Characterizing Generic Global Rigidity

Global rigidity has applications in chemistry, where various technologies measure interatomic distances. From this data one may try to infer the geometry of the configuration [11, inter alia]. This inference is only well posed if the associated framework is globally rigid. Moreover, testing for generic global rigidity can be used as part of a divide-and-conquer strategy for this inference [18]. Similar problems arise in the field of sensor networks [7, inter alia].

1.1. Definitions and results. Definition 1.1. A graph Γ is a set of v vertices V(Γ) and e edges E(Γ), where E(Γ) is a set of two-element subsets of V(Γ). We will typically drop the graph Γ from this notation. A configuration of Γ in E d is a mapping from V(Γ) to Euclidean space E d . A framework ρ in E d is a graph Γ together with a configuration of Γ in E d ; we will also say that ρ is a framework of Γ. Let C d (Γ) denote the space of frameworks with a given graph Γ and dimension d. For ρ ∈ C d (Γ) and u ∈ V(Γ), let ρ(u) denote the image of u under the configuration of ρ. For a given graph Γ and dimension d, the length-squared function : C d (Γ) → R e is the function assigning to each edge of Γ its squared edge length in the framework. In particular, the component of (ρ) in the direction of an edge {u, w} is |ρ(u) -ρ(w)| 2 . Definition 1.2. A framework ρ in E d is congruent to another framework if they are related by an element of the group Eucl(d) of rigid motions of E d (rotations, reflections, and translations). We say that ρ is globally rigid if ρ is the only framework of Γ in E d with the same edge lengths, up to congruence. Equivalently, ρ is globally rigid iff -1 ( (ρ))/ Eucl(d) consists of [ρ], the congruence class of frameworks which contains ρ. Definition 1.3. A framework ρ ∈ C d (Γ) is locally rigid if there exists a neighborhood U of ρ in C d (Γ) such that ρ is the only framework in U with the same set of edge lengths, up to congruence; equivalently, [ρ] is isolated in -1 ( (ρ))/ Eucl (d).

(This property is usually just called rigidity but we will use term local rigidity in this paper to make explicit its distinction from global rigidity.) Definition 1.4. A framework is generic if the coordinates of its configuration do not satisfy any non-trivial algebraic equation with rational coefficients.

We first deal with graphs with very few vertices. Asimow and Roth proved that a generic framework ρ in E d of a graph Γ with d + 1 or fewer vertices is globally rigid if Γ is a complete graph (i.e., a simplex), otherwise it is not even locally rigid [1, Corollary 4]. Therefore in the rest of the paper we may assume that our graph has d + 2 or more vertices. In particular, this implies that a generic framework does not lie in a proper affine subspace of E d .

Since Eucl(d) acts freely on frameworks that do not lie in a proper affine subspace of E d , for such a framework ρ, the fiber -1 ( (ρ)) always has dimension at least dim(Eucl(d)) (which is d+1 2 ). In particular, a generic framework with at least d + 1 vertices satisfies this condition. The intuition behind the following definition and theorem is that the dimension of a generic fiber of an algebraic map f is the same as the kernel of the Jacobian df x of f at a generic point x, and for a generic locally rigid framework the kernel of df x only contains the tangents to the action of Eucl(d).

Definition 1.5. Let d ρ be the rigidity matrix of ρ, the Jacobian of at the framework ρ; by definition, this is a linear map from C d (Γ) to R e . A framework ρ ∈ C d (Γ) of a graph Γ with d + 1 or more vertices is infinitesimally rigid if (1) rank

Theorem 1.6 (Asimow-Roth [1]). If a generic framework ρ ∈ C d (Γ) of a graph Γ with d + 1 or more vertices is locally rigid in E d , then it is infinitesimally rigid. Otherwise rank d ρ is lower than vd -d+1 2 . Since the rank of the linearization an algebraic map is the same (and maximal) at every generic point, local rigidity in E d is a generic property. (See also Lemma 5.6.) That is, either all generic frameworks in C d (Γ) are locally rigid, or none of them are. Thus we can call this condition generic local rigidity in E d and consider it as a property of the graph Γ.

We next define some concepts we need to state the condition for generic global rigidity.

Definition 1.7. An equilibrium stress vector of a framework ρ of Γ is a real valued function ω on the (undirected) edge

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