On gonality-tight graphs

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.