the geodesics whose domain is ${\mathcal{C}}(S \backslash g_i)$ or the geodesic connecting $g_{i-1}$ and $g_{i+1}$. We can turn a hierarchy into a path in ${\mathcal{P}}(S)$ by looked at all edges in turn, as pictured in Figure \[Hierarchy picture\]. We will often blur the line between the hierarchy being a path in the pants graph or a collection of geodesics - and refer to both as the hierarchy between $\alpha$ and $\beta$.

Let $\xi(S) =3$. We make a path in ${\mathcal{P}}_{rel}(S)$ using a similar technique. Take two pants decompositions in ${\mathcal{P}}_{rel}(S)$, $\alpha = \{\alpha_0, \alpha_1, \alpha_2\}$ and $\beta = \{\beta_0, \beta_1, \beta_2\}$. Connect $\alpha_0$ to $\beta_0$ with a geodesic $g_{\alpha\beta}$ in ${\mathcal{C}}(S)$, we still refer to this as the main geodesic. For every non-domain separating curve $w \in g$, connect $w^{-1}$ to $w^{+1}$ with a geodesic, $h$, in ${\mathcal{C}}(S \backslash w)$ where $w^{-1}$ and $w^{+1}$ are the curves before and after $w$ in $g$. If $w = \alpha_0$ then $w^{-1} = \alpha_1$ and if $w = \beta_0$ then $w^{+} = \beta_1$. Now for each non-domain separating curve $z \in h$ connect $z^{-1}$ to $z^{+1}$ with a geodesic in ${\mathcal{C}}(S \backslash (w \cup z))$, where $z^{-1}$ and $z^{+1}$ are the curves before 