### 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 2p(n) for a polynomial p.

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 2p(n) for a polynomial p.

Original language | English |
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Title of host publication | SOFSEM 2019: Theory and Practice of Computer Science |

Subtitle of host publication | 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings |

Editors | Barbara Catania, Rastislav Královic, Jerzy Nawrocki, Giovanni Pighizzini |

Publisher | Springer, Cham |

Pages | 81-93 |

ISBN (Electronic) | 978-3-030-10801-4 |

ISBN (Print) | 978-3-030-10800-7 |

DOIs | |

Publication status | Published - 2019 |

### Publication series

Series | Lecture Notes in Computer Science |
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Volume | 11376 |

ISSN | 0302-9743 |

### Cite this

Bodlaender, H. L., van der Wegen, M., & van der Zanden, T. (2019). Stable Divisorial Gonality is in NP. In B. Catania, R. Královic, J. Nawrocki, & G. Pighizzini (Eds.),

*SOFSEM 2019: Theory and Practice of Computer Science: 45th International Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 27-30, 2019, Proceedings*(pp. 81-93). Springer, Cham. Lecture Notes in Computer Science, Vol.. 11376 https://doi.org/10.1007/978-3-030-10801-4_8