Brill-Noether theory seeks to answer the basic question of when an abstract complex curve $C$ of genus $g$ comes equipped with a nondegenerate degree-$d$ morphism to $P^r$.

When $C$ is general in the moduli space $M_g$, an answer is provided by a celebrated theorem of Griffiths and Harris, which establishes that $C$'s behavior is as "expected" whenever $\rho =g-(r+1)(g-d+r)$ is positive, and that there are no morphisms when $\rho$ is negative.

Payne, et al. recently gave a proof of the nonexistence statement using a combinatorial analysis of metric graphs of a particular combinatorial type. These belong to a moduli space $M_g^{\mathrm{trop}}$ for stable metric graphs, which admits a stratification that is dual to that of the Deligne-Mumford space of stable curves.

We will discuss work in progress with Melo, Neves, and Viviani aimed at understanding the Brill-Noether-type behavior of metric graphs of other combinatorial types, focusing on a remarkable infinite family of these.