each open annulus $A$. The length of the edge $\Sigma(A)$ is the length $s - r$ defined in Definition \[def:open-annuli\].

### Completed skeleta

A *completed semistable vertex set* is defined the same as a semistable vertex set except we also allow ourselves to include some points of type I. These type I points are infinitely far away from the finite skeleton. If $V$ is a completed semistable vertex set, then the set of type II points in $V$ form a semistable vertex set by themselves.

The skeleton associated to a completed semistable vertex set is called a *completed skeleton*. It is defined similarly. The main difference is that the addition of type I points turn some of the open discs of ${X^{\rm an}}\setminus V$ into punctured discs. The skeleton of a punctured disc is an edge of infinite length.

We typically use the letter $\Gamma$ in this paper for a finite skeleton and $\Sigma$ for a completed skeleton.

### Skeleta associated to toric embeddings

Let $X$ be a smooth projective curve and let ${\varphi}\colon X \to Y$ be a closed embedding of $X$ into a toric variety $Y$. Let $T$ be the dense torus in $Y$. Let $X^\circ = {\varphi}^{-1}(T)$.

