Flat foldings of plane graphs with prescribed angles and edge lengths

Zachary Abel, Erik D. Demaine, Martin L. Demaine, David Eppstein, Anna Lubiw & Ryuhei Uehara
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 an infinitesimally thick line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each...
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