Chapter

Algorithms and Computation

Volume 7074 of the series Lecture Notes in Computer Science pp 574-583

Folding Equilateral Plane Graphs

  • Zachary AbelAffiliated withCarnegie Mellon UniversityMIT Department of Mathematics
  • , Erik D. DemaineAffiliated withCarnegie Mellon UniversityMIT Computer Science and Artificial Intelligence Laboratory
  • , Martin L. DemaineAffiliated withCarnegie Mellon UniversityMIT Computer Science and Artificial Intelligence Laboratory
  • , Sarah EisenstatAffiliated withCarnegie Mellon UniversityMIT Computer Science and Artificial Intelligence Laboratory
  • , Jayson LynchAffiliated withCarnegie Mellon UniversityMIT Computer Science and Artificial Intelligence Laboratory
  • , Tao B. SchardlAffiliated withCarnegie Mellon UniversityMIT Computer Science and Artificial Intelligence Laboratory
  • , Isaac Shapiro-EllowitzAffiliated withCarnegie Mellon UniversityUniversity of Massachusetts Boston

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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, 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.