Enumerating Constrained Non-crossing Minimally Rigid Frameworks

@article{Avis2008EnumeratingCN,
  title={Enumerating Constrained Non-crossing Minimally Rigid Frameworks},
  author={David Avis and Naoki Katoh and Makoto Ohsaki and Ileana Streinu and Shin-ichi Tanigawa},
  journal={Discrete \& Computational Geometry},
  year={2008},
  volume={40},
  pages={31-46}
}
Abstract In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In… 
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References

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Enumerating Non-crossing Minimally Rigid Frameworks
TLDR
An algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks (simply called non-Crossing Laman frameworks) on a given generic set of n points is presented.
Enumerating Planar Minimally Rigid Graphs
TLDR
It is obtained that the set of all planar Laman graphs on a given point set is connected by flips which remove an edge and then restore the Laman property with the addition of a non-crossing edge.
Finding and Maintaining Rigid Components
We give the first complete analysis that the complexity of finding and maintaining rigid components of planar bar-and-joint frameworks and arbitrary d-dimensional body-and-bar frameworks, using a
Reverse Search for Enumeration
Acute Triangulations of Polygons
TLDR
A combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other.
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. This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by
Enumerating Triangulation Paths
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TLDR
A combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets that yields very efficient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom.
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An algorithm is provided that enumerates all pseudo-triangulation zigzag paths of a given point set with respect to a given line in O(n 2) time per path and O( n 2) space, where n is the number of points.
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