  A foundational result in origami mathematics is Kawasaki and Justin's simple,
efficient characterization of flat foldability for unassigned single-vertex
crease patterns (where each crease can fold mountain or valley) on flat
material. This result was later generalized to cones of material, where the
angles glued at the single vertex may not sum to $360^\circ$. Here we
generalize these results to when the material forms a complex (instead of a
manifold), and thus the angles are glued at the single vertex in the structure
of an arbitrary planar graph (instead of a cycle). Like the earlier
characterizations, we require all creases to fold mountain or valley, not
remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly
for flat material and strongly for complexes). Equivalently, we efficiently
characterize which combinatorially embedded planar graphs with prescribed edge
lengths can fold flat, when all angles must be mountain or valley (not unfolded
flat). Our algorithm runs in $O(n \log^3 n)$ time, improving on the previous
best algorithm of $O(n^2 \log n)$.
