Minimally globally rigid graphs

Enumerating Planar Minimally Rigid Graphs

We present an algorithm for enumerating without repetitions all the planar (noncrossing) minimally rigid (Laman) graphs embedded on a given generic set of n points. Our algorithm is based on the Reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n) time and O(n) space, or, with a slightly different implementation, in O(n) time and O(n) space. In particular, we obtai...

Globally rigid frameworks and rigid tensegrity graphs in the plane

On Minimally Highly Vertex-Redundantly Rigid Graphs

A graph G = (V,E) is called k-rigid in Rd if |V | ≥ k + 1 and after deleting any set of at most k − 1 vertices the resulting graph is rigid in Rd. A k-rigid graph G is called minimally k-rigid if the omission of an arbitrary edge results in a graph that is not k-rigid. B. Setvatius showed that a 2-rigid graph in R2 has at least 2|V |−1 edges and this bound is sharp. We extend this lower bound f...

The Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most ( 2n−4 n−2 ) ≈ 4n . We also exhibit several fami...

Combining Globally Rigid Frameworks

Here it is shown how to combine two generically globally rigid bar frameworks in d-space to get another generically globally rigid framework. The construction is to identify d+1 vertices from each of the frameworks and erase one of the edges that they have in common.