Prescribed Arc Graphs

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