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On Unfolding Lattice Polygons/Trees and Diameter4 Trees
 In Proc. 12th Annual International Computing and Combinatorics Conference (COCOON), 186–195
, 2006
"... Abstract. We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such th ..."
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Abstract. We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such that each edge points “away ” from a designated leaf node. A polygon can be convexified if it can be reconfigured to a convex polygon. A lattice tree (resp. polygon) is a tree (resp. polygon) containing only edges from a square or cubic lattice. We first show that a 2D lattice chain or a 3D lattice tree can be straightened efficiently in O(n) moves and time, where n is the number of tree edges. We then show that a 2D lattice tree can be straightened efficiently in O(n 2) moves and time. Furthermore, we prove that a 2D lattice polygon or a 3D lattice polygon with simple shadow can be convexified efficiently in O(n 2) moves and time. Finally, we show that two special classes of diameter4 trees in two dimensions can always be straightened.
Minimal Locked Trees
"... Abstract. Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomialtime characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different mea ..."
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Abstract. Locked tree linkages have been known to exist in the plane since 1998, but it is still open whether they have a polynomialtime characterization. This paper examines the properties needed for planar trees to lock, with a focus on finding the smallest locked trees according to different measures of complexity, and suggests some new avenues of research for the problem of algorithmic characterization. First we present a locked linear tree with only eight edges. In contrast, the smallest previous locked tree has 15 edges. We further show minimality by proving that every locked linear tree has at least eight edges. We also show that a sixedge tree can interlock with a fouredge chain, which is the first locking result for individually unlocked trees. Next we present several new examples of locked trees with varying minimality results. Finally, we provide counterexamples to two conjectures of [12], [13] by showing the existence of two new types of locked tree: a locked orthogonal tree (all edges horizontal and vertical) and a locked equilateral tree (all edges unit length). 1
On the Maximum Span of FixedAngle Chains
"... Soss proved that it is NPhard to find the maximum flat span of a fixedangle polygonal chain: the largest distance achievable between the endpoints in a planar embedding. These fixedangle chains can serve as models of protein backbones. The corresponding problem in 3D is open. We show that two spe ..."
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Soss proved that it is NPhard to find the maximum flat span of a fixedangle polygonal chain: the largest distance achievable between the endpoints in a planar embedding. These fixedangle chains can serve as models of protein backbones. The corresponding problem in 3D is open. We show that two special cases of particular relevance to the protein model are solvable in polynomial time: when all link lengths are equal, and all angles are equal, the maximum 3D span is achieved in a flat configuration and can be computed in constant time. When all angles are equal (but the link lengths arbitrary), the maximum 3D span is in general nonplanar but can be found in polynomial time. 1
On the Maximum Span of FixedAngle Chains
"... Soss proved that it is NPhard to find the maximum flat span of a fixedangle polygonal chain: the largest distance achievable between the endpoints in a planar embedding. These fixedangle chains can serve as models of protein backbones. The corresponding problem in 3D is open. We show that two spe ..."
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Soss proved that it is NPhard to find the maximum flat span of a fixedangle polygonal chain: the largest distance achievable between the endpoints in a planar embedding. These fixedangle chains can serve as models of protein backbones. The corresponding problem in 3D is open. We show that two special cases of particular relevance to the protein model are solvable in polynomial time: when all link lengths are equal, and all angles are equal, the maximum 3D span is achieved in a flat configuration and can be computed in constant time. When all angles are equal (but the link lengths arbitrary), the maximum 3D span is in general nonplanar but can be found in polynomial time. 1
On Unfolding Trees and Polygons on Various Lattices
"... We consider the problem of unfolding lattice trees and polygons in hexagonal or triangular lattice in two dimensions. We show that a hexagonal/triangular lattice chain (resp. tree) can be straightened in O(n) (resp. O(n 2)) moves and time, and a hexagonal/triangular lattice polygon can be convexifie ..."
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We consider the problem of unfolding lattice trees and polygons in hexagonal or triangular lattice in two dimensions. We show that a hexagonal/triangular lattice chain (resp. tree) can be straightened in O(n) (resp. O(n 2)) moves and time, and a hexagonal/triangular lattice polygon can be convexified in O(n 2) moves and time. We hope that the techniques we used shed some light on solving the more general conjecture that a unit tree in two dimensions can always be straightened. 1
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"... We consider the problem of unfolding lattice polygons embedded on the surface of some classes of lattice polyhedra. We show that an unknotted lattice polygon embedded on a lattice orthotube or orthotree can be convexified in O(n) moves and time, and a lattice polygon embedded on a lattice Tower of H ..."
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We consider the problem of unfolding lattice polygons embedded on the surface of some classes of lattice polyhedra. We show that an unknotted lattice polygon embedded on a lattice orthotube or orthotree can be convexified in O(n) moves and time, and a lattice polygon embedded on a lattice Tower of Hanoi or Manhattan Tower can be convexified in O(n 2) moves and time. 1