the Bounded Geodesic Image Theorem for any geodesic that doesn’t contain $v$ or $w$.

Whenever a domain separating curve, $c$, shows up in a relative 3-archy in ${\mathcal{P}}_{rel}(S)$, the section of the relative 3-archy containing $c$ has length $2$. Therefore, when referring to a curve along a geodesic within a relative 3-archy we will assume it is non-domain separating since this type of curve adds the most length to the relative 3-archy. This also just makes the proof cleaner.

**Case 1:** Let $v$ be a vertex where all three main geodesics intersect. If $v$ is a domain separating curve then each edge of the triangle $\alpha\beta\gamma$ contains the point $p_v$, so the triangle is $0$-centered. Now assume $v$ is not a domain separating curve. Let $v_{\alpha\beta}^{-1}$ and $v_{\alpha\beta}^{+1}$ be the curves that are directly before and after $v$ on $g_{\alpha\beta}$. Similarly define $v_{\alpha\gamma}^{-1}$, $v_{\alpha\gamma}^{+1}$, $v_{\beta\gamma}^{-1}$, and $v_{\beta\gamma}^{+1}$. Consider the geodesics associated with $v$ in each relative 3-archy edge; in other words, all geodesics in the relative 3-archy that contribute to defining the path where $v$ is a part of every pants decomposition. Let $x_{\alpha\beta}$ be the curve in $[v_{\alpha\beta}^{-1}, v_{\alpha\beta}^{+1}]_{S \backslash v}$ that is adjacent to $v_{\alpha\beta}^{-1}$; similarly define $x_{\alpha\gamma}$. 