$\{c, c_0\}$.

\[main thm 1\] For a surface $S = S_{0,5}, S_{1,2}$, ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic.

For $x, y \in {\mathcal{P}}(S)$ define ${\mathcal{L}}(x,y)$ to be the collection of hierarchy paths between $x$ and $y$. These are connected because each hierarchy path is connected and all contain $x$ and $y$. By Theorem \[hierarchy k-centered\] and Lemma \[centered to thin\] we have that for all $x, y, z \in {\mathcal{P}}(S)$ $${\mathcal{L}}(x, y) \subset N_{4*8,900}({\mathcal{L}}(x,z) \cup {\mathcal{L}}(z,y)).$$ If $d(x,y) \leq 1$ then any hierarchy between $x$ and $y$ is just the edge $\{ xy\}$, so ${\mathcal{L}}(x,y) = \{x, y\}$. Thus, both conditions of Proposition \[subset hyperbolic\] are satisfied. Therefore by applying Proposition \[subset hyperbolic\] we get ${\mathcal{P}}(S)$ is $2,691,437$-thin hyperbolic.

Relative Hyperbolicity of Pants Graphs Complexity 3
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In this section we turn our attention to relative pants graphs and their hyperbolicity constant.

\[relative hierarchy k-centered\] Take $S$ such that $\xi(S) = 3$. The relative 3-archy triangles in ${\mathcal{P}}_{rel}(S)$ are $6,191,300$-centered.

Take three pants decompositions of $S$, say $\alpha = \{\alpha_0, \alpha_1, \alpha_2 \}$, $\beta = \{\beta_0, \beta_1, \beta_2\}$, and $\gamma = \{\gamma_0, \gamma_1, \gamma_2\}$. Form the triangle $\alpha\beta\gamma$ such that each edge in the triangle is a relative 3-archy in ${\mathcal{P}}_{rel}(S)$. Let 