TOPICS
Search

Generically Globally Rigid


A d-dimensional framework is a pair (G,p) where G=(V,E) is a graph with vertex set V and edge set E and p:V->R^d is a map that assigns a point in R^d to each vertex of G. The length of an edge uv in (G,p) is the Euclidean distance between p(u) and p(v). Call (G,p) a realization of G in R^d. The framework is said to be generic if the set of the coordinates of its points is algebraically independent over Q, and is said to be globally rigid if every other realization (G,q) of G in R^d in which corresponding edges have the same length is congruent to (G,p), i.e., the graph G and its edge lengths in (G,p) uniquely determine the pairwise distances of all vertices in (G,p).

After defining the above terminology, G is generically globally rigid in R^d if every (equivalently, if some) generic realization of G in R^d is globally rigid (Garamvölgyi et al. 2021).

If G is globally rigid in R^d on n>=d+2 vertices, then G is fully reconstructible in C^d (Garamvölgyi et al. 2021).


See also

Fully Reconstructible Graph

Explore with Wolfram|Alpha

References

Bernstein, D. I. and Gortler, S. J. "K_(5,5) is Fully Reconstructible in C^3." Submitted to Disc. Math., 2022.Garamvölgyi, D.; Gortler, S. J.; and Jordán, J. "Globally Rigid Graphs Are Fully Reconstructible." 10 May 2021. https://arxiv.org/abs/2105.04363.

Cite this as:

Weisstein, Eric W. "Generically Globally Rigid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GenericallyGloballyRigid.html

Subject classifications