Combining Globally Rigid Frameworks

Globally rigid frameworks and rigid tensegrity graphs in the plane

On Global Rigidity and Universal Rigidity of Generic Bar Frameworks with Few Vertices

Gale transform was successfully used to classify polytopes with few vertices; that is, r-dimensional polytopes with r + k vertices, where k is small. In this paper, we use Gale transform to classify r-dimensional generic bar frameworks with r + k vertices, where k ≤ 4; as universally rigid, globally rigid or nonglobally rigid.

Operations preserving the global rigidity of graphs and frameworks in the plane

A straight-line realization of (or a bar-and-joint framework on) graph G in R is said to be globally rigid if it is congruent to every other realization of G with the same edge lengths. A graph G is called globally rigid in R if every generic realization of G is globally rigid. We give an algorithm for constructing a globally rigid realization of globally rigid graphs in R2. If G is triangle-re...

Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

A 2-dimensional framework (G, p) is a graph G = (V,E) together with a map p : V → R2. We view (G, p) as a straight line realization of G in R2. Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length. A pair of vertices {u, v} is globally linked in G if the distance between the points corresponding to u and v is the same in all pairs of equival...

Necessary Conditions for the Global Rigidity of Direction-Length Frameworks

It is an intriguing open problem to give a combinatorial characterisation or polynomial algorithm for determining when a graph is globally rigid in R. This means that any generic realisation is uniquely determined up to congruence when each edge represents a fixed length constraint. Hendrickson gave two natural necessary conditions, one involving connectivity and the other redundant rigidity. I...