On gonality-tight graphs

TL;DR

This paper characterizes a large family of graphs with a specific gonality sequence property, providing a structure theorem and an inductive construction method involving complete and banana graphs.

Contribution

It offers a new characterization and structural understanding of graphs with a second gonality exactly one greater than the first, under a stronger condition.

Findings

Graphs satisfying the condition are constructed via an inductive process.

A structure theorem describes these graphs as glued combinations of complete and banana graphs.

The paper fully characterizes the gonality sequence for this family of graphs.

Abstract

We address a question posed by Fessler-Jensen-Kelsey-Owen regarding graphs whose second gonality is greater than the first by exactly 1. We answer the question affirmatively under a stronger condition, thereby characterising the entire gonality sequence for a large family of graphs. We prove a structure theorem for the graphs satisfying the condition, and show that they are all obtained via an inductive process by gluing together complete and banana graphs under certain rules.