Optimal attack and reinforcement of a network
Journal of the ACM (JACM)
The algebraic geometry of motions of bar-and-body frameworks
SIAM Journal on Algebraic and Discrete Methods
The union of matroids and the rigidity of frameworks
SIAM Journal on Discrete Mathematics
Handbook of discrete and computational geometry
Rigid Components in Molecular Graphs
Algorithmica
Pin-Collinear Body-and-Pin Frameworks and the Molecular Conjecture
Discrete & Computational Geometry
A rooted-forest partition with uniform vertex demand
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
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A body-and-hinge framework is a structure consisting of rigid bodies connected by hinges in d-dimensional space. The generic infinitesimal rigidity of a body-and-hinge framework has been characterized in terms of the underlying graph independently by Tay and Whiteley as follows: A graph G can be realized as an infinitesimally rigid body-and-hinge framework by mapping each vertex to a body and each edge to a hinge if and only if ({d+1/2}-1)G contains {d+1/2} edge-disjoint spanning trees, where ({d+1/2}-1)G is the graph obtained from $G$ by replacing each edge by (d+1/2-1) parallel edges. In 1984 they jointly posed a question about whether their combinatorial characterization can be further applied to a nongeneric case. Specifically, they conjectured that G can be realized as an infinitesimally rigid body-and-hinge framework if and only if G can be realized as that with the additional "hinge-coplanar" property, i.e., all the hinges incident to each body are contained in a common hyperplane. This conjecture is called the Molecular Conjecture due to the equivalence between the infinitesimal rigidity of "hinge-coplanar" body-and-hinge frameworks and that of bar-and-joint frameworks derived from molecules in 3-dimension. In 2-dimensional case this conjecture has been proved by Jackson and Jordán in 2006. In this paper we prove this long standing conjecture affirmatively for general dimension. Also, as a corollary, we obtain a combinatorial characterization of the 3-dimensional bar-and-joint rigidity matroid of the square of a graph.