2.1 Real and complex frameworks

Let $G = (V,E)$ be a graph on n vertices and $d \geq 1$ some fixed integer. A d-dimensional realization of G is a pair $(G,p)$, where $p : V \rightarrow \mathbb {R}^d$ maps the vertices of G into Euclidean space. We call such a map p (or equivalently, a point $p = (p_v)_{v \in V}$ in $\mathbb {R}^{nd}$) a configuration, and we say that the pair $(G,p)$ is a framework. Two d-dimensional frameworks $(G,p)$ and $(G,q)$ are equivalent if $\left \lVert p(u) - p(v)\right \rVert = \left \lVert q(u) - q(v)\right \rVert $ for every edge $uv \in E$ and congruent if the same holds for every pair of vertices $u,v \in V$. Here, $\left \lVert \cdot \right \rVert $ denotes the Euclidean norm.

A framework is (locally) rigid if every continuous motion of the vertices which preserves the edge lengths takes it to a congruent framework and globally rigid if every equivalent framework is congruent to it.

We say that a configuration $p \in \mathbb {R}^{nd}$ is generic if its $n \cdot d$ coordinates are algebraically independent over $\mathbb {Q}$. It is known that, in any fixed dimension d, both local and global rigidity are generic properties of the underlying graph, in the sense that either every generic d-dimensional framework is locally/globally rigid or none of them are (see [Reference Connelly5, Reference Gortler, Healy and Thurston10]). Thus, we say that a graph is rigid (respectively, globally rigid) in d dimensions if every (or equivalently, if some) generic d-dimensional realization of the graph is rigid (respectively, globally rigid).

Let $G = (V,E)$ be a graph on n vertices and $d \geq 1$. The function $m_{d,G} : \mathbb {R}^{nd} \rightarrow \mathbb {R}^E$ mapping each realization of G to the sequence of its Euclidean squared edge lengths is called the rigidity map or edge measurement map of G in d dimensions. That is, for a d-dimensional realization $(G,p)$ of G, the coordinate of $m_{d,G}(p)$ corresponding to the edge $uv \in E$ is $\left \lVert p(u) - p(v)\right \rVert ^2$.

Analogously to the real case, we define a d-dimensional complex framework to be a pair $(G,p)$, where $G = (V,E)$ is a graph and $p : V \rightarrow \mathbb {C}^d$ is a complex mapping. Given a framework $(G,p)$ in $\mathbb {C}^d$ and a pair of vertices $u,v$ in G, we define the complex squared distance of u and v in $(G,p)$ by

$$ \begin{align*} m_{uv}(p) = \left(p(u) - p(v)\right)^{T} \cdot \left(p(u) - p(v)\right) = \sum_{k=1}^d (p(u)_k - p(v)_k)^2, \end{align*} $$

where k indexes over the d dimension-coordinates. Note that in this definition we do not use conjugation, and thus, this is not the usual distance of $p(u)$ and $p(v)$. We also note that the mapping $m_{uv} : \mathbb {C}^{nd} \rightarrow \mathbb {C}$ depends on G and d, but since these will always be clear from the context, we shall omit them from our notation.

For an edge $e = uv$ of G, we say that $m_{uv}(p)$ is the complex squared length of the edge in $(G,p)$. For real frameworks, this coincides with the usual (Euclidean) squared length, so we can extend $m_{d,G}$ to a $\mathbb {C}^{nd} \rightarrow \mathbb {C}^E$ function by letting

$$ \begin{align*} m_{d,G}(p) = \big(m_{uv}(p)\big)_{uv \in E}. \end{align*} $$

We say, as in the real case, that two frameworks $(G,p)$ and $(G,q)$ are equivalent if $m_{d,G}(p) = m_{d,G}(q)$, and they are congruent if $m_{d,K_V}(p) = m_{d,K_V}(q)$, where $K_V$ is the complete graph on the vertex set V. A configuration $p \in \mathbb {C}^{nd}$ is, again, generic, if the coordinates of p are algebraically independent over $\mathbb {Q}$. A point $p \in \mathbb {R}^{nd}$ is generic as a real configuration precisely if it is generic as a complex one.

Although we will not explicitly need them, we note that one can define the analogues of rigidity and global rigidity for complex frameworks. It turns out that, as in the real case, the (global) rigidity of generic complex frameworks only depends on the underlying graph, and the corresponding graph property of being ‘(globally) rigid in $\mathbb {C}^d$’ is equivalent to being (globally) rigid in $\mathbb {R}^d$; see [Reference Gortler and Thurston11, Reference Gortler, Theran and Thurston12].

It follows from the definitions that globally rigid graphs are rigid. The following much stronger necessary conditions of global rigidity are due to Hendrickson [Reference Hendrickson13]. We say that a graph is redundantly rigid in a given dimension if it remains rigid after deleting any edge. A graph is k-connected for some $k \geq 2$ if it has at least $k+1$ vertices and it remains connected after deleting any set of less than k vertices.

In $d = 1,2$ dimensions, the conditions of Theorem 2.1 are, in fact, sufficient for global rigidity [Reference Jackson and Jordán14]. This fails in the $d \geq 3$ case, and a combinatorial characterization of globally rigid graphs in these dimensions is a major open question.

2.2 The rigidity matrix and the rigidity matroid

The rigidity matroid of a graph G is a matroid defined on the edge set of G which reflects the rigidity properties of all generic realizations of G. For a general introduction to matroid theory, we refer the reader to [Reference Oxley18]. Let $(G,p)$ be a realization of a graph $G=(V,E)$ in $\mathbb {R}^d$. The rigidity matrix of the framework $(G,p)$ is the matrix $R(G,p)$ of size $|E|\times d|V|$, where, for each edge $v_iv_j\in E$, in the row corresponding to $v_iv_j$, the entries in the d columns corresponding to vertices $v_i$ and $v_j$ contain the d coordinates of $(p(v_i)-p(v_j))$ and $(p(v_j)-p(v_i))$, respectively, and the remaining entries are zeros. In other words, it is $1/2$ times the Jacobian of the rigidity map $m_{d,G}$. The rigidity matrix of $(G,p)$ defines the rigidity matroid of $(G,p)$ on the ground set E by linear independence of rows. It is known that any pair of generic frameworks $(G,p)$ and $(G,q)$ have the same rigidity matroid. We call this the d-dimensional rigidity matroid ${\mathcal{R}}_d(G)=(E,r_d)$ of the graph G. We can define the rigidity matrix $R(G,p)$ for complex frameworks in the same way as in the real case. This, again, allows us to define the rigidity matroid of the framework. It is not difficult to show that the rigidity matroid of a generic framework in $\mathbb {C}^d$ is, again, the d-dimensional rigidity matroid $\mathcal {R}_d(G)$.

We denote the rank of ${\mathcal{R}}_d(G)$ by $r_d(G)$. A graph $G=(V,E)$ is $\mathcal {R}_d$-independent if $r_d(G)=|E|$, and it is an $\mathcal {R}_d$-circuit if it is not $\mathcal {R}_d$-independent, but every proper subgraph $G'$ of G is $\mathcal {R}_d$-independent. We note that in the literature such graphs are sometimes called M-independent in $\mathbb {R}^d$ and M-circuits in $\mathbb {R}^d$, respectively. An edge e of G is an $\mathcal {R}_d$-bridge in G if $r_d(G-e)=r_d(G)-1$ holds. Equivalently, e is an $\mathcal {R}_d$-bridge in G if it is not contained in any subgraph of G that is an $\mathcal {R}_d$-circuit.

Gluck characterized rigid graphs in terms of their rank.

Let ${\cal M}$ be a matroid on ground set E with rank function r. We can define a relation on the pairs of elements of E by saying that $e,f\in E$ are equivalent if $e=f$ or there is a circuit C of ${\cal M}$ with $\{e,f\}\subseteq C$. This defines an equivalence relation. The equivalence classes are the connected components of ${\cal M}$. The matroid is said to be connected if there is only one equivalence class and separable otherwise. We shall use the fact that ${\cal M}$ is separable if and only if there is a partition $E=E_1\cup E_2$ of E into two nonempty subsets for which

$$ \begin{align*} r({\cal M})=r({\cal M}_1)+r({\cal M}_2), \end{align*} $$

holds, where ${\cal M}_i$ denotes the restriction of ${\cal M}$ to $E_i$, $i=1,2$.

Given a graph $G=(V,E)$, the subgraphs induced by the edge sets of the connected components of ${\mathcal{R}}_d(G)$ are the $\mathcal {R}_d$-connected components of G. The graph is said to be $\mathcal {R}_d$-connected if ${\mathcal{R}}_d(G)$ is connected, and $\mathcal {R}_d$-separable otherwise. See Figure 1 for an example of an $\mathcal {R}_3$-separable graph.

Figure 1

A $3$-connected, redundantly rigid and ${\mathcal{R}}_3$-separable graph. This graph satisfies $r_3(G)= 36 = 27 + 9 =r_3(G^o)+r_3(K_5)$, where $G^o$ is the outer ring of $K_5$’s and $K_5$ is the subgraph induced by the black (filled) vertices.

In one and two dimensions, $\mathcal {R}_d$-connectivity has close ties with (global) rigidity, as shown by the following result.

It is known that part (b) of Theorem 2.3 is not true in $d \geq 3$ dimensions. On the other hand, we shall show that part (a) remains valid in d dimensions for all $d \geq 3$ (Theorem 3.5). We note that this is essentially a strengthening of the second part of Hendrickson’s theorem (Theorem 2.1), which can be seen as follows. A graph is redundantly rigid in $\mathbb {R}^d$ if and only if it is rigid in $\mathbb {R}^d$ and contains no $\mathcal {R}_d$-bridges, and $\mathcal {R}_d$-connected graphs contain no $\mathcal {R}_d$-bridges (indeed, e is an $\mathcal {R}_d$-bridge in G if and only if $\{e\}$ is an $\mathcal {R}_d$-connected component of G). Since a globally rigid graph is clearly rigid, Theorem 3.5 implies that such a graph (on at least $d+2$ vertices) is redundantly rigid as well.

Let $G = (V,E)$ be a graph on n vertices and $(G,p)$ a framework in $\mathbb {C}^d$. The elements of $\ker (R(G,p)) \subseteq \mathbb {C}^{nd}$ are the infinitesimal motions of $(G,p)$, while the elements of $\ker (R(G,p)^T) \subseteq \mathbb {C}^E$ are the equilibrium stresses (or stresses, for short) of $(G,p)$. We shall also use the notation $S(G,p) = \ker (R(G,p)^T)$. Let us denote the column space of $R(G,p)$ by $\operatorname {\mathrm {span}}(R(G,p))$. By basic linear algebra, $S(G,p)$ is the orthogonal complementFootnote ¹ of $\operatorname {\mathrm {span}}(R(G,p))$ in $\mathbb {C}^E$. Since the elements of $S(G,p)$ capture the row dependences of $R(G,p)$, we have that G is $\mathcal {R}_d$-independent if and only if for every generic realization $(G,p)$ in $\mathbb {C}^d$, we have $S(G,p) = \{0\}$, and G is an $\mathcal {R}_d$-circuit if and only if every generic realization $(G,p)$ in $\mathbb {C}^d$ has a unique (up to scalar multiple) nonzero equilibrium stress $\omega $ and $\omega $ is nonzero on every edge of G.

Suppose that G is $\mathcal {R}_d$-separable and let $E = E_1 \cup E_2$ be a partition of E into nonempty subsets such that for the graphs $G_i$ induced by $E_i, i=1,2$, we have $r_d(G) = r_d(G_1) + r_d(G_2)$. It is not difficult to see that, in this case for any generic realization $(G,p)$ in $\mathbb {C}^d$, we have $\operatorname {\mathrm {span}}(R(G,p)) = \operatorname {\mathrm {span}}(R(G_1,p)) \oplus \operatorname {\mathrm {span}}(R(G_2,p))$ as linear subspaces of $\mathbb {C}^E$, under the identification $\mathbb {C}^E = \mathbb {C}^{E_1} \times \mathbb {C}^{E_2}$. This also implies $S(G,p) = S(G_1,p) \oplus S(G_2,p)$ under the same identification.

We close this section by recalling the well-known coning operation on a graph. Given a graph G, the cone of G is obtained by adding a new vertex v to G, along with new edges from v to every vertex of G. Coning provides a transfer between various rigidity properties in d and $d+1$ dimensions, as shown by the following results.

2.3 Affine maps and conics at infinity

We say that a framework $(G,p)$ in $\mathbb {C}^d$ has full affine span if the affine span of the image of the vertices under p is all of $\mathbb {C}^d$. A configuration $q \in \mathbb {C}^{nd}$, viewed as a point $q = (q_v)_{v \in V}$, is an affine image of p if $q_v = Ap_v + b, v \in V$ for some matrix $A \in \mathbb {C}^{d \times d}$ and vector $b \in \mathbb {C}^d$. We say that p and q are strongly congruent if q can be obtained as the affine image of p under a rigid motion, i.e., an affine map $x \mapsto Ax + b$ such that $A^TA$ is the identity matrix.

Two frameworks $(G,p)$ and $(G,q)$ in $\mathbb {R}^d$ are strongly congruent if and only if they are congruent. This is not always the case for frameworks in $\mathbb {C}^d$. However, congruent frameworks that have full affine span are strongly congruent; see [Reference Gortler and Thurston11, Corollary 8].

Let $(G,p)$ be a framework in $\mathbb {C}^d$. We say that the edge directions of $(G,p)$ lie on a conic at infinity if there is a nonzero symmetric matrix Q such that, for every edge $uv$ of G, $(p(u) - p(v))^T Q (p(u) - p(v)) = 0$. The following lemma is implied by results of Connelly (see, e.g., [Reference Connelly5, Proposition 4.2]). For completeness, we give a proof.

Proof. The ‘if’ direction is immediate. In the other direction, suppose that $m_{d,G}(q) = m_{d,G}(p)$. Let $x \mapsto Ax + b$ be an affine transformation that sends p to q. It follows from the definitions that, for any pair of vertices $u,v \in V$,

$$ \begin{align*} m_{uv}(q) = (p(u) - p(v))^T A^T A (p(u) - p(v)). \end{align*} $$

Therefore we have

(2.1)$$ \begin{align} m_{uv}(q) - m_{uv}(p) = (p(u) - p(v))^T (A^T A - I) (p(u) - p(v)). \end{align} $$

By assumption, for every edge $uv \in E$, the left-hand side of equation (2.1) is zero. Since the edge directions of $(G,p)$ do not lie on a conic at infinity, this implies $A^TA - I= 0$ so that the left-hand side is zero for every pair of vertices $u,v \in V$, which is what we wanted to show.

The following lemma is stated in [Reference Connelly5] for frameworks in $\mathbb {R}^d$, but the same proof works for frameworks in $\mathbb {C}^d$.

The following lemma is folklore.

Finally, we shall use the following property of globally rigid graphs which is easy to deduce from previous results on global rigidity and maximum rank stress matrices. We sketch the proof and refer the reader to [Reference Connelly5, Reference Gortler, Healy and Thurston10] for the definitions and key theorems.

2.4 Algebraic geometry background

We briefly recall the notions from algebraic geometry that we shall use. For a more detailed exposition, see [Reference Gortler, Theran and Thurston12, Appendix A] or [Reference Garamvölgyi and Jordán8, Section 2.2]. We say that a subset $X \subseteq \mathbb {C}^m$ is a variety if it is the set of simultaneous vanishing points of some polynomials $f_1,\ldots ,f_k \in \mathbb {C}[x_1,\ldots ,x_m]$. The varieties in $\mathbb {C}^m$ form the closed sets of the so-called Zariski topology. For an arbitrary set $X \subseteq \mathbb {C}^m$, we shall use $\overline {X}$ to denote its closure in the Zariski topology on $\mathbb {C}^m$.

Let $X \subseteq \mathbb {C}^m$ be a variety. We denote by $I(X) \subseteq \mathbb {C}[x_1,\ldots ,x_m]$ the set of polynomials that vanish on X. We say that X is irreducible if it cannot be written as the proper union of a finite number of varieties, and it is defined over $\mathbb {Q}$ if $I(X)$ has a generating set consisting of polynomials with rational coefficients. The dimension of an irreducible variety X is the largest number k such that there exists a chain $X_0 \subsetneq X_1 \subsetneq \cdots \subsetneq X_k = X$ of irreducible varieties. From this definition, the following useful fact is immediate: If $X,Y \subseteq \mathbb {C}^m$ are irreducible varieties of the same dimension with $X \subseteq Y$, then $X = Y$. We shall also use the following result.

Let $X \subseteq \mathbb {C}^m$ be an irreducible variety. At each point $x \in X$, we define the Zariski tangent space of X at x, denoted by $T_x X$, to be the kernel of the Jacobian matrix of a set of generating polynomials of $I(X)$, evaluated at x. Thus, $T_x X$ is a linear subspace of $\mathbb {C}^m$. We say that x is smooth if $\dim (T_x X) = \dim (X)$. If X is homogeneous (i.e., it can be defined by homogeneous polynomials, or equivalently, $tx \in X$ for every $x \in X$ and $t \in \mathbb {C}$) and $x \in X$ is a smooth point, we define the Gauss fiber corresponding to x to be the set $\{y \in X: y \text { is smooth and } T_y X = T_x X\}$.Footnote ²

Let $X \subseteq \mathbb {C}^m$ be a variety defined over $\mathbb {Q}$. We say that a point $x \in X$ is generic in X if the only polynomials with rational coefficients satisfied by x are those in $I(X)$. Note that a framework $(G,p)$ in $\mathbb {C}^d$ is generic if and only if p is generic as a point of the variety $\mathbb {C}^{nd}$. If X is an irreducible variety defined over $\mathbb {Q}$, then every generic point of X is smooth. We shall also need the following result.

2.5 The measurement variety

Recall that for a graph $G = (V,E)$, we denote its d-dimensional edge measurement map by $m_{d,G} : \mathbb {C}^{nd} \rightarrow \mathbb {C}^E$.

We shall frequently use the following lemma on generic points. It follows by applying [Reference Gortler, Theran and Thurston12, Lemmas 4.4, A.7, A.8] to the varieties $\mathbb {C}^{nd}$, $M_{d,G}$ and the map $m_{d,G}$.

It is known that the measurement variety, being the Zariski-closure of the image of an irreducible variety defined over $\mathbb {Q}$, is also an irreducible variety defined over $\mathbb {Q}$. It follows from the definition and basic topological considerations that if $E' \subseteq E$ is a subset of edges inducing a subgraph $G'$ of G, then $M_{d,G'} = \overline {\pi _{E'}(M_{d,G})}$, where $\pi _{E'} : \mathbb {C}^E \rightarrow \mathbb {C}^{E'}$ is the projection onto the coordinate axes corresponding to $E'$; see [Reference Garamvölgyi and Jordán8, Lemma 3.8].

In what follows, we shall frequently compare the measurement varieties of different graphs, say $G = (V,E)$ and $H = (V',E')$, that have the same number of edges. Since $M_{d,G}$ and $M_{d,H}$ lie in different ambient spaces, to compare them we must specify an identification between $\mathbb {C}^E$ and $\mathbb {C}^{E'}$. To this end, we introduce the following notation. Let $\psi : E \rightarrow E'$ be a bijection between the edge sets of G and H. We write that $M_{d,G} =_\psi M_{d,H}$ if $\Psi (M_{d,G}) = M_{d,H}$, where $\Psi : \mathbb {C}^{E} \rightarrow \mathbb {C}^{E'}$ is the mapping induced by $\psi $ in the natural way. Similarly, we write $M_{d,G} \subseteq _\psi M_{d,H}$ if $\Psi (M_{d,G}) \subseteq M_{d,H}$.

The following results show that $\mathcal {R}_d(G)$ is ‘encoded’ in the measurement variety in some sense. This has been observed before; see, e.g., [Reference Garamvölgyi and Jordán8, Reference Gortler, Theran and Thurston12, Reference Rosen, Sidman and Theran19]. Using the terminology of the latter paper, the situation can be summarized by saying that the algebraic matroid corresponding to the variety $M_{d,G}$ is $\mathcal {R}_d(G)$.

Lemma 2.13. Let G be a graph on n vertices. Then

$$ \begin{align*} \dim(M_{d,G})=r_{d}(G). \end{align*} $$

In particular, for $n \geq d+1$ we have $\dim (M_{d,G}) \leq nd - \binom {d+1}{2}$ and equality holds if and only if G is rigid in $\mathbb {R}^d$. Moreover, G is $\mathcal {R}_d$-independent if and only if $M_{d,G} = \mathbb {C}^E$.

The following result shows that the measurement variety also encodes the space of stresses of generic frameworks. This follows from the fact that $S(G,p)$ is the orthogonal complement of $\operatorname {\mathrm {span}}(R(G,p))$ in $\mathbb {C}^E$ using standard results in differential geometry; see [Reference Gortler, Healy and Thurston10, Lemma 2.21] or [Reference Gortler, Theran and Thurston12, Lemma 4.10].

The lemma implies that if $(G,p)$ and $(G,q)$ are generic frameworks in $\mathbb {C}^d$, then $S(G,p)\not = S(G,q)$ if and only if the Gauss fibers corresponding to $m_{d,G}(p)$ and $m_{d,G}(q)$ are different. We shall use this corollary later.

2.6 Unlabeled reconstruction

In what follows, it will be convenient to use the following notions. We say that two frameworks $(G,p)$ and $(H,q)$ in $\mathbb {C}^d$ are length-equivalent (under the bijection $\psi $) if there is a bijection $\psi $ between the edge sets of G and H such that, for every edge e of G, the complex squared length of e in $(G,p)$ is equal to the complex squared length of $\psi (e)$ in $(H,q)$. For an edge bijection $\psi : E(G) \rightarrow E(H)$ and a graph isomorphism $\varphi : V(G) \rightarrow V(H)$, we say that $\psi $ is induced by $\varphi $ if for every edge $e = uv \in E(G)$ we have $\psi (e) = \varphi (u)\varphi (v)$.

In this paper, we shall mainly focus on the following stronger property, where the condition on the number of vertices of H is omitted.

Note that, since we assume $(G,p)$ to be generic, its edge lengths are pairwise distinct, and hence, the bijection $\psi $ is unique in the above definitions. We also point out that, when considering full reconstructibility, it is natural to only consider graphs without isolated vertices since from any framework we can create other length-equivalent frameworks by adding isolated vertices. On the other hand, in the case of strong reconstructibility it is sensible to consider graphs with isolated vertices. In fact, it follows immediately from the definitions that G is fully reconstructible in $\mathbb {C}^d$ if and only if every graph obtained from G by adding zero or more isolated vertices is strongly reconstructible in $\mathbb {C}^d$.

As Theorem 2.19 below shows, both strong and full reconstructibility of a generic framework can be characterized in terms of a certain uniqueness condition on the measurement variety $M_{d,G}$ of the underlying graph, and in fact it is this formulation of reconstructibility that we shall most commonly use throughout the paper. This also implies that these reconstructibility notions are generic properties of a graph in the sense that if there is a generic framework $(G,p)$ in $\mathbb {C}^d$ which is strongly (respectively, fully) reconstructible, then every generic realization of G in $\mathbb {C}^d$ is strongly (respectively, fully) reconstructible. This motivates the following definition.

The ‘strongly reconstructible’ part of Theorem 2.19 is [Reference Garamvölgyi and Jordán8, Theorem 3.4]. The same proof works for the ‘fully reconstructible’ version after omitting the condition on the number of vertices of H.

We close this section by recalling the main result of [Reference Gortler, Theran and Thurston12].

In the next section, we shall strengthen this result by proving that globally rigid graphs on at least $d+2$ vertices are, in fact, fully reconstructible in $\mathbb {C}^d$ for $d\geq 2$. The cases $d = 1,2$ were already settled in [Reference Garamvölgyi and Jordán8] by verifying the following equivalence.

In Section 4, we shall give examples showing that, for $d \geq 3$, there are fully reconstructible graphs in $\mathbb {C}^d$ (on at least $d+2$ vertices) that are not globally rigid in $\mathbb {R}^d$. On the other hand, we do not know any example of a strongly reconstructible graph in $\mathbb {C}^d$ on at least $d+2$ vertices that is not fully reconstructible in $\mathbb {C}^d$, although it seems likely that such a graph exists.

3.1 The structure of the measurement variety

The next lemma implies that the measurement variety of an $\mathcal {R}_d$-separable graph G is the product of the measurement varieties of its $\mathcal {R}_d$-connected components. A special case of this statement when G contains an $\mathcal {R}_d$-bridge was proved in [Reference Garamvölgyi and Jordán8, Theorem 3.13].

Proof. For clarity, we shall write m and M instead of $m_{d,G}$ and $M_{d,G}$ in the following. For any $q \in \mathcal {A}$, if q has full affine span, then by Lemma 2.7, we have $S(G,p) = S(G,q)$.

Let $\mathcal {A}^g$ denote the set of frameworks in $\mathcal {A}$ that are generic. Since $(G,p)$ is generic, this set is nonempty, and since $\mathcal {A}$ is irreducible (being a linear space), Lemma 2.10 implies $\overline {\mathcal {A}^g} = \mathcal {A}$. Now for any $q \in \mathcal {A}^g$, we have $S(G,q) = S(G,p)$ by Lemma 2.7, and it follows by Lemma 2.15 that $T_{m(q)} M = T_{m(p)} M$, or in other words, $m(q) \in F$.

This shows that $m(\mathcal {A}^g) \subseteq F$. Taking Zariski-closures and using the continuity of m with respect to the Zariski topology, we have

$$ \begin{align*} \overline{F} \supseteq \overline{m(\mathcal{A}^g)} = \overline{m(\overline{\mathcal{A}^g})} = \overline{m(\mathcal{A})}, \end{align*} $$

as desired.

Although we shall not use this fact, we note that, by [Reference Gortler, Theran and Thurston12, Lemma 4.6], $m_{d,G}(\mathcal {A})$ is a linear space, and in particular it is closed.

Let G be a graph and $d \geq 1$. We say that a Gauss fiber F of $M_{d,G}$ is generic if it contains a point that is generic in $M_{d,G}$. It will also be convenient to use the following notion in the next proof. Let $d \geq 2$, and let n denote the number of vertices of G. We say that a framework $(G,p)$ in $\mathbb {C}^d$ is a lifting of the framework $(G,q)$ in $\mathbb {C}^{d-1}$ if $(G,q)$ is obtained by projecting the image of each vertex in $(G,p)$ onto the first $d-1$ coordinate axes. If $(G,q)$ is generic, then the generic liftings of $(G,q)$ form a dense subset of the space of liftings of $(G,q)$. This follows from the basic fact that for any finite set $S \subseteq \mathbb {C}$ that is algebraically independent over $\mathbb {Q}$, the numbers $x \in \mathbb {C}$ for which $S \cup \{x\}$ is also algebraically independent form a dense subset of $\mathbb {C}$.

Proof. For clarity, we shall suppress the role of $\psi $ and assume that $M_{d,G}$ and $M_{d,H}$ are in the same ambient space $\mathbb {C}^E$ so that $M_{d,G} = M_{d,H}$. Let R be the ring of polynomials over $\mathbb {C}$ with variables indexed by E and let $I(M_{d-1,G}),I(M_{d-1,H}) \subseteq R$ be the set of polynomials vanishing on $M_{d-1,G}$ and $M_{d-1,H}$, respectively. Moreover, let $R_{\mathbb {Q}},I_{\mathbb {Q}}(M_{d-1,G})$ and $I_{\mathbb {Q}}(M_{d-1,H})$ denote the subset of $R,I(M_{d-1,G})$ and $I(M_{d-1,H})$, respectively, consisting of polynomials with rational coefficients.

Theorem 2.1 implies that G has no $\mathcal {R}_d$-bridges, so in particular it cannot be $\mathcal {R}_d$-independent. By Theorem 2.14, it follows that H is not $\mathcal {R}_d$-independent either, and thus, part a) of Lemma 3.3 implies that there is a generic point $x \in M_{d-1,H} \subseteq M_{d,H}$ such that there is an infinite number of generic Gauss fibers $F \subseteq M_{d,H}$ with $x \in \overline {F}$. Part b) of the same lemma then implies that x is in $M_{d-1,G}$. From this and the fact that x is generic in $M_{d-1,H}$, we have the following chain of containments:

(3.1)$$ \begin{align} I_{\mathbb{Q}}(M_{d-1,G}) \subseteq \{f \in R_{\mathbb{Q}}: f(x) = 0\} = I_{\mathbb{Q}}(M_{d-1,H}). \end{align} $$

Since both $M_{d-1,G}$ and $M_{d-1,H}$ are defined over $\mathbb {Q}$, $I_{\mathbb {Q}}(M_{d-1,G})$ and $I_{\mathbb {Q}}(M_{d-1,H})$ generate $I(M_{d-1,G})$ and $I(M_{d-1,H})$, respectively. Thus, equation (3.1) also implies $I(M_{d-1,G}) \subseteq I(M_{d-1,H})$, which is equivalent to $M_{d-1,H} \subseteq M_{d-1,G}$.

3.2 Globally rigid graphs are $\mathcal {R}_d$-connected

We are ready to prove the first main result of this section.

Theorem 3.5 was known to hold in $\mathbb {R}^1$ (where global rigidity, $2$-connectivity and $\mathcal {R}_1$-connectivity are equivalent) and in $\mathbb {R}^2$, see [Reference Jackson and Jordán14] (c.f. Theorem 2.3 above). We note that in [Reference Jordán and Tanigawa17] it is conjectured that globally rigid graphs are ‘nondegenerate’ in $\mathbb {R}^d$, a condition that is stronger than $\mathcal {R}_d$-connectivity.

Underlying the proof of Theorem 3.5 is the following structural observation on the measurement variety of G, which we give without details. Since G is globally rigid, [Reference Gortler, Healy and Thurston10, Theorem 4.4] and [Reference Gortler, Theran and Thurston12, Lemma 4.24, Remark 4.25] imply that for any generic Gauss fiber F of $M_{d,G}$ we have $\dim (\overline {F}) = \binom {d+1}{2}$. On the other hand, it follows from the definitions that if $M_{d,G} = M_{d,G_1} \times M_{d,G_2}$, then $F = F_1 \times F_2$, where $F_1$ and $F_2$ are some generic Gauss fibers of $M_{d,G_1}$ and $M_{d,G_2}$, respectively. Using Lemma 3.2, it can be shown that, under the assumptions on $G_1$ made in the proof of Theorem 3.5, we have $\dim (\overline {F_1}) \geq \binom {d+1}{2}$. Since $\dim (\overline {F_2}) \geq 1$, this gives

$$ \begin{align*} \binom{d+1}{2} = \dim(\overline{F}) = \dim(\overline{F_1}) + \dim(\overline{F_2})> \binom{d+1}{2}, \end{align*} $$

a contradiction.

3.3 Globally rigid graphs are fully reconstructible

Our goal in this subsection is to prove the following result, which gives an affirmative answer to [Reference Gortler, Theran and Thurston12, Question 7.5].

Theorem 4.7, below, will lead to examples that show that global rigidity is not necessary for full reconstructibility.

Our proof of Theorem 3.6 uses Theorem 3.5. In fact, as the following theorem shows, the former result is a strengthening of the latter.

This theorem is similar in spirit to [Reference Garamvölgyi and Jordán8, Theorem 5.21], which states that if G is strongly reconstructible in $\mathbb {C}^d$ on at least $d+2$ vertices and without isolated vertices, then it cannot contain an $\mathcal {R}_d$-bridge. Example 4.6 in the next section shows that the converse of Theorem 3.7 is not true: The $6$-ring depicted in Figure 4 is not even strongly reconstructible in $\mathbb {C}^3$, despite being $\mathcal {R}_3$-connected, $4$-connected and redundantly rigid in $\mathbb {R}^3$.

Figure 2

This graph G is $4$-connected, redundantly rigid in $\mathbb {R}^3$, and ${\mathcal{R}}_3$-separable. It satisfies $r_3(G)= 105 = 96 + 9 =r_3(G^o)+r_3(K_5)$, where $G^o$ is the outer ring of $K_5$’s.

Figure 3

This graph is also $4$-connected, redundantly rigid in $\mathbb {R}^3$, and ${\mathcal{R}}_3$-separable.

Figure 4

A graph that is $4$-connected, redundantly rigid in $\mathbb {R}^3$ and ${\mathcal{R}}_3$-connected but not globally rigid in $\mathbb {R}^3$.

We now turn our attention to Theorem 3.6. As our argument is quite involved, we start by giving a brief outline of the proof. By Theorem 2.19, to show that the globally rigid graph G is fully reconstructible, we need to show that, whenever $M_{d,G} =_\psi M_{d,H}$ for some graph H without isolated vertices and some edge bijection $\psi : E(G) \rightarrow E(H)$, we have that H is isomorphic to G (and the isomorphism induces the appropriate edge bijection). Let n and $n'$ denote the number of vertices of G and H, respectively. If $n = n'$, then we are done by the strong reconstructibility of G (Theorem 2.20). If $n' < n$, then $M_{d,H}$ must necessarily be of lower dimension than $M_{d,G}$, which is impossible. The only remaining possibility to be ruled out is that $n'>n$; note that in this case $M_{d,G} =_\psi M_{d,H}$ (and in particular the equality of dimensions) implies that H is locally flexible in $\mathbb {R}^d$.

We rule this out as follows. From $M_{d,G} =_\psi M_{d,H}$ and $M_{d-1,H} \subseteq _\psi M_{d-1,G}$ (which follows from Corollary 3.4), we get bounds on $k_d$ and $k_{d-1}$, the generic dimension of the space of infinitesimal motions of H in d and $d-1$ dimensions, respectively. Using Theorem 3.5, we also get that H is $\mathcal {R}_d$-connected and in particular connected. The crux of our argument is Corollary 3.12, which states that for a connected graph, $k_d - \binom {d+1}{2}$ (the ‘d-dimensional degrees of freedom’ of the graph) is larger by a multiplicative factor than $k_{d-1} - \binom {d}{2}$, the $(d-1)$-dimensional degrees of freedom of the graph. Applying this to H, we shall get a contradiction with the bounds on $k_d$ and $k_{d-1}$ obtained previously.

Before proving Corollary 3.12, we need a number of technical results. The first is about the structure of a certain variety, while the second gives a concrete example of an infinitesimal motion in d dimensions that, generically, cannot be decomposed into two $(d-1)$-dimensional infinitesimal motions. The subsequent lemmas use this result to obtain bounds on the generic dimension of infinitesimal motions in $\mathbb {C}^i$ for all $1 \leq i \leq d$.

Proof. a) Let H be an arbitrary hyperplane in $\mathbb {C}^E$ whose orthogonal complement is generated by some nonzero $\omega \in \mathbb {C}^E$. The configurations $p \in \mathbb {C}^{n}$ for which $\omega $ is an equilibrium stress of $(G,p)$ form a proper linear subspace of $\mathbb {C}^{n}$. In particular, we can find a generic framework $(G,p)$ in $\mathbb {C}^1$ which does not have $\omega $ as an equilibrium stress. By Lemma 2.15, $\omega $ is not in the orthogonal complement of the tangent space of $M_{1,G}$ at $m_{1,G}(p)$. It follows that this tangent space is not contained in H, which implies that $M_{1,G}$ is not contained in H either, as desired.

b) By direct calculationFootnote ⁴ we have that $f = m_{2,G} \circ \alpha $ where $\alpha : \mathbb {C}^{2n} \rightarrow \mathbb {C}^{2n}$ is defined by

$$ \begin{align*} p = \left(p^{(1)}_v,p^{(2)}_v\right)_{v \in V} \longmapsto \left(\frac{p^{(1)}_v + p^{(2)}_v}{2}, \frac{p^{(1)}_v - p^{(2)}_v}{2\sqrt{-1}} \right)_{v \in V}. \end{align*} $$

Since $\alpha $ is a linear automorphism of $\mathbb {C}^{2n}$, this implies that $f(\mathbb {C}^{2n}) = m_{2,G}(\mathbb {C}^{2n})$, and thus $\overline {f(\mathbb {C}^{2n})} = M_{2,G}$. Now by part a), $M_{1,G}$ is not contained in any linear hyperplane in $\mathbb {C}^E$. Since $M_{1,G} \subseteq M_{2,G}$, the same holds for $M_{2,G}$.

For the next two lemmas, we introduce the following notation. Let $d \geq 3$ and let $G = (V,E)$ be a graph on n vertices. For a framework $(G,p)$ in $\mathbb {C}^d$, let $W_1(G,p) \leq \mathbb {C}^{nd}$ denote the set of infinitesimal motions of $(G,p)$ that are supported on the first $d-1$ coordinates, that is, infinitesimal motions of the form $\left (q_v,0\right )_{v \in V}$, where $q_v \in \mathbb {C}^{d-1}$ for each vertex v. Similarly, let $W_2(G,p)$ denote the set of infinitesimal motions that are supported on the last $d-1$ coordinates, and let $W(G,p)$ be the subspace of $\mathbb {C}^{nd}$ spanned by $W_1(G,p) \cup W_2(G,p)$.

Lemma 3.9. Let $d \geq 3$ and let $G = (V,E)$ be a graph on n vertices that is not $\mathcal {R}_{d-2}$-independent. Then for any generic configuration $p = \left (p_v^{(1)},\dots ,p^{(d)}_v\right )_{v \in V} \in \mathbb {C}^{nd}$, the infinitesimal rotation $\varphi = \varphi (p)$ of $(G,p)$ defined by

$$ \begin{align*} \varphi_v = (- p_v^{(d)}, 0, \ldots, 0, p^{(1)}_v) \hspace{2em} \forall v \in V \end{align*} $$

is not in the subspace $W(G,p)$. In particular, $W(G,p)$ is a proper subspace of the space of infinitesimal motions of $(G,p)$.

Proof. Consider an arbitrary realization $(G,p)$ in $\mathbb {C}^d$, and let $\widetilde {p} \in \mathbb {C}^{n(d-2)}$ denote the projection of p onto the middle $d-2$ coordinate axes. Suppose that $\varphi (p)$ can be written as $\varphi (p) = q + r$, where $q \in W_1(G,p)$ and $r \in W_2(G,p)$. Then $q_v = (- p_v^{(d)}, \widetilde {q}_v, 0)$ for every vertex $v \in V$, where $\widetilde {q} = (\widetilde {q}_v)_{v \in V} \in \mathbb {C}^{n(d-2)}$. Since q is an infinitesimal motion, we have that

$$\begin{align*}\left(p_v - p_u\right) \cdot \left(q_v - q_u\right) = 0, \hspace{2em} \forall uv \in E.\end{align*}$$

Using the above description of q and rearranging gives

(3.2)$$ \begin{align} (\widetilde{p}_v - \widetilde{p}_u)\cdot (\widetilde{q}_v - \widetilde{q}_u) = (p_v^{(1)} - p_u^{(1)})(p_v^{(d)} - p_u^{(d)}), \hspace{2em} \forall uv \in E. \end{align} $$

We shall show that, for a generic realization $(G,p)$ of G in $\mathbb {C}^d$, equation (3.2) cannot hold for any vector $\left (\widetilde {q}_v\right )_{v \in V}$. We start by observing that $(G,\widetilde {p})$ is a generic framework in $\mathbb {C}^{d-2}$, so $\operatorname {\mathrm {rank}}(R(G,\widetilde {p}))$ is equal to the rank $r = r_{d-2}(G)$ of G, which is the maximal rank of the rigidity matrix of any framework in $\mathbb {C}^{d-2}$. Note that

$$\begin{align*}\big((\widetilde{p}_v - \widetilde{p}_u)\cdot (\widetilde{q}_v - \widetilde{q}_u)\big)_{uv \in E} = R(G,\widetilde{p})\widetilde{q},\end{align*}$$

so it is sufficient to show that the vector $z = \left ((p_v^{(1)} - p_u^{(1)})(p_v^{(d)} - p_u^{(d)}\right )_{uv \in E}$ is not contained in $\operatorname {\mathrm {span}}(R(G,\widetilde {p}))$. This is the same as saying $\operatorname {\mathrm {rk}}(M(p)) = r + 1$, where $M(p)$ is obtained by appending the column vector z to $R(G,\widetilde {p})$, which is further equivalent to the nonvanishing of some $(r+1) \times (r+1)$ subdeterminant of $M(p)$. The entries of $M(p)$ are polynomial functions in the coordinates of $(G,p)$ with rational coefficients and consequently so are these $(r+1) \times (r+1)$ subdeterminants.

Since $(G,p)$ is generic, it is sufficient to show that at least one of the polynomial functions describing these subdeterminants is not identically zero; in other words, that there is some framework $(G,p')$ for which the corresponding matrix $M(p')$ has rank $r+1$. In fact, we can find such a framework by only modifying the first and last coordinates of $p_v$ for each $v \in V$. Since G is not $\mathcal {R}_{d-2}$-independent, $r < |E|$, and thus, $\operatorname {\mathrm {span}}(R(G,\widetilde {p}))$ is contained in some linear hyperplane $H \subseteq \mathbb {C}^E$. By part b) of Lemma 3.8, we can find some vectors $(x_v)_{v \in V}$ and $(y_v)_{v \in V}$ such that the vector $\left ((x_v - x_u)(y_v-y_u)\right )_{uv \in E}$ is not contained in H. It follows that the framework $(G,p')$ defined by $p^{\prime }_v = (x_v,\widetilde {p}_v,q_v), v \in V$ satisfies $\operatorname {\mathrm {rk}}(M(p')) = r+1$, as desired.

Lemma 3.10. Let $d \geq 3$, and let G be a graph on n vertices that is not $\mathcal {R}_{d-2}$-independent. For $i = 1,\ldots ,d$, let $k_i$ denote the dimension of the space of infinitesimal motions of a generic realization of G in $\mathbb {C}^i$. Then for $i = 1,\ldots ,d-2$, we have

$$\begin{align*}k_{i+2} - k_{i+1} \geq k_{i+1} - k_{i} + 1.\end{align*}$$

Proof. The assumption that G is not $\mathcal {R}_{d-2}$-independent implies that it is not $\mathcal {R}_i$-independent for all $1 \leq i \leq d-2$. Thus, it is sufficient to prove for $i = d-2$. By Lemma 3.9, there is a generic realization $(G,p)$ in $\mathbb {C}^d$ such that $W(G,p)$ is a proper subset of the space of infinitesimal motions of $(G,p)$. Note that $\dim (W_1(G,p)) = \dim (W_2(G,p)) = k_{d-1}$, and by the choice of $(G,p)$, we have $\dim (W(G,p)) \leq k_d - 1$. Moreover, the subspace $W' = W_1(G,p) \cap W_2(G,p)$ consists of the infinitesimal motions of $(G,p)$ that are supported on the middle $d-2$ coordinates. This implies $\dim (W') = k_{d-2}$. By basic linear algebra we have

$$ \begin{align*} \dim(W') + \dim(W(G,p)) = \dim(W_1(G,p)) + \dim(W_2(G,p)). \end{align*} $$

Substituting the above equalities and inequality and then rearranging gives

$$ \begin{align*} k_{d} - k_{d-1} \geq k_{d-1} - k_{d-2} + 1, \end{align*} $$

as desired.

If G is $\mathcal {R}_{d-2}$-independent, then the conclusion of Lemma 3.10 does not hold. In this case, G is $\mathcal {R}_{d-1}$-independent and $\mathcal {R}_d$-independent as well, so we have $k_i = ni - |E|$ for $d - 2 \leq i \leq d$ so that $k_{d} - k_{d-1} = k_{d-1} - k_{d-2}$.

The following combinatorial lemma lets us turn the recursive bound on $k_i$ given in Lemma 3.10 into a lower bound that only depends on $k_d$ and $k_{d-1}$.

Proof. Consider the numbers $l_i = k_i - \binom {i+1}{2}$ for $i = 1,\ldots ,d$. Our goal is to prove $l_i \geq (d-i)(y-x) + x$. By definition, we have $l_{d-1} - l_{d} = y - x$. We claim that $l_i - l_{i+1} \geq l_{i+1} - l_{i+2}$ holds for all $i = 1, \ldots , d-2$. Indeed, using the bound on $k_i$ we obtain

$$ \begin{align*} l_i = k_i - \binom{i+1}{2} &\geq 2k_{i+1} - k_{i+2} + 1 - \binom{i+1}{2} \\ &= (2 \binom{i+2}{2} - \binom{i+3}{2} - \binom{i+1}{2} + 1) + (2l_{i+1} - l_{i+2}) \\ &=2l_{i+1} - l_{i+2}, \end{align*} $$

where we used the fact that for all $a \geq 1$, $\binom {a}{2} + \binom {a+2}{2} = 2\binom {a+1}{2} + 1$. This also gives

$$ \begin{align*} l_i - l_{i+1} \geq l_{i+1} - l_{i+2} \geq l_{i+2} - l_{i+3} \geq \cdots \geq l_{d-1} - l_{d} = y-x. \end{align*} $$

The statement now follows easily by induction on $j = d-i$. For $j=1$, we need $l_{d-1} \geq y$, which is actually satisfied with equality. Now let us assume $1 < j < d$. Then by induction we have

$$ \begin{align*} l_i \geq l_{i+1} + y - x \geq (d - i - 1)(y-x) + x + y - x = (d-i)(y-x) + x, \end{align*} $$

as desired.

If G is a connected graph, then its one-dimensional generic realizations have a one-dimensional space of infinitesimal motions. For such graphs, combining Lemma 3.10 with the bound on $k_1$ given by Lemma 3.11 gives the following corollary.

Corollary 3.12. Let $d \geq 3$ be an integer and G a connected graph, and suppose that G is not $\mathcal {R}_{d-2}$-independent. Let $k_{d-1}$ and $k_d$ denote the dimension of the space of infinitesimal motions of a generic realization of G in $\mathbb {C}^{d-1}$ and in $\mathbb {C}^d$, respectively. Finally, let $\operatorname {\mathrm {dof}}_{d-1}(G) = k_{d-1} - \binom {d}{2}$ and $\operatorname {\mathrm {dof}}_{d}(G) = k_d - \binom {d+1}{2}$ denote the ‘degrees of freedom’ of G in $d-1$ and d dimensions, respectively. Then we have

$$\begin{align*}\operatorname{\mathrm{dof}}_d(G) \geq \frac{d-1}{d-2}\operatorname{\mathrm{dof}}_{d-1}(G).\end{align*}$$

Now we are ready to prove our main theorem.

Proof of Theorem 3.6.

The $d=2$ case follows from Theorem 2.21, so we only prove for $d \geq 3$. Let H be a graph on $n'$ vertices, without isolated vertices and such that $M_{d,G} =_\psi M_{d,H}$ under some edge bijection $\psi $. Then by Corollary 3.4, we also have $M_{d-1,H} \subseteq _\psi M_{d-1,G}$. Observe that, since G is globally rigid, it is $\mathcal {R}_d$-connected by Theorem 3.5, and thus, so is H by Theorem 2.14 and the equality of the measurement varieties. In particular, H is a connected graph, and it is not $\mathcal {R}_d$-independent and thus not $\mathcal {R}_{d-2}$-independent.

Let $s = n'-n$, and let $k_i$ denote the dimension of the space of infinitesimal motions of a generic realization of H in $\mathbb {C}^i$ for $1 = 1,\ldots ,d$. By considering the dimension of $M_{d,G}$ (using Lemma 2.13) and using the assumption that G is (globally) rigid, we get $n'd - k_d = nd - \binom {d+1}{2}$, implying $k_d = sd + \binom {d+1}{2}$. Similarly, from $M_{d-1,H} \subseteq _\psi M_{d-1,G}$, we have $n'(d-1) - k_{d-1} \leq n(d-1) - \binom {d}{2},$ so that $k_{d-1} \geq s(d-1) + \binom {d}{2}$. Note that here we used the fact that if G is (globally) rigid in $\mathbb {R}^d$, then it is rigid in $\mathbb {R}^{d-1}$; see, e.g., [Reference Jordán, Whiteley, Goodman, O’Rourke and Tóth16, Theorem 63.2.11].

Since $k_d \geq \binom {d+1}{2}$, we must have $s \geq 0$. On the other hand, since H is connected and not $\mathcal {R}_{d-2}$-independent, Corollary 3.12 implies that

$$\begin{align*}sd = \operatorname{\mathrm{dof}}_d(H) \geq \frac{d-1}{d-2}\operatorname{\mathrm{dof}}_{d-1}(H) \geq \frac{d-1}{d-2}s(d-1).\end{align*}$$

Reordering, we get

$$\begin{align*}sd(d-2) \geq s(d-1)^2,\end{align*}$$

which is equivalent to $s \leq 0$. It follows that $s = 0$, so G and H have the same number of vertices. By Theorem 2.20, G is strongly reconstructible in $\mathbb {C}^d$, so $\psi $ is induced by a graph isomorphism $\varphi : G \rightarrow H$, as desired.

Applying Theorem 3.6 to a globally rigid subgraph of a graph, we obtain the following corollary.

The $d = 2$ case of Corollary 3.13 can be found in [Reference Garamvölgyi and Jordán8, Corollary 5.2].

To close this section, let us briefly return to Lemma 3.10. Let G be a graph and, as before, let $k_d$ denote the dimension of the set of infinitesimal motions of a generic realization of G in $\mathbb {C}^d$. Using the fact that, for a graph G on at least $d+2$ vertices we have $r_d(G) = nd - k_d$, we can rephrase the $i = d-2$ case of Lemma 3.10 in the following way.

Corollary 3.14. Let $d \geq 3$, and let G be a graph that is not $\mathcal {R}_{d-2}$-independent. Then the rank of G satisfies

(3.3)$$ \begin{align} r_{d}(G) - r_{d-1}(G) \leq r_{d-1}(G) - r_{d-2}(G) - 1. \end{align} $$

This bound is the best possible in the sense that if G is rigid in $\mathbb {R}^{d}$, then equation (3.3) is satisfied with equality. We note that Corollary 3.14 can be interpreted as a statement about the so-called secant defect of $M_{1,G}$, similar to Zak’s theorem on superadditivity [Reference Fantechi7, Reference Zak25]; see [Reference Gortler, Healy and Thurston10, Section 4.3] for a related discussion.

4.1 New examples of H-graphs

Following [Reference Jordán, Király and Tanigawa15], we say that a graph G is an H-graph in $\mathbb {R}^d$ if it is $(d+1)$-connected and redundantly rigid in $\mathbb {R}^d$ (i.e., it satisfies the necessary conditions of Theorem 2.1), but it is not globally rigid in $\mathbb {R}^d$. There are no H-graphs for $d=1,2$, but for $d\geq 3$ they exist, and finding more examples may lead to a better understanding of higher-dimensional global rigidity. For a long time, the complete bipartite graph $K_{5,5}$ was the only known H-graph in $\mathbb {R}^3$ (identified in [Reference Connelly4]), until infinite families had been found in [Reference Jordán, Király and Tanigawa15].

Theorem 3.5 can be used to give new examples of H-graphs which are $\mathcal {R}_3$-separable. These also demonstrate that redundant rigidity and $(d+1)$-connectivity together do not imply $\mathcal {R}_d$-connectivity in the $d \geq 3$ case.

We note that some of the H-graphs obtained in [Reference Jordán, Király and Tanigawa15] (for example, the ‘$6$-ring’ depicted in Figure 4) show that $4$-connectivity, redundant rigidity and $\mathcal {R}_3$-connectivity together do not imply global rigidity in $\mathbb {R}^3$.

It is also interesting to note that every known H-graph in $\mathbb {R}^3$, except $K_{5,5}$, has a $4$-separator.

The following related question also seems to be open.

It is easy to see that coning increases vertex connectivity by one so that the cone of a $(d+1)$-connected graph is $(d+2)$-connected. It is also folklore (and follows from Theorem 2.4(a) and Lemma 5.3 in the next section) that if G is redundantly rigid in $\mathbb {R}^d$, then its cone graph $G^v$ is redundantly rigid in $\mathbb {R}^{d+1}$. Together with Theorem 2.4(b) these imply that the cone of a H-graph in $\mathbb {R}^d$ is a H-graph in $\mathbb {R}^{d+1}$. We can use this fact to construct further families of H-graphs. However, these graphs will not be $\mathcal {R}_{d+1}$-separable (see Theorem 5.4). Instead, we can use higher-dimensional body-hinge graphs [Reference Jordán, Király and Tanigawa15] to generalize the $\mathcal {R}_{3}$-separable construction of Figure 2 to $d\geq 4$. We omit the details.

4.2 Unlabeled reconstructibility and small separators

In [Reference Gortler, Theran and Thurston12, Question 7.2], the authors asked whether every graph G that is $3$-connected and redundantly rigid in $\mathbb {R}^d$ is determined by its measurement variety, that is, whether $M_{d,G} =_\psi M_{d,H}$ under some edge bijection $\psi $ implies that G and H are isomorphic (note that here we do not require that the isomorphism induces $\psi $). Such a graph was called ‘weakly reconstructible in $\mathbb {C}^d$’ in [Reference Garamvölgyi and Jordán8]. In the other direction, in [Reference Garamvölgyi and Jordán8, Section 7], the authors asked whether every graph on at least $d+2$ vertices that is strongly reconstructible in $\mathbb {C}^d$ for some $d \geq 3$ is globally rigid in $\mathbb {R}^d$ or (more weakly) whether it is $(d+1)$-connected.

In this subsection, we provide negative answers to each of these questions for $d\geq 3$. Throughout this section, we shall use the following (folklore) result which also appears in the proof of [Reference Garamvölgyi and Jordán8, Theorem 5.21].

The graphs considered in Example 4.5 are all $\mathcal {R}_3$-separable. The next example is of an $\mathcal {R}_3$-connected graph that is not strongly reconstructible in $\mathbb {C}^3$.

Finally, we construct examples of fully reconstructible graphs with small separators. In the next proof, we shall use the following fact. Let $G_1, G_2$ be rigid graphs in $\mathbb {R}^d$ on at least $d+1$ vertices, and let G be obtained from $G_1$ and $G_2$ by identifying k pairs of vertices. Then if $0 \leq k \leq d - 1$, we have $r_d(G) = d|V(G)| - \binom {d+1}{2} - \binom {d-k+1}{2}$, and if $k \geq d$, then G is rigid. This follows, e.g., from the ‘gluing lemma’ [Reference Whiteley23, Lemma 11.1.9], or it can be seen directly by considering the infinitesimal motions of a generic realization of G in $\mathbb {R}^d$.