Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks

Abstract

We show that any graph that is generically globally rigid in R d has a realization in R d that is both generic and universally rigid. This also implies that the graph also must have a realization in R d that is both infinitesimally rigid and universally rigid; such a realization serves as a certificate of generic global rigidity.Our approach involves an algorithm by Lovász, Saks and Schrijver that, for a sufficiently connected graph, constructs a general position orthogonal representation of the vertices, and a result of Alfakih that shows how this representation leads to a stress matrix and a universally rigid framework of the graph. * An earlier version of this paper [15] had the title "Generic global and universal rigidity".

Maximum likelihood thresholds via graph rigidity

“…See Figure 2 for an illustration of Theorem 1.15. There is a link between globally rigid graphs and PSD equilibrium stresses, established in [18]. This will allow us to obtain bounds on the MLT using global rigidity.…”

“…Section 3 also develops the proof of Theorem 1.17 and direct consequences. The technical tools we use, from [2] and [18], arose in the study of universal rigidity. In Section 6, we combine Theorem 1.17 with results on graph rigidity in dimension 2 to completely solve the MLT problem for small values of mlt(G) and gcr(G).…”

Characterizing the universal rigidity of generic tensegrities

“…and since Z is non-singular ψ| T is an injective linear map. By (13), T and ρ∈ ˜ I d ρ ⊗ R d ρ ×d ρ have the same dimensions, and hence ψ| T is an isomorphism.…”

“…We say that (G, σ, p) is universally rigid if (G, σ, p) is globally rigid in R d for every integer d ≥ d. Clearly, universal rigidity implies global rigidity but the converse implication does not hold in general as indicated in Fig. 1 (See, e.g., [13] for further interaction between two rigidity concepts. )…”