10.20382/JOCG.V9I1A3
Abel, Zachary
Demaine, Erik D.
Demaine, Martin L.
Eppstein, David
Lubiw, Anna
Uehara, Ryuhei
Flat foldings of plane graphs with prescribed angles and edge lengths
Journal of Computational Geometry
2018
2015-01-11
2015-01-11
2018-02-27
2019-04-16
en
Article
1-14-191
10.20382/jocg.v9i1
74–93 Pages
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 vertex should sum to $360^\circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.
Journal of Computational Geometry, Vol 9, No 1 (2018)