  We address the "inverse problem" for discrete geometry, which consists in
determining whether, given a discrete structure of a type that does not in
general imply geometrical information or even a topology, one can associate
with it a unique manifold in an appropriate sense, and constructing the
manifold when it exists. This problem arises in a variety of approaches to
quantum gravity that assume a discrete structure at the fundamental level; the
present work is motivated by the semiclassical sector of loop quantum gravity,
so we will take the discrete structure to be a graph and the manifold to be a
spatial slice in spacetime. We identify a class of graphs, those whose vertices
have a fixed valence, for which such a construction can be specified. We define
a procedure designed to produce a cell complex from a graph and show that, for
graphs with which it can be carried out to completion, the resulting cell
complex is in fact a PL-manifold. Graphs of our class for which the procedure
cannot be completed either do not arise as edge graphs of manifold cell
decompositions, or can be seen as cell decompositions of manifolds with
structure at small scales (in terms of the cell spacing). We also comment
briefly on how one can extend our procedure to more general graphs.
