Kawasaki's Theorem - Local and Global Flat-foldability

Local and Global Flat-foldability

Kawasaki's theorem, applied to each of the vertices of an arbitrary crease pattern, determines whether the crease pattern is locally flat-foldable, meaning that the part of the crease pattern near the vertex can be flat-folded. However, there exist crease patterns that are locally flat-foldable but that have no global flat folding that works for the whole crease pattern at once. Tom Hull (1994) conjectured that global flat-foldability could be tested by checking Kawasaki's theorem at each vertex of a crease pattern, and then also testing bipartiteness of an undirected graph associated with the crease pattern, but this conjecture was disproven by Bern & Hayes (1996), who showed that the problem of testing global flat-foldability is NP-complete.