Folding equilateral plane graphs
Name
AbelDeDe11.pdf
Description
Accepted version
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430.75 KB
Format
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Author(s)
Abel, Zachary R.
Demaine, Erik D.
Demaine, Martin L.
Eisenstat, Sarah
Lynch, Jayson
Schardl, Tao B.
Shapiro-Ellowitz, Isaac
Date Issued
February 2013
Journal
International journal of computational geometry & applications
Publisher
World Scientific Pub Co Pte Lt
Citation
Abel, Zachary, et al., "Folding equilateral plane graphs." International journal of computational geometry & applications 23, 2 (February 2013): p. 75-92 doi 10.1142/S0218195913600017 ©2013 Author(s)
Version
Author's final manuscript
Abstract
We consider two types of folding applied to equilateral plane graph linkages. First, under continuous folding motions, we show how to reconfigure any linear equilateral tree (lying on a line) into a canonical configuration. By contrast, it is known that such reconfiguration is not always possible for linear (nonequilateral) trees and for (nonlinear) equilateral trees. Second, under instantaneous folding motions, we show that an equilateral plane graph has a noncrossing linear folded state if and only if it is bipartite. Furthermore, we show that the equilateral constraint is necessary for this result, by proving that it is strongly NP-complete to decide whether a (nonequilateral) plane graph has a linear folded state. Equivalently, we show strong NP-completeness of deciding whether an abstract metric polyhedral complex with one central vertex has a noncrossing flat folded state. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete. ©2013
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://doi.org/10.1142/S0218195913600017
