that $\pi_Y(v) \neq \emptyset$ for every vertex of $v$ of $g$. There is a constant $M$ depending only on $\xi(S)$ such that $${\mathrm{diam}}_Y(g) \leq M.$$

It can be shown that $M$ is at most $100$ for all surfaces [@Webb].

Hyperbolicity of Pants Graph for Complexity 2
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In this section we explore the hyperbolicity constant for the pants graph of surfaces with complexity $2$. Before we state any results, some notation must be discussed. Throughout the paper we denote $[a, b]_\Sigma$ as a geodesic in ${\mathcal{C}}(\Sigma)$ connecting $a$ to $b$, for any surface $\Sigma$. If a geodesic satisfying this is contained in a hierarchy (or relative 3-archy, in later sections) being discussed, $[a,b]_\Sigma$ denotes the geodesic in the hierarchy.

\[hierarchy k-centered\] For $S = S_{0,5}, S_{1,2}$, hierarchy triangles in ${\mathcal{P}}(S)$ are $8,900$-centered.

Let $S = S_{0,5}$ or $S_{1,2}$. Take three pants decompositions $\alpha = \{\alpha_0, \alpha_1\}$, $\beta = \{\beta_0, \beta_1\}$, and $\gamma = \{\gamma_0, \gamma_1\}$ in $S$. Consider the triangle $\alpha\beta\gamma$ in ${\mathcal{P}}(S)$ where the edges are taken to be hierarchies instead of geodesics. There are three cases:

1.  All three main geodesics have a curve in common.

2.  Any two of the main geodesics share a curve, but not 