  We study arc graphs and curve graphs for surfaces of infinite topological
type. First, we define an arc graph relative to a finite number of (isolated)
punctures and prove that it is a connected, uniformly hyperbolic graph of
infinite diameter; this extends a recent result of J. Bavard to a large class
of punctured surfaces.
  Second, we study the subgraph of the curve graph spanned by those elements
which intersect a fixed separating curve on the surface. We show that this
graph has infinite diameter and geometric rank 3, and thus is not hyperbolic.

Geometric Topology