pants decompositions if they are related by an elementary move. Pants decompositions $\alpha$ and $\beta$ differ by an elementary move if one curve, $c$, from $\alpha$ can be deleted and replaced by a curve that intersects $c$ minimally to obtain $\beta$, see Figure \[elementary moves\].

We equip both graphs with the metric where each edge is length 1. Then ${\mathcal{C}}(S)$ and ${\mathcal{P}}(S)$ are complete geodesic metric spaces.

The hyperbolicity of these graphs have been studied before.

\[curve hyp\] For any hyperbolic surface $S$, ${\mathcal{C}}(S)$ is $17$-centered hyperbolic.

Brock and Farb showed:

For any hyperbolic surface $S$, ${\mathcal{P}}(S)$ is hyperbolic if and only if $\xi(S) \leq 2$.

Relative graphs
---------------

Let $S$ be a hyperbolic surface such that $\xi(S) \geq 3$. We say that a curve $c \in {\mathcal{C}}(S)$ is *domain separating* if $S \backslash c$ has two components of positive complexity. Each domain separating curve $c$ determines a set in ${\mathcal{P}}(S)$, $X_c = \{\alpha \in {\mathcal{P}}(S)  | c \in \alpha \}$. To form the *relative pants graph*, denoted ${\mathcal{P}}_{rel}(S)$, we add a point $p_c$ for each domain separating curve and an edge from $p_c$ to each vertex in $X_c$, where each edge has length $1$. Effectively, we have made the 