### Flat foldings of plane graphs with prescribed angles and edge lengths

#### Abstract

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.

#### Full Text:

PDFDOI: http://dx.doi.org/10.20382/jocg.v9i1a3

ISSN: 1920-180X