Computing graph gonality is hard

TL;DR

This paper proves that computing the two main notions of graph gonality, divisorial and stable gonality, is computationally hard (NP-hard and APX-hard), indicating significant complexity in these graph parameters.

Contribution

The paper establishes the NP-hardness and APX-hardness of computing both divisorial and stable gonality, providing the first complexity results for these graph invariants.

Findings

Computing divisorial gonality is NP-hard.

Computing stable gonality is NP-hard.

Both problems are APX-hard.

Abstract

There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree of a finite harmonic morphism from a refinement of G to a tree, as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and sgon(G) are NP-hard by a reduction from the maximum independent set problem and the vertex cover problem, respectively. Both constructions show that computing gonality is moreover APX-hard.