Bounds on higher graph gonality

TL;DR

This paper establishes new bounds on higher graph gonalities, generalizing existing bounds for first gonality, and explores the computational complexity of calculating these invariants.

Contribution

It introduces generalized bounds for higher gonalities of graphs and proves NP-hardness for computing the second gonality with certain restrictions.

Findings

New bounds on higher graph gonalities

Generalizations of known bounds for first gonality

NP-hardness of computing second gonality

Abstract

We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving independence number, and lower bounds on gonality by scramble number. We apply our bounds to study the computational complexity of computing higher gonalities, proving that it is NP-hard to compute the second gonality of a graph when restricting to multiplicity-free divisors.