A Proof of the Molecular Conjecture

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space R d , where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in R d if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G. Hendrickson [9] proved that redundant rigidity is a necessary condition for generic global rigidity, as is (d+1)-connectivity.Recent results in [2] and [10] have shown that when the configuration p is generic and d = 2, redundant rigidity and 3-connectivity are also sufficient -a good combinatorial characterization that only depends on G and can be checked in polynomial time. It appears that a similar result for d = 3 is beyond the scope of present techniques However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in R d for any d ≥ 1, as shown in [19], namely generic bodyand-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in R d , for any d ≥ 1, if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm on the graph to determine the generic global rigidity of body-and-bar frameworks in any dimension.

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“…A proof for this conjecture has recently been announced [15], so this 'Conjecture' may now be a Theorem. -tree-connected.…”

Section: Body-hinge and Molecular Frameworkmentioning

confidence: 99%

Generic global rigidity of body–bar frameworks

Connelly

Jordán

Whiteley

2013

Journal of Combinatorial Theory, Series B

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“…Refining the genericity conditions to prove that they include such practical occurrences is in general a very difficult problem. A remarkable and important success of this nature is the recent proof of the Molecular Conjecture [19]. Our contribution here is of a similar flavor, in the sense that it eliminates a certain type of genericity assumption.…”

Section: And 2(b)mentioning

confidence: 81%

“…Remaining open questions include: (a) characterizing body-and-hinge and (b) body-and-pin periodic frameworks, and (c) verifying whether the Molecular conjecture [19] holds as well in the periodic case. Besides intrinsic theoretical interest, such results would have immediate applications, as these structures appear in modeling families of natural crystals.…”

Section: Discussionmentioning

confidence: 99%

“…The problem of completing the combinatorial characterization for barand-joint frameworks in arbitrary dimensions remains the most conspicuous open question in rigidity theory. However, some restricted classes of finite frameworks have been shown to have similar Maxwell-Laman counting characterizations: body-and-bar and body-and-hinge frameworks [17,33,34,35,39], panel-andhinge frameworks [19] and (d − 2)-dimensional plate-and-bar frameworks [33,35,36].…”

Section: And 2(b)mentioning

confidence: 99%

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Periodic body-and-bar frameworks

Borcea

Streinu

Tanigawa

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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Abstractions of crystalline materials known as periodic body-and-bar frameworks are made of rigid bodies connected by fixed-length bars and subject to the action of a group of translations. In this paper, we give a Maxwell-Laman characterization for generic minimally rigid periodic body-and-bar frameworks. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games.

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“…The counts in Theorem 1 (and the corresponding pebble game algorithms) also characterize regular rigid body-hinge frameworks in 3-space [18,19], where a body-hinge framework with multigraph G is called regular if its rigidity matrix has maximal rank among all body-hinge realizations of G. Moreover, the recent Molecular Theorem [8] confirmed that Tay's counts also characterize regular rigid molecular linkages, where a molecular linkage is a body-hinge framework with the special geometry that all hinges of each body are concurrent in a single point. This result solved the more than 20 year old Molecular Conjecture [18].…”

Section: Detecting Flexibility In Body-bar and Molecular Linkagesmentioning

confidence: 99%

On the Symmetric Molecular Conjectures

Porta

Ros

Schulze

et al. 2013

Computational Kinematics

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A molecular linkage consists of a set of rigid bodies pairwise connected by revolute hinges, so that all hinge lines of each body are concurrent in a single point. It is an important problem in biochemistry, as well as in robotics, to efficiently analyze the motions and configuration spaces of such linkages. The well-developed mathematical theory of generic rigidity of body-bar frameworks permits a rigidity and flexibility analysis of molecular linkages via fast combinatorial algorithms. However, recent work in rigidity theory has shown that symmetry (a common feature of many molecular and mechanical linkages) can lead to additional motions in body-bar and molecular frameworks. These motions typically maintain the original symmetry of the structure throughout the path, and they can often be detected via simple combinatorial counts. In this paper, we outline how these symmetry-based mathematical counts and methods can be used to efficiently predict the motions of symmetric molecular linkages, and we provide numerical companion configuration spaces supporting the symmetric Molecular Conjectures in rigidity theory.

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A Proof of the Molecular Conjecture

Cited by 63 publications

References 29 publications

Generic global rigidity of body–bar frameworks

Generic global rigidity of body–bar frameworks

Periodic body-and-bar frameworks

On the Symmetric Molecular Conjectures

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