Stable Divisorial Gonality is in NP

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.

On the Gonality of Metric Graphs

“…Over the last years, the gonality of metric graphs has been the subject of research in combinatorics and computer science: the gonality of Γ is bounded from below by the treewidth of for any model ( , ℓ) of Γ [vDdBG20], and also-like for Riemann surfaces [YY80]-by an expression involving its volume and the smallest positive eigenvalue of the Laplace operator of Γ [AK16]. Moreover, like treewidth, gonality is NP-complete [Gij15,BvdWvdZ19].…”