Folding equilateral plane graphs
Author(s)Abel, Zachary Ryan; Demaine, Erik D.; Demaine, Martin L.; Eisenstat, Sarah Charmian; Lynch, Jayson R.; Schardl, Tao Benjamin; Shapiro-Ellowitz, Isaac; ... Show more Show less
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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, such reconfiguration is known to be impossible 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. Not only is the equilateral constraint necessary for this result, but we show 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 with a specified “outside region”. By contrast, the analogous problem for a polyhedral manifold with one central vertex (single-vertex origami) is only weakly NP-complete.
22nd International Symposium, ISAAC 2011, Yokohama, Japan, December 5-8, 2011. Proceedings
DepartmentMassachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Department of Mathematics
Algorithms and Computation
Springer Berlin / Heidelberg
Abel, Zachary et al. “Folding Equilateral Plane Graphs.” Algorithms and Computation. Ed. Takao Asano et al. LNCS Vol. 7074. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. 574–583.
Author's final manuscript