  Let a $A$ be the 1-skeleton of a triangulated topological annulus. We
establish bounds on the combinatorial modulus of a refinement $A'$, formed by
attaching new vertices and edges to $A$, that depend only on the refinement and
not on the structure of $A$ itself. This immediately applies to showing that a
disk triangulation graph may be refined without changing its combinatorial
type, provided the refinement is not too wild. We also explore the type problem
in terms of disk growth, proving a parabolicity condition based on a
superlinear growth rate, which we also prove optimal. We prove our results with
no degree restrictions in both the EEL and VEL settings and examine type
problems for more general complexes and dual graphs.
