set $X_c$ have diameter $2$ in the relative pants graph.

Brock and Masur have shown:

For $S$ such that $\xi(S) = 3$, ${\mathcal{P}}_{rel}(S)$ is hyperbolic.

Paths in the Pants Graph
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Here we describe how we will get a path in ${\mathcal{P}}(S)$ if $\xi(S) =2$ or ${\mathcal{P}}_{rel}(S)$ if $\xi(S) = 3$. The paths for ${\mathcal{P}}(S)$ are hierarchies and were originally introduced by Masur and Minsky in [@MMII] (in more generality than we will use here); the paths in ${\mathcal{P}}_{rel}(S)$ are motivated by hierarchies.

Take two pants decompositions, $\alpha = \{ \alpha_0, \alpha_1\}$ and $\beta = \{ \beta_0, \beta_1\}$, in ${\mathcal{P}}(S)$ where $S = S_{0,5}$ or $S_{1,2}$. To create a hierarchy between $\alpha$ and $\beta$ first connect $\alpha_0$ and $\beta_0$ with a geodesic path in ${\mathcal{C}}(S)$. This geodesic is referred to as the *main geodesic*, $g_{\alpha\beta} = \{ \alpha_0 = g_0, \ldots, g_n = \beta_0\}$. For each $g_i$, $0 \leq i \leq n$, connect $g_{i-1}$ to $g_{i+1}$ by a geodesic, $\gamma_i$, in ${\mathcal{C}}(S\backslash g_i)$, where $g_{-1} = \alpha_1$ and $g_{n+1} = \beta_1$. The collection of all of these geodesics is a *hierarchy* between $\alpha$ and $\beta$, generally pictured as in Figure \[Hierarchy picture\]. We often refer to the geodesic $\gamma_i$ as 