NP-completeness conjecture for stable divisorial gonality

Fix a positive integer $r$. The $r^{th}$ Stable Divisorial Gonality problem asks whether a graph has stable divisorial gonality at most a given integer. Stable divisorial gonality conjecture. The $r^{th}$ Stable Divisorial Gonality problem is NP-complete. NP-hardness is established in the paper, while membership in NP is known for $r=1$ but remains open for general $r$.