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
An energy-driven approach to linkage unfolding
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On straightening low-diameter unit trees
GD'05 Proceedings of the 13th international conference on Graph Drawing
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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We consider the problems of unfolding 3D lattice polygons embedded on the surface of some classes of lattice polyhedra, and of unfolding 2D orthogonal trees. During the unfolding process, all graph edges are preserved and no edge crossings are allowed. Let nbe the number of edges of the given polygon or tree. We show that a lattice polygon embedded on an open lattice orthotube can be convexified in O(n) moves and time, and a lattice polygon embedded on a lattice Tower of Hanoi, a lattice Manhattan Tower, or an orthogonally-convex lattice polyhedron can be convexified in O(n2) moves and time. The main technique in our algorithms is to fold up the lattice polygon from the end blocks of the given lattice polyhedron. On the other hand, we show that a 2-monotone orthogonal tree on the plane can be straightened in O(n2) moves and time. We hope that our results shed some light on solving the more general conjectures, which we proposed, that a 3D lattice polygon embedded on any lattice polyhedron can always be convexified, and any 2D orthogonal tree can always be straightened.