Our knowledge about single-vertex flat foldability from Section 12.2 (p. 198) allows us to solve a more general problem involving local foldability of vertices in a general crease pattern (on a flat piece of paper). A vertex in a crease pattern or mountain–valley pattern is locally flat foldable if we obtain flat foldability when we cut out any region containing the vertex, portions of its incident edges, and no other vertices or edges. We can make the canonical choice for the cut-out region of a small disk centered at the vertex. Thus local flat foldability of a vertex is equivalent to single-vertex foldability in a crease pattern or mountain–valley pattern obtained by restricting the larger pattern to the creases incident to the vertex.

We can easily determine whether every vertex in a crease pattern or a mountain–valley pattern is locally flat foldable: just test each vertex according to the algorithms in Section 12.2. The catch is that while every vertex in a crease pattern may be locally flat foldable, each vertex might require its own local mountain–valley assignment. The more interesting question, the local flat foldability problem, is to determine whether a general crease pattern has one, global mountain–valley assignment for which every vertex is locally flat foldable. Bern and Hayes (1996) showed how to solve this problem by finding one such mountain–valley assignment, or determining that none exist, in linear time.