Polygonal chains cannot lock in 4D
Computational Geometry: Theory and Applications
Proceedings of the nineteenth annual symposium on Computational geometry
A combinatorial approach to planar non-colliding robot arm motion planning
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
On Unfolding 3D Lattice Polygons and 2D Orthogonal Trees
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
On unfolding lattice polygons/trees and diameter-4 trees
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
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A polygonal chain is a sequence of consecutively joined edges embedded in space. A k-chain is a chain of k edges. A polygonal tree is a set of edges joined into a tree structure embedded in space. A unit tree is a tree with only edges of unit length. A chain or a tree is simple if non-adjacent edges do not intersect. We consider the problem about the reconfiguration of a simple chain or tree through a series of continuous motions such that the lengths of all tree edges are preserved and no edge crossings are allowed. A chain or tree can be straightened if all its edges can be aligned along a common straight line such that each edge points “away” from a designed leaf node. Otherwise it is called locked. Graph reconfiguration problems have wide applications in contexts including robotics, molecular conformation, rigidity and knot theory. The motivation for us to study unit trees is that for instance, the bonding-lengths in molecules are often similar, as are the segments of robot arms.