on a metric graph $\Gamma$ is called a *break divisor* if there exists a graph model $G$ of $\Gamma$ and edges $e_1,\dots,e_g$ of $G$ forming the complement of a spanning tree such that $B = x_1 + \dots + x_g$ where $x_i \in e_i$.

Break divisors were first introduced by Mikhalkin and Zharkov [@MZ] and were used by An, Baker, Kuperberg, and Shokrieh [@ABKS] to give a geometric proof of Kirchhoff’s Matrix-Tree Theorem.

Mumford curves
--------------

A smooth, projective curve $X$ over $K$ is called a *Mumford curve* if the genus of $X$ is equal to the genus (i.e. the first Betti number) of its skeleton.

While the question of which curves admit fully or totally faithful tropicalizations is still open, it is known that only Mumford curves admit smooth tropicalizations.

[[@J Theorem A]]{} Let $X$ be a smooth projective curve. Then the following are equivalent

1.  $X$ is a Mumford curve.

2.  There exists an embedding $\varphi \colon X \to Y$ for a toric variety $Y$ such that $\operatorname{Trop}_{\varphi}(X)$ is smooth.

This theorem shows that, at least for the results of Section \[sec:res-of-singularities\], we have to consider Mumford curves. The question of whether general smooth algebraic curves admit fully 