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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References
Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008). https://doi.org/10.2140/ant.2008.2.613
Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007). https://doi.org/10.1016/j.aim.2007.04.012
Baker, M., Shokrieh, F.: Chip-firing games, potential theory on graphs, and spanning trees. J. Comb. Theory Ser. A 120(1), 164–182 (2013). https://doi.org/10.1016/j.jcta.2012.07.011
Bodewes, J.M., Bodlaender, H.L., Cornelissen, G., van der Wegen, M.: Recognizing hyperelliptic graphs in polynomial time. In: Brandstädt, A., Köhler, E., Meer, K. (eds.) Graph-Theoretic Concepts in Computer Science, pp. 52–64 (2018). (extended abstract of http://arxiv.org/abs/1706.05670)
Bodlaender, H.L., van der Wegen, M., van der Zanden, T.C.: Stable divisorial gonality is in NP (2018). http://arxiv.org/abs/1808.06921
Caporaso, L.: Gonality of algebraic curves and graphs. In: Frühbis-Krüger, A., Kloosterman, R., Schütt, M. (eds.) Algebraic and Complex Geometry, vol. 71, pp. 77–108. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-05404-9_4
Cornelissen, G., Kato, F., Kool, J.: A combinatorial Li-Yau inequality and rational points on curves. Math. Ann. 361(1–2), 211–258 (2015). https://doi.org/10.1007/s00208-014-1067-x
Corry, S., Perkinson, D.: Divisors and Sandpiles: An Introduction to Chip-Firing. American Mathematical Society, Providence (2018)
van Dobben de Bruyn, J.: Reduced divisors and gonality in finite graphs. Bachelor thesis, Leiden University (2012). https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/bachvandobbendebruyn.pdf
van Dobben de Bruyn, J., Gijswijt, D.: Treewidth is a lower bound on graph gonality (2014). http://arxiv.org/abs/1407.7055v2
Gijswijt, D.: Computing divisorial gonality is hard (2015). http://arxiv.org/abs/1504.06713
Hladký, J., Král’, D., Norine, S.: Rank of divisors on tropical curves. J. Comb. Theory Ser. A 120(7), 1521–1538 (2013). https://doi.org/10.1016/j.jcta.2013.05.002
Papadimitriou, C.H.: On the complexity of integer programming. J. ACM 28(4), 765–768 (1981). https://doi.org/10.1145/322276.322287
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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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