Stable gonality is computable

TL;DR

This paper proves that determining the stable gonality of a graph is in NP and provides an algorithm to compute it, linking graph complexity to number theory applications.

Contribution

It establishes the computational complexity of stable gonality and presents an algorithm for its calculation.

Findings

Deciding stable gonality is in NP.

Provided an algorithm with exponential time complexity.

Linked stable gonality to number theory problems.

Abstract

Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number of edges mapped to each edge of the tree. This parameter is related to treewidth, but unlike treewidth, it distinguishes multigraphs from their underlying simple graphs. Stable gonality is relevant for problems in number theory. In this paper, we show that deciding whether the stable gonality of a given graph is at most a given integer $k$ belongs to the class NP, and we give an algorithm that computes the stable gonality of a graph in $O((1.33n)^nm^m \text{poly}(n,m))$ time.