On gonality-tight 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, that we refer to as quasi-banana graphs. We prove a structure theorem for those graphs, and show that they are all obtained via an inductive process by gluing together complete and banana graphs under certain rules.

38 pages, 13 figures. Fixed multiple incorrect references caused by a recurring \cref issue