Stable divisorial gonality is in NP

TL;DR

This paper proves that determining whether a graph's stable divisorial gonality is at most a given value is in NP, establishing its NP-completeness, and provides bounds on the number of subdivisions needed.

Contribution

It shows that stable divisorial gonality decision problem is NP-complete and introduces a verifiable certificate via Integer Linear Programming.

Findings

Deciding stable divisorial gonality is in NP.

Stable divisorial gonality is NP-complete.

Number of subdivisions needed is bounded by an exponential function.

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 $p$.