Given a compact surface \(\Sigma \) with boundary and a relation \(\Gamma \) on \(\pi _0(\partial \Sigma )\) , we define the prescribed arc graph \(\mathcal {A}(\Sigma ,\Gamma )\) to be the full subgraph of the arc graph \(\mathcal {A}(\Sigma )\) containing only classes of arcs between boundary components in \(\Gamma \) . We prove that \(\mathcal {A}(\Sigma ,\Gamma )\) is connected and infinite-diameter (if \(\Sigma \) is not the sphere with three boundary components), and classify when it is \(\delta \) -hyperbolic: in particular, \(\mathcal {A}(\Sigma ,\Gamma )\) is \(\delta \) -hyperbolic if and only if \(\Gamma \) is not bipartite except in some sporadic cases, where \(\delta \) may be chosen uniformly.
