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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, 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.

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3-Colorability of Pseudo-Triangulations

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195915500168 ◽

2015 ◽

Vol 25 (04) ◽

pp. 283-298

Author(s):

Oswin Aichholzer ◽

Franz Aurenhammer ◽

Thomas Hackl ◽

Clemens Huemer ◽

Alexander Pilz ◽

...

Keyword(s):

Polynomial Time ◽

Linear Time ◽

Plane Graphs ◽

Np Completeness ◽

Complete Problem ◽

Polynomial Time Algorithms ◽

Complexity Results ◽

The Family ◽

General Plane ◽

Np Complete

Deciding 3-colorability for general plane graphs is known to be an NP-complete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudo-triangulations, which are a generalization of triangulations, and prove NP-completeness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudo-triangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudo-triangulations with maximum face degree four are always 3-colorable. An according 3-coloring can be found in linear time. Some complexity results relating to the rank of pseudo-triangulations are also given.

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