similarly define $v_{\alpha \gamma}$ and $v_{\beta \gamma}$.

Let $c_0$ be the curve directly preceding $c$ in $[v_{\alpha\beta}, c]_S$ and let $c^{-1}$ be the curve directly preceding $c_0$. Consider a geodesic in ${\mathcal{C}}(S \backslash c_0)$ which connects $c^{-1}$ to $c$, define $c_1$ to be the curve directly preceding $c$ in this geodesic. We will show $\{c, c_0, c_1\}$ is a center of our relative 3-archy triangle $\alpha\beta\gamma$.

Let $v_{\beta\gamma}^{-1}$ be the curve before $v_{\beta\gamma}$ in $g_{\beta\gamma}$ and $v_{\beta\gamma}'$ be the curve adjacent to $v_{\beta\gamma}$ in the geodesic contained in the relative 3-archy connecting $\beta$ to $\gamma$ whose domain is ${\mathcal{C}}(S \backslash v_{\beta\gamma}^{-1})$. Now connect $\{v_{\beta\gamma}, v_{\beta\gamma}^{-1}, v_{\beta\gamma}'\}$ to $\{c, c_0, c_1 \}$ with a relative 3-archy, $H$. Our goal is to bound the length of $H$.

Using the exact argument as in Theorem \[hierarchy k-centered\] case 3, for each $w \in [c, v_{\beta\gamma}]_S$ which is non-separating, the geodesic in $H$ whose domain is ${\mathcal{C}}(S\backslash w)$ has length no more than $5M$. Let $w^{-1}$ and $w^{+1}$ be the curves before and after $w$ in $[c, v_{\beta\gamma}]_S$ and let $[w^{-1}, w^{+1}]_{S \backslash w}$ be the geodesic coming from $H$. Take $z \in [w^{-1}, w^{+1}]_{S \backslash w}$ and consider the geodesic in $H$ with 