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When can a plane graph with prescribed edge lengths and prescribed angles(from among $\{0,180^\circ, 360^\circ$\}) be folded flat to lie in aninfinitesimally thin line, without crossings? This problem generalizes theclassic theory of single-vertex flat origami with prescribed mountain-valleyassignment, which corresponds to the case of a cycle graph. We characterizesuch flat-foldable plane graphs by two obviously necessary but also sufficientconditions, proving a conjecture made in 2001: the angles at each vertex shouldsum to $360^\circ$, and every face of the graph must itself be flat foldable.This characterization leads to a linear-time algorithm for testing flatfoldability of plane graphs with prescribed edge lengths and angles, and apolynomial-time algorithm for counting the number of distinct folded states.

Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths