In this case, the above Theorem would put a bound on the run-time of the algorithm in the cases that $S = S_{0,5}, S_{1,2}$.

We now turn our attention to the relatively hyperbolic cases.

For a surface $S = S_{3,0}, S_{1,3}, S_{0,6}$, ${\mathcal{P}}_{rel}(S)$ is $2,606,810,489$-thin hyperbolic.

To show both of our main theorems, we construct a family of paths that is very closely related to hierarchies, introduced in [@MMII]. We show that this family of paths satisfies the thin triangle condition which, by a theorem of Bowditch, allows us to conclude the whole space is hyperbolic [@Bow]. A key tool used throughout is the Bounded Geodesic Image Theorem [@MMII]. This theorem allows us to control the length of geodesics in subspaces.

This method cannot be made to generalize to pants graphs in general since any pants graph of a surface with complexity higher than $3$ is not strongly relatively hyperbolic [@BM]. Although, this method may be able to be used for other graphs which are variants on the pants graph.

One might consider approaching this problem by finding the sectional curvature of Teichmüller space and using the quasi-isometry to inform on the hyperbolicity constant of the pants graph. If the 