by Harvey in [@Harvey], is a graph whose vertices are homotopy classes of essential simple closed curves on $S$ and there is an edge between two vertices if the curves can be realized disjointly, up to isotopy. From here on when we talk about curves we really mean a representative of the homotopy class of an essential, non-peripheral, simple closed curve. When $\xi(S) = 1$, the definition of the curve graph is slightly altered in order to have a non-trivial graph: the vertices have the same definition, but there is an edge between two curves if they have minimal intersection number. We can similarly define the *arc and curve graph*, ${\mathcal{A}}{\mathcal{C}}(S)$, where a vertex is either a homotopy class of curves or homotopy class of arcs and the edges represent disjointness. This definition is the same for all surfaces such that $\xi(S) > 0$.

A related graph associated to a surface is the pants graph. We call a maximal set of disjoint curves on a surface a *pants decomposition*. For $\xi(S) \geq 1$ the *pants graph*, denoted ${\mathcal{P}}(S)$, of a surface $S$ is a graph whose vertices are homotopy classes of pants decompositions and there exists an edge between two 