Recently, V. Alexandrov proposed an intriguing sufficient condition for rigidity, which we will call transverse rigidity. We show that transverse rigidity is actually equivalent to the known sufficient condition for rigidity called prestress stabilit
y. Indeed this leads to a novel interpretation of the prestress condition.
We prove that universal second-order rigidity implies universal prestress stability and that triangulated convex polytopes in three-space (with holes appropriately positioned) are prestress stable.
We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework $(K(n,m),p,q)$. This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general position.
Even without any of these assumptions, in complete generality, we extend these ideas to obtain an efficient algorithm, based on a sequence of linear programs, that determines whether an input framework of a complete bipartite graph is universally rigid or not.
In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo-Euclidean space ($R^d$ with a metric of indefinite signature). We show that a graph is gener
ically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed.
