thank Gunther Cornelissen and Nils Donselaar for helpful discussions.
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Bodlaender, H.L., van der Wegen, M., van der Zanden, T.C. (2019). Stable Divisorial Gonality is in NP. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_8
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