Abstract
Tanigawa (2016) showed that vertexredundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertexredundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension \(d>2\). We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in \({\mathbb {R}}^d\) on at most d vertices obviously implies their global rigidity, it is nontrivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent ddimensional realisations of a single graph in \({\mathbb {R}}^{2d}\) to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic bodybar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite bodybar frameworks.
Introduction
A ddimensional barjoint framework is a pair (G, p), where G is a simple graph and p is a map which assigns a point in \({\mathbb {R}}^d\) to each vertex of G. We think of (G, p) as a straight line realisation of G in \({\mathbb {R}}^d\), where the edge lengths are measured by the standard Euclidean metric. Loosely speaking, (G, p) is (locally) rigid if any edgelength preserving continuous motion of the vertices of (G, p) is necessarily a congruent motion (i.e., a motion corresponding to an isometry of \({\mathbb {R}}^d\)). Moreover, (G, p) is globally rigid if it is the only framework in dspace with the same graph and edge lengths, up to congruent motions. It is well known that both rigidity and global rigidity are generic properties, in the sense that a generic realisation of a graph G in \({\mathbb {R}}^d\) is rigid (globally rigid) if and only if every generic realisation of G in \({\mathbb {R}}^d\) is rigid (globally rigid) [1, 7, 8]. Therefore, a graph G is called rigid (globally rigid) if some (equivalently any) generic realisation of G is rigid (globally rigid).
The celebrated Laman’s theorem from 1970 (which had previously been discovered by PollaczekGeiringer in 1927 [17]), gives a combinatorial characterisation of the rigid graphs in \({\mathbb {R}}^2\) [15]. Extending this result to higher dimensions is a fundamental open problem in distance geometry [27]. Similarly, a combinatorial characterisation of the globally rigid graphs in \({\mathbb {R}}^2\) has been obtained by Jackson and Jordán in 2005 [9], but the problem of extending this result to higher dimensions also remains open. For the special class of bodybar frameworks [27], however, complete combinatorial characterisations for rigidity and global rigidity have been established in all dimensions in [26] and [6], respectively.
Tanigawa recently proved the following result, which is an important new tool to investigate the global rigidity of frameworks in \({\mathbb {R}}^d\).
Theorem 1.1
([24]) Let G be a rigid graph in \({\mathbb {R}}^d\) and suppose \(Gv\) remains rigid for every vertex v of G. Then G is globally rigid in \({\mathbb {R}}^d\).
In particular, the following combinatorial characterisation of globally rigid bodybar frameworks in \({\mathbb {R}}^d\) by Connelly et al. [6] easily follows from this result.
Theorem 1.2
([6, 24]) A generic bodybar framework is globally rigid in \({\mathbb {R}}^d\) if and only if it is rigid in \({\mathbb {R}}^d\) and it remains rigid after the removal of any edge.
In Sects. 4 and 5, we obtain analogues of these results for infinite periodic frameworks under fixed lattice representations. Due to their applications in fields such as crystallography, materials science, and engineering, the rigidity and flexibility of periodic structures has seen an increased interest in recent years (see e.g. [3, 4, 13, 16, 19, 21]). In particular, combinatorial characterisations of generic rigid and globally rigid periodic barjoint frameworks under fixed lattice representations in \({\mathbb {R}}^2\) were obtained in [21] and [13], respectively. Analogous to the situation for finite frameworks, extensions of these results to higher dimensions remain key open problems in the field. In fact, while it is well known that periodic local rigidity is a generic property in each dimension, it is currently not even known whether periodic global rigidity is a generic property for any \(d> 2\).
For the special class of periodic bodybar frameworks, Ross gave a combinatorial characterisation for generic local rigidity in \({\mathbb {R}}^3\) [20], and this result was recently extended to all dimensions by Tanigawa in [25] (see also Theorem 5.3). As an application of the main result of Sect. 4 (Theorem 4.5), we give the first combinatorial characterisation of generic globally rigid periodic bodybar frameworks in all dimensions in Sect. 5 (Theorem 5.2). We note that the proof of Theorem 5.2 does not rely on periodic global rigidity being a generic property in \({\mathbb {R}}^d\), and it also does not require the notion of stress matrices [5, 6]. It is a consequence of Theorem 5.2 that global rigidity of periodic bodybar frameworks is a generic property in each dimension.
Preliminaries
\(\Gamma \)Labelled Graphs and Periodic Graphs
Let \(\Gamma \) be a group isomorphic to \({\mathbb {Z}}^k\) for some integer \(k>0\). A \(\Gamma \)labelled graph is a pair \((G,\psi )\) of a finite directed (multi) graph G and a map \(\psi :E(G)\rightarrow \Gamma \).
For a given \(\Gamma \)labelled graph \((G,\psi )\), one may construct a kperiodic graph \(\tilde{G}\) by setting \(V(\tilde{G})=\{\gamma v_i: v_i\in V(G),\, \gamma \in \Gamma \}\) and \(E(\tilde{G})=\{\{\gamma v_i, \psi (v_iv_j) \gamma v_j\}: (v_i, v_j)\in E(G), \,\gamma \in \Gamma \}\). This \(\tilde{G}\) is called the covering of \((G, \psi )\), and \(\Gamma \) is the periodicity of \(\tilde{G}\), which acts naturally on \(V({\tilde{G}})\) and \(E(\tilde{G})\). The graph \((G,\psi )\) is also called the quotient \(\Gamma \)labelled graph of \(\tilde{G}\).
To guarantee that the covering of \((G, \psi )\) is a simple graph, we assume that \((G,\psi )\) has no parallel edges with the same label when oriented in the same direction. Moreover, we assume that \((G,\psi )\) has no loops. This is because a loop in \((G,\psi )\) (with a nontrivial label) does not give rise to any constraint when we study the rigidity and flexibility of the covering \(\tilde{G}\) under fixed lattice representations, as will become clear below.
Note that the orientation of \((G,\psi )\) is only used as a reference orientation and may be changed, provided that we also modify \(\psi \) so that if an edge has a label \(\gamma \) in one direction, then it has the label \(\gamma ^{1}\) in the other direction. The resulting \(\Gamma \)labelled graph still has the same covering \(\tilde{G}\).
It is also often useful to modify \((G,\psi )\) by using the switching operation. A switching at \(v\in V(G)\) by \(\gamma \in \Gamma \) changes \(\psi \) to \(\psi '\) defined by \(\psi '(e)=\gamma \psi (e)\) if e is directed from v, \(\psi '(e)=\gamma ^{1} \psi (e)\) if e is directed to v, and \(\psi '(e)=\psi (e)\) otherwise. It is easy to see that a switching operation performed on a vertex in \((G,\psi )\) does not alter the covering \(\tilde{G}\), up to isomorphism.
Given a \(\Gamma \)labelled graph \((G,\psi )\), we define a walk in \((G,\psi )\) as an alternating sequence \(v_1,e_1,v_2,\ldots , e_k,v_{k+1}\) of vertices and edges such that \(v_i\) and \(v_{i+1}\) are the endvertices of \(e_i\). For a closed walk \(C=v_1,e_1,v_2,\ldots , e_k,v_1\) in \((G,\psi )\), let \(\psi (C)=\prod _{i=1}^k\psi (e_i)^{\text {sign}(e_i)}\), where \(\text {sign}(e_i)=1\) if \(e_i\) has forward direction in C, and \(\text {sign}(e_i)=1\) otherwise. For a subgraph H of G define \(\Gamma _H\) as the subgroup of \(\Gamma \) generated by the elements \(\psi (C)\), where C ranges over all closed walks in H. The rank of H is defined to be the rank of \(\Gamma _H\). Note that the rank of G may be less than the rank of \(\Gamma \), in which case the covering graph \(\tilde{G}\) contains an infinite number of connected components.
Periodic BarJoint Frameworks
Recall that a pair (G, p) of a simple graph \(G=(V,E)\) and a map \(p:V\rightarrow {\mathbb {R}}^d\) is called a (barjoint) framework in \({\mathbb {R}}^d\). A periodic framework is a special type of infinite framework defined as follows.
Let \(\tilde{G}=(\tilde{V},\tilde{E})\) be a kperiodic graph with periodicity \(\Gamma \), and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be a nonsingular homomorphism with \(k\le d\), where L is said to be nonsingular if \(L(\Gamma )\) has rank k. A pair \((\tilde{G},\tilde{p})\) of \(\tilde{G}\) and \(\tilde{p}:\tilde{V}\rightarrow {\mathbb {R}}^d\) is said to be an Lperiodic framework in \({\mathbb {R}}^d\) if
We also say that a pair \((\tilde{G}, \tilde{p})\) is kperiodic in \({\mathbb {R}}^d\) if it is Lperiodic for some nonsingular homomorphism \(L:\Gamma \rightarrow {\mathbb {R}}^d\). Note that the rank k of the periodicity may be smaller than d.
An Lperiodic framework \((\tilde{G}, \tilde{p})\) is generic if the set of coordinates is algebraically independent over the rationals modulo the ideal generated by the equations (1).
A \(\Gamma \)labelled framework is defined to be a triple \((G,\psi , p)\) of a finite \(\Gamma \)labelled graph \((G,\psi )\) and a map \(p:V(G)\rightarrow {\mathbb {R}}^d\). Given a nonsingular homomorphism \(L:\Gamma \rightarrow {\mathbb {R}}^d\), the covering of \((G,\psi ,p)\) is the Lperiodic framework \((\tilde{G}, \tilde{p})\), where \(\tilde{G}\) is the covering of G and \(\tilde{p}\) is uniquely determined from p by (1). \((G,\psi ,p)\) is also called the quotient \(\Gamma \)labelled framework of \((\tilde{G}, \tilde{p})\).
We say that a \(\Gamma \)labelled framework \((G,\psi ,p)\) is generic if the set of coordinates in p is algebraically independent over the rationals. Note that an Lperiodic framework \((\tilde{G}, \tilde{p})\) is generic if and only if the quotient \((G,\psi ,p)\) of \((\tilde{G}, \tilde{p})\) is generic.
Periodic BodyBar Frameworks
A ddimensional bodybar framework consists of disjoint fulldimensional rigid bodies in \({\mathbb {R}}^d\) connected by disjoint bars, and may be considered as a special type of barjoint framework, as we will describe below. The rigidity and flexibility of bodybar frameworks has been studied extensively (see e.g. [6, 20, 26, 27]), as they have important applications in fields such as engineering, robotics, materials science, and biology. The underlying graph of a bodybar framework is a multigraph \(H=(V(H),E(H))\) with no loops, where each vertex in V(H) corresponds to a rigid body, and each edge in E(H) corresponds to a rigid bar. To represent a bodybar framework as a barjoint framework, we extract the bodybar graph \(G_H\) from the multigraph H as follows (see also [24], for example). \(G_H\) is the simple graph with vertex set \(V_H\) and edge set \(E_H\), where

\(V_H\) is the disjoint union of vertex sets \(B_H^v\) for each \(v\in V(H)\), with \(B_H^v\) defined as \(B_H^v=\{v_1,v_2, \ldots , v_{d+1}\} \cup \{v_e : e \in E(H) \text { is incident to } v\}\);

\(E_H= \bigl (\bigcup _{v\in V(H)} K(B_H^v) \bigr ) \cup \{e'=u_ev_e:e=uv \in E(H)\}\), where \(K(B_H^v)\) is the complete graph on \(B_H^v\).
For each \(v\in V(H)\), the vertices of \(B_H^v\) induce a complete subgraph of \(G_H\), which is referred to as the body associated with v. A barjoint framework \((G_H,p)\) with \(p:V_H \rightarrow {\mathbb {R}}^d\) is called a bodybar realisation of H in \({\mathbb {R}}^d\). See Fig. 1 for an example.
To define a periodic bodybar framework, we start with a \(\Gamma \)labelled graph \((H,\psi )\), as defined in Sect. 2.1. However, we now allow \((H,\psi )\) to have loops with nontrivial labels, as well as parallel edges with equal labels when oriented in the same direction. Thus, \((H,\psi )\) defines a kperiodic multigraph \(\tilde{H}\) which has no loops but may have parallel edges. We now use the procedure described above to construct the kperiodic bodybar graph \(G_{{\tilde{H}}}\) from the multigraph \(\tilde{H}\), with the slight modification that for any edge \(e\in E(\tilde{H})\) joining a vertex v with \(\gamma v\) for some \(\gamma \ne \text {id}\), we add two vertices \(v_{e^}\) and \(v_{e^+}\) (instead of just one vertex \(v_e\)) to \(B^v_{\tilde{H}}\), and define \(e'\) to be the edge \(v_{e^} \gamma v_{e^+}\) (instead of \(v_{e} \gamma v_{e}\)). This guarantees that the quotient \(\Gamma \)labelled graph of the bodybar graph \(G_{{\tilde{H}}}\) has no loops.
An Lperiodic barjoint framework \((G_{\tilde{H}},{\tilde{p}})\) with \({\tilde{p}}:V_{\tilde{H}}\rightarrow {\mathbb {R}}^d\) is called an Lperiodic bodybar realisation of \({\tilde{H}}\) in \({\mathbb {R}}^d\).
Rigidity and Global Rigidity
Let \(G=(V,E)\) be a graph. Two barjoint frameworks (G, p) and (G, q) in \({\mathbb {R}}^d\) are said to be equivalent if
They are congruent if
A barjoint framework (G, p) is called globally rigid if every framework (G, q) in \({\mathbb {R}}^d\) which is equivalent to (G, p) is also congruent to (G, p).
Analogously, following [13], we define an Lperiodic barjoint framework \((\tilde{G}, \tilde{p})\) in \({\mathbb {R}}^d\) to be Lperiodically globally rigid if every Lperiodic framework in \({\mathbb {R}}^d\) which is equivalent to \((\tilde{G}, \tilde{p})\) is also congruent to \((\tilde{G}, \tilde{p})\). Note that if the rank of the periodicity is equal to zero, then Lperiodic global rigidity coincides with the global rigidity of finite frameworks.
A key notion to analyse Lperiodic global rigidity is Lperiodic rigidity. A framework \((\tilde{G},\tilde{p})\) is called Lperiodically rigid if there is an open neighborhood N of \(\tilde{p}\) in which every Lperiodic framework \((\tilde{G},\tilde{q})\) which is equivalent to \((\tilde{G},\tilde{p})\) is also congruent to \((\tilde{G},\tilde{p})\). If \((\tilde{G},\tilde{p})\) is not Lperiodically rigid, then it is called Lperiodically flexible.
A barjoint framework \((\tilde{G},\tilde{p})\) is called Lperiodically vertexredundantly rigid, or Lperiodically 2rigid in short, if for every vertex orbit \({\tilde{v}}\) of \(\tilde{G}\), the framework \((\tilde{G}\tilde{v},\tilde{p}_{V(\tilde{G})\tilde{v}})\) is Lperiodically rigid.
Characterisation of LPeriodic Rigidity
A key tool to analyse the rigidity or global rigidity of finite frameworks is the lengthsquared function and its Jacobian, called the rigidity matrix. We may use the same approach to analyse periodic rigidity or periodic global rigidity (see also [13]).
For a \(\Gamma \)labelled graph \((G,\psi )\) and \(L:\Gamma \rightarrow {\mathbb {R}}^d\), we define \(f_{G,L}:{\mathbb {R}}^{dV(G)}\rightarrow {\mathbb {R}}^{E(G)}\) to be the function that assigns to every \(p\in {\mathbb {R}}^{dV(G)}\) the tuple of squared edge lengths of the \(\Gamma \)labelled framework \((G,\psi , p)\) (for a given order of the edges). That is, for \(p\in {\mathbb {R}}^{dV(G)}\), we have
For a finite set V, the complete \(\Gamma \) labelled graph \(K(V,\Gamma )\) on V is defined to be the \(\Gamma \)labelled graph with vertex set V and edge set \(\{((u, v);\gamma ): u,v \in V, \,\gamma \in \Gamma \}\), where \(((u, v);\gamma )\) denotes the directed edge (u, v) with label \(\gamma \). We simply denote \(f_{K(V,\Gamma ), L}\) by \(f_{V,L}\). By (1) we have the following fundamental fact.
Proposition 2.1
Let \((\tilde{G}, \tilde{p})\) be an Lperiodic framework and let \((G, \psi , p)\) be a quotient \(\Gamma \)labelled framework of \((\tilde{G}, \tilde{p})\). Then \((\tilde{G}, \tilde{p})\) is Lperiodically globally rigid (resp. rigid) if and only if for every \(q\in {\mathbb {R}}^{dV(G)}\) (resp. for every q in an open neighborhood of p in \({\mathbb {R}}^{dV(G)}\)), \(f_{G,L}(p)=f_{G,L}(q)\) implies \(f_{V(G),L}(p)=f_{V(G), L}(q)\).
We may therefore say that a \(\Gamma \)labelled framework \((G,\psi ,p)\) is Lperiodically globally rigid (or rigid) if for every \(q\in {\mathbb {R}}^{dV(G)}\) (resp. for every q in an open neighborhood of p in \({\mathbb {R}}^{dV(G)}\)), \(f_{G,L}(p)=f_{G,L}(q)\) implies \(f_{V(G),L}(p)=f_{V(G), L}(q)\), and we may focus on characterising the Lperiodic global rigidity (or rigidity) of \(\Gamma \)labelled frameworks. If \((G,\psi ,p)\) is not Lperiodically rigid, then it is called L periodically flexible. A \(\Gamma \)labelled framework \((G,\psi ,p)\) is Lperiodically 2rigid if for every vertex v of G, the \(\Gamma \)labelled framework \((Gv,\psi _{Gv},p_{V(G)v})\) is Lperiodically rigid. We have the following basic result for analysing Lperiodic rigidity.
Theorem 2.2
([13, 19]) Let \((G,\psi , p)\) be a generic \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) with \(V(G)\ge d+1\) and rank k periodicity \(\Gamma \), and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. Then \((G,\psi ,p)\) is Lperiodically rigid if and only if
where \(df_{G,L}_p\) denotes the Jacobian of \(f_{G,L}\) at p.
For combinatorial characterisations of generic Lperiodically rigid or globally rigid \(\Gamma \)labelled frameworks in \({\mathbb {R}}^2\), we refer the reader to [13, 21] and [13], respectively. A combinatorial characterisation of generic Lperiodically rigid bodybar frameworks in \({\mathbb {R}}^d\) has been established in [25] (see also Theorem 5.3).
Rigidity Implies Global Rigidity for Small Graphs
We first prove the following periodic counterpart of [2, Lem. 1].
Lemma 3.1
Let \((G,\psi )\) be the \({\mathbb {Z}}^k\)labelled graph with vertices \(v_1,\ldots , v_n\) and no edges, and let \(L:{\mathbb {Z}}^k\rightarrow {\mathbb {R}}^d\) be a nonsingular homomorphism. Further, let \((G,\psi ,p)\) and \((G,\psi ,q)\) be two \({\mathbb {Z}}^k\)labelled frameworks whose coverings are the Lperiodic frameworks \(({\tilde{G}},{\tilde{p}})\) and \(({\tilde{G}}, \tilde{q})\) in \({\mathbb {R}}^d\).
We denote \(p_{\gamma , i}={\tilde{p}}(\gamma v_i)=p(v_i)+L(\gamma )\) and \(q_{\gamma , i}={\tilde{q}}(\gamma v_i)=q(v_i)+L(\gamma )\) for \(i=1,\ldots , n\) and \(\gamma \in {\mathbb {Z}}^k\). Let \({\bar{p}}_{\gamma , i}:[0,1]\rightarrow {\mathbb {R}}^{2d}\) be the following continuous maps for \(i=1,\ldots , n\):
Then \({\bar{p}}_{\gamma ,i}(0)=(p_{\gamma ,i},0^d)\) and \(\bar{p}_{\gamma ,i}(1)=(q_{\gamma ,i},0^d)\), where \(0^d\) denotes the ddimensional zero vector. Further, \({\bar{p}}_{\gamma ,i}(t)\bar{p}_{\gamma ',j}(t)\) is monotone and \({\bar{p}}_{\gamma ,i}(t)=\bar{p}_{0^k,i}(t)+(L(\gamma ),0^d)\) for every \(i,j\in \{1,\ldots ,n\}\) and \(\gamma ,\gamma '\in {\mathbb {Z}}^k\).
Proof
We only prove the last equation as the other statements follow directly from [2, Lem. 1]. Observe that
holds for every \(i\in \{1,\ldots ,n\}\) and \(\gamma \in {\mathbb {Z}}^k\). \(\square \)
Lemma 3.1 implies the following theorem.
Theorem 3.2
Let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be a nonsingular homomorphism and let \((G,\psi ,p)\) be a \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) which is not Lperiodically globally rigid. Then the framework \((G,\psi ,(p,0^d))\) with \((p,0^d):V(G)\rightarrow \mathbb {R}^{2d}\) is \((L,0^d)\)periodically flexible in \({\mathbb {R}}^{2d}\), where \((L,0^d):\Gamma \rightarrow {\mathbb {R}}^{2d}\) maps \(\gamma \in \Gamma \) to \((L(\gamma ),0^d)\).
Proof
Since \((G,\psi ,p)\) is not Lperiodically globally rigid, it follows from Proposition 2.1 that there exists a \(\Gamma \)labelled framework \((G,\psi ,q)\) whose covering \((\tilde{G},{\tilde{q}})\) is equivalent but not congruent to the covering \(({\tilde{G}},{\tilde{p}})\) of \((G,\psi , p)\). By Lemma 3.1, there exists a continuous deformation between \(({\tilde{G}},(\tilde{p},0^d))\) and \((\tilde{G},(\tilde{q},0^d))\) in \({\mathbb {R}}^{2d}\) that maintains the lattice \((L,0^d)\) and, by the monotonicity of the distances, also maintains the edge lengths. Therefore, this map proves that \((G,\psi ,(p,0^d))\) is \((L,0^d)\)periodically flexible. \(\square \)
Let \((G,\psi ,p)\) be a \(\Gamma \)labelled framework in \({\mathbb {R}}^d\), \({\tilde{G}}\) be the covering of \((G,\psi )\), and \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be a nonsingular homomorphism. Suppose \(V(G)\le dk+1\). Observe that for \(D\ge d\), the points \(({\tilde{q}}(v),0^{Dd})\), \(v\in V({\tilde{G}})\), of the \((L,0^{Dd})\)periodic framework \(({\tilde{G}}, ({\tilde{q}},0^{Dd}))\) in \({\mathbb {R}}^D\) affinely span a space of dimension at most \(V(G)+k1\le d\). Now suppose that \((G,\psi ,p)\) is Lperiodically rigid in \({\mathbb {R}}^d\). Then it also has to be Lperiodically globally rigid in \({\mathbb {R}}^d\). If not, then during its nontrivial continuous motion in \({\mathbb {R}}^{2d}\), which is guaranteed to exist by Theorem 3.2, the points of the corresponding covering frameworks span an at most ddimensional subspace, a contradiction. Hence we have the following corollary of Theorem 3.2.
Corollary 3.3
Let \((G,\psi ,p)\) be a \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) with rank k periodicity and \(L:\Gamma \rightarrow {\mathbb {R}}^d\). Suppose that \((G,\psi ,p)\) is Lperiodically rigid and \(V(G)\le dk+1\). Then \((G,\psi ,p)\) is also Lperiodically globally rigid.
2Rigidity Implies Global Rigidity
In this section we extend Theorem 1.1 to periodic frameworks by showing that Lperiodic 2rigidity, together with a rank condition on the \(\Gamma \)labelled graph in the case when the framework is dperiodic in \({\mathbb {R}}^d\), implies Lperiodic global rigidity. We need several lemmas. The first one is [11, Prop. 13].
Lemma 4.1
Let \(f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}^d\) be a polynomial map with rational coefficients and p be a generic point in \({\mathbb {R}}^d\). Suppose that \(df_p\) is nonsingular. Then for every \(q\in f^{1}(f(p))\) we have \(\overline{{\mathbb {Q}}(p)}=\overline{{\mathbb {Q}}(q)}\), where \(\overline{{\mathbb {Q}}(p)}\) and \(\overline{{\mathbb {Q}}(q)}\) denote the algebraic closures of \({\mathbb {Q}}(p)\) and \({\mathbb {Q}}(q)\), respectively.
Let \(\Gamma \) be a group isomorphic to \({\mathbb {Z}}^k\), \(t=\max {\{dk, 1\}}\), \((G,\psi )\) be a \(\Gamma \)labelled graph with \(V(G)\ge t\), and \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. For simplicity we suppose that the linear span of \(L(\Gamma )\) is \(\{0\}^{dk}\times {\mathbb {R}}^k\), the linear subspace spanned by the last k coordinates. We pick any t vertices \(v_1,\ldots , v_t\), and define the augmented function of \(f_{G,L}\) by \(\hat{f}_{G,L}:=(f_{G,L}, g)\), where \(g:{\mathbb {R}}^{dV}\rightarrow {\mathbb {R}}^{d+{t\atopwithdelims ()2}}\) is a rational polynomial map given by
where \(p\in {\mathbb {R}}^{dV}\) and \(p_i(v_j)\) denotes the ith coordinate of \(p(v_j)\). Augmenting \(f_{G,L}\) by g corresponds to “pinning down” some coordinates to eliminate trivial continuous motions. The following lemma is [13, Prop. 3.6].
Lemma 4.2
Let \((G,\psi ,p)\) be a \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) with rank k periodicity and \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be a nonsingular homomorphism such that \(L(\Gamma )\subset \{0\}^{dk}\times {\mathbb {R}}^k\). Suppose that p is generic and \(V(G)\ge \max {\{dk, 1\}}\). Then
We also need an adapted version of [13, Lem. 4.5], which is a periodic generalisation of an observation made in [10, 24]. To state this lemma, we require the following definition.
Let \((G,\psi )\) be a \(\Gamma \)labelled graph and let v be a vertex of G. Suppose that every edge incident to v is directed from v. For each pair of nonparallel edges \(e_1=vu\) and \(e_2=vw\) in \((G,\psi )\), let \(e_1\cdot e_2\) be the edge from u to w with label \(\psi (vu)^{1}\psi (vw)\). We define \((G_v,\psi _v)\) to be the \(\Gamma \)labelled graph obtained from \((G,\psi )\) by removing v and inserting \(e_1\cdot e_2\) (unless it is already present) for every pair of nonparallel edges \(e_1, e_2\) incident to v.
Lemma 4.3
Let \((G,\psi ,p)\) be a generic \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) with rank k periodicity \(\Gamma \) and with \(V(G)\ge dk+2\) and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. Suppose that the covering \(({\tilde{G}},{\tilde{p}})\) has a vertex v with at least \(d+1\) neighbours \(\gamma _0v_0,\gamma _1v_1,\ldots ,\gamma _dv_d\), where \(v,v_i\in V(G)\) and \(\gamma _i\in \Gamma \), so that the points \({\tilde{p}}(\gamma _0v_0), {\tilde{p}}(\gamma _1v_1),\ldots , {\tilde{p}}(\gamma _dv_d)\) affinely span \({\mathbb {R}}^d\). Suppose further that

\((Gv,\psi _{Gv}, p')\) is Lperiodically rigid in \({\mathbb {R}}^d\), with notation \(p'=p_{V(G)v}\), and

\((G_v, \psi _v, p')\) is Lperiodically globally rigid in \({\mathbb {R}}^d\).
Then \((G, \psi , p)\) is Lperiodically globally rigid in \({\mathbb {R}}^d\).
Proof
We assume (by rotating the whole space if necessary) that \(L(\Gamma )=\{0\}^{dk}\times {\mathbb {R}}^k\). We pin the framework \((G,\psi ,p)\) and take any \(q\in \hat{f}_{G,L}^{1}(\hat{f}_{G,L}(p))\). Since \(V(G)\ge dk+2>\max {\{dk, 1\}}\), we may assume that v is not “pinned” (i.e., v is different from the vertices selected when augmenting \(f_{G,L}\) to \(\hat{f}_{G,L}\)). Our goal is to show that \(p=q\).
Let \(p'\) and \(q'\) be the restrictions of p and q to \(V(G)v\), respectively. Since \((Gv,\psi _{Gv},p')\) is Lperiodically rigid, we have \({\text {rank}} df_{Gv,L}_{p'}=dV(Gv)d{dk\atopwithdelims ()2}\) by Theorem 2.2. Then, by Lemma 4.2, we further have \({\text {rank}} d\hat{f}_{Gv,L}_{p'}=dV(Gv)\). Thus we can take a spanning subgraph H of \(Gv\) such that \(d\hat{f}_{H,L}_{p'}\) has linearly independent rows and is hence nonsingular. Since \(q'\in \hat{f}_{H,L}^{1}(\hat{f}_{H,L}(p'))\), it follows from Lemma 4.1 that \(\overline{{\mathbb {Q}}(p')}=\overline{{\mathbb {Q}}(q')}\). This in turn implies that \(q'\) is generic.
Consider the edges \(e_0=vv_0, e_1=vv_1,\ldots , e_{d}=vv_d\) in \((G,\psi )\) (all assumed to be directed from v) with respective labels \(\psi (e_0)=\gamma _0\), \(\psi (e_1)=\gamma _1\), \(\ldots \), \(\psi (e_d)=\gamma _d\). Note that we may have \(v_i=v_j\) for some i, j. By switching, we may further assume that \(\gamma _0=\text {id}\). For each \(1\le i\le d\), let
and let P and Q be the \(d\times d\)matrices whose ith column is \(x_i\) and \(y_i\), respectively. Note that since \(p(v_i)+L(\gamma _i)p(v_0)=\tilde{p}(\gamma _iv_i)\tilde{p}(v_0)\), and \(q(v_i)+L(\gamma _i)q(v_0)=\tilde{q}(\gamma _iv_i)\tilde{q}(v_0)\), and \(p',q'\) are generic, \(x_1,\ldots , x_d\) and \(y_1,\ldots , y_d\) are, respectively, linearly independent, and hence P and Q are both nonsingular.
Let \(x_v=p(v)p(v_0)\) and \(y_v=q(v)q(v_0)\). We then have \(\Vert x_v\Vert =\Vert y_v\Vert \) since G has the edge \(vv_0\) with \(\psi (vv_0)=\text {id}\). Due to the existence of the edge \(e_i\) we also have
where we used \(\Vert x_v\Vert =\Vert y_v\Vert \). Denoting by \(\delta \) the ddimensional vector whose ith coordinate is equal to \(\Vert x_i\Vert ^2\Vert y_i\Vert ^2\), the above d equations can be summarized as
which is equivalent to
By putting this into \(\Vert x_v\Vert ^2=\Vert y_v\Vert ^2\), we obtain
where \(I_d\) denotes the \(d\times d\) identity matrix.
Note that each entry of P is contained in \({\mathbb {Q}}(p')\), and each entry of Q is contained in \({\mathbb {Q}}(q')\). Since \(\overline{{\mathbb {Q}}(p')}=\overline{{\mathbb {Q}}(q')}\), this implies that each entry of \(PQ^{1}\) is contained in \(\overline{{\mathbb {Q}}(p')}\). On the other hand, since p is generic, the set of coordinates of p(v) (and hence those of \(x_v\)) is algebraically independent over \(\overline{{\mathbb {Q}}(p')}\). Therefore, by regarding the lefthand side of (3) as a polynomial in \(x_v\), the polynomial must be identically zero. In particular, we get
Thus, \(PQ^{1}\) is orthogonal. In other words, there is some orthogonal matrix S such that \(P=SQ\), and we get
for every \(1\le i\le d\). Therefore, \(q'\in f_{G_v, L}^{1}( f_{G_v, L}(p'))\). Since \((G_v, \psi _v,p)\) is Lperiodically globally rigid, this in turn implies that \(f_{Vv, L}(p')=f_{Vv, L}(q')\). Thus we have \(p'=q'\).
Since \(\{p(v_i)+L(\gamma _i): 0\le i\le d\}\) affinely spans \({\mathbb {R}}^d\), there is a unique extension of \(p':V(G)v\rightarrow {\mathbb {R}}^d\) to \(r:V(G)\rightarrow {\mathbb {R}}^d\) such that \(f_{G, L}(r)=f_{G,L}(p)\). Thus we obtain \(p=q\). \(\square \)
Finally, we need the following special case of [13, Lem. 3.1].
Lemma 4.4
Let \((G, \psi , p)\) be a generic \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) with \(V(G)\ge 2\), rank k periodicity \(\Gamma \), and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be a nonsingular homomorphism. If \((G, \psi , p)\) is Lperiodically globally rigid, then the rank of \((G,\psi )\) is equal to k.
Lemma 4.4 is easily seen to be true, because if the rank of \((G,\psi )\) is less than k, then the covering \((\tilde{G}, {\tilde{p}})\) of \((G,\psi ,p)\) has infinitely many connected components, each of which may be ‘flipped’ individually in a periodic fashion to obtain an Lperiodic framework \(({\tilde{G}}, {\tilde{q}})\) which is equivalent, but not congruent to \(({\tilde{G}}, {\tilde{p}})\). Thus, \((G, \psi , p)\) is not Lperiodically globally rigid.
This is illustrated by the two equivalent but noncongruent 2periodic frameworks in \({\mathbb {R}}^2\) shown in Fig. 2 whose \(\Gamma \)labelled graph \((G,\psi )\) has rank 1. Note, however, that \((G,\psi )\) is Lperiodically 2rigid, since it is Lperiodically rigid and the removal of any vertex results in a trivial framework with one vertex orbit and no edges (recall also Theorem 2.2).
It follows that in the case when \(k=d\), Lperiodic 2rigidity is not sufficient for Lperiodic global rigidity. In this case we need the added assumption that \({\text {rank}}(G,\psi )=d\). In the case when \(k<d\) and \({\text {rank}}(G,\psi )<k\), \((G,\psi , p)\) can also not be Lperiodically globally rigid, by Lemma 4.4. However, in this case, \((G,\psi , p)\) is also not Lperiodically 2rigid.
Theorem 4.5
Let \((G, \psi , p)\) be a generic \(\Gamma \)labelled framework in \({\mathbb {R}}^d\) with rank k periodicity \(\Gamma \), and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. If \((G, \psi , p)\) is Lperiodically 2rigid, and if \((G,\psi )\) is also of rank d in the case when \(k=d\), then \((G, \psi , p)\) is Lperiodically globally rigid in \({\mathbb {R}}^d\).
Proof
We use induction on V(G). If \(V(G)\le dk+1\), then \((G, \psi , p)\) is Lperiodically globally rigid by the Lperiodic rigidity of \((G, \psi , p)\) and Corollary 3.3.
Now suppose that \(V(G)\ge dk+2\), and let \(({\tilde{G}}, {\tilde{p}})\) be the covering of \((G,\psi , p)\). By our assumption, \((Gv,\psi _{Gv},p_{V(G)v})\) is Lperiodically rigid for any vertex \(v\in V(G)\).
Suppose first that \(V(G)= dk+2\). Then \((Gv,\psi _{Gv},p_{V(G)v})\) is also Lperiodically globally rigid by Corollary 3.3. We claim that for any occurrence of any \(v\in V(G)\) in the covering \({\tilde{G}}\), the affine span of the set \(\{\tilde{p}(w): vw \in E(\tilde{G})\}\) is all of \({\mathbb {R}}^d\).
If \(d=k\) (and hence \(V(G)=dk+2=2\)) the claim follows from the fact that \({\text {rank}}(G,\psi )=d\), by our assumption.
If \(d>k\) (and hence \(V(G)=dk+2>2\)), then we suppose for a contradiction that the claim is not true. Then the removal of a neighbour of v (and of all vertices belonging to that same vertex orbit) results in an Lperiodic framework with at least two distinct orbits of points (since \(V(G)>2\)) and, by our genericity assumption, this framework has the property that all the points connected to \(\tilde{p}(v)\) affinely span a space of dimension at most \(d2\), so that \(\tilde{p}(v)\) can be rotated about this \((d2)\)dimensional axis. Since all copies of points in the same orbit can then also be rotated in a periodic fashion and the affine span of the nonmoving points is \((d1)\)dimensional (as a kperiodic configuration with \(dk\) vertex orbits), we obtain a contradiction to the Lperiodic 2rigidity of \(({\tilde{G}},\tilde{p})\).
Thus, the affine span of the points \(\{\tilde{p}(w) : vw \in E(\tilde{G})\}\) is indeed all of \({\mathbb {R}}^d\) as claimed, and it follows from Lemma 4.3 that \((G,\psi , p)\) is Lperiodically globally rigid.
We may therefore assume that \(V(G)>dk+2\). We show that \((G_v, \psi _v, p')\) is Lperiodically 2rigid for any \(v\in V(G)\). Suppose for a contradiction that this is not true. Then there is a vertex u whose removal results in an Lperiodically flexible framework. As the neighbours of one occurrence of v in \(\tilde{G}\) induce a complete graph in \(\tilde{G_v}\) (where any pair of vertices from the same vertex orbit may always be considered adjacent due to the fixed lattice representation), adding v together with its incident edges to \((G_vu,\psi _v_{G_vu},p_{V(G)\{u,v\}})\) still yields an Lperiodically flexible framework. This is a contradiction, as \((Gu,\psi _{Gu},p_{V(G)u})\) is an Lperiodically rigid \(\Gamma \)labelled spanning subframework of the framework obtained from \((G_vu, \psi _v_{G_vu}, p_{V(G)\{u,v\}})\) by adding v and its incident edges.
Thus \((G_v, \psi _v, p')\) is Lperiodically 2rigid as claimed. Moreover, since \((G,\psi )\) is 2connected by the Lperiodic 2rigidity of \((G,\psi ,p)\), it follows from the definition of \((G_v,\psi _v)\) that \(\Gamma _G=\Gamma _{G_v}\). Thus, if \((G,\psi )\) is of rank d then so is \((G_v,\psi _v)\). It now follows from the induction hypothesis that \((G_v, \psi _v, p')\) is Lperiodically globally rigid. Moreover, by the same argument as above for the case when \(V(G)=dk+2>2\), the affine span of the points \(\{\tilde{p}(w):vw \in E(\tilde{G})\}\) is all of \({\mathbb {R}}^d\). Thus, by Lemma 4.3, \((G, \psi , p)\) is Lperiodically globally rigid. \(\square \)
Global Rigidity of BodyBar Frameworks
Using Theorem 4.5 in combination with Lemma 4.4 and the following Lemma 5.1 (which is [13, Lem. 3.7]) we can now easily prove an extension of Theorem 1.2 to periodic bodybar frameworks. We need the following definitions.
An Lperiodic framework \((\tilde{G},{\tilde{p}})\) with rank k periodicity \(\Gamma \) is said to be L periodically barredundantly rigid if \(({\tilde{G}} {\tilde{e}},{\tilde{p}})\) is Lperiodically rigid for every edge orbit \({\tilde{e}}\) of \(\tilde{G}\).
Similarly, an Lperiodic bodybar realisation \((G_{\tilde{H}},{\tilde{p}})\) of a multigraph \({\tilde{H}}\) is Lperiodically barredundantly rigid if for every edge orbit \({\tilde{e}}\) of \({\tilde{H}}\), the framework \((G_{\tilde{H}} \tilde{e}, {\tilde{p}})\) is Lperiodically rigid. (Recall the definition of a bodybar realisation in Sect. 2.3.)
Lemma 5.1
([13]) Let \(({\tilde{G}},{\tilde{p}})\) be a generic Lperiodic framework in \({\mathbb {R}}^d\) with rank k periodicity \(\Gamma \), and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. Suppose also that the quotient \(\Gamma \)labelled graph \((G,\psi )\) of \(\tilde{G}\) has \(V(G)\ge d+1\) if \(k\ge 1\) and \(V(G)\ge d+2\) if \(k=0\). If \(({\tilde{G}}, {\tilde{p}})\) is Lperiodically globally rigid, then \(({\tilde{G}}, {\tilde{p}})\) is Lperiodically barredundantly rigid.
The following extension of Theorem 1.2 gives a combinatorial characterisation of generic Lperiodically globally rigid bodybar frameworks in \({\mathbb {R}}^d\).
Theorem 5.2
Let \((G_{\tilde{H}},{\tilde{p}})\) be a generic Lperiodic bodybar realisation of the multigraph \({\tilde{H}}\) in \({\mathbb {R}}^d\) with rank k periodicity \(\Gamma \), and let \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. Then \((G_{\tilde{H}},{\tilde{p}})\) is Lperiodically globally rigid in \({\mathbb {R}}^d\) if and only if \((G_{\tilde{H}},{\tilde{p}})\) is Lperiodically barredundantly rigid in \({\mathbb {R}}^d\), and the quotient \(\Gamma \)labelled graph of \(G_{{\tilde{H}}}\) is of rank d in the case when \(k=d\).
Proof
It immediately follows from Lemma 5.1 that Lperiodic barredundant rigidity is necessary for a generic Lperiodic bodybar realisation to be Lperiodically globally rigid. Moreover, it follows from Lemma 4.4 that in the case when \(k=d\), the rank of the quotient \(\Gamma \)labelled graph of \(G_{{\tilde{H}}}\) must be equal to d for a generic Lperiodic bodybar realisation to be Lperiodically globally rigid. It is also easy to see that if a generic Lperiodic bodybar realisation is Lperiodically barredundantly rigid, then it is Lperiodically 2rigid, since the edges connecting the bodies are all disjoint. The result now follows from Theorem 4.5. \(\square \)
Note that generic Lperiodic barredundant rigidity can easily be checked in polynomial time based on the combinatorial characterisation of generic Lperiodic rigidity of bodybar frameworks in \({\mathbb {R}}^d\) conjectured by Ross in [20, Conj. 5.1] and proved by Tanigawa in [25, Thm. 7.2]. Using our notation and a simplified expression for the dimension of the space of trivial motions for a kperiodic framework in \({\mathbb {R}}^d\), this result may be restated as follows.
Theorem 5.3
([25]) Let \((G_{\tilde{H}},{\tilde{p}})\) be a generic Lperiodic bodybar realisation of the multigraph \({\tilde{H}}\) in \({\mathbb {R}}^d\) with rank k periodicity \(\Gamma \), and \(L:\Gamma \rightarrow {\mathbb {R}}^d\) be nonsingular. Then \((G_{\tilde{H}},{\tilde{p}})\) is Lperiodically rigid in \({\mathbb {R}}^d\) if and only if the quotient \(\Gamma \)labelled graph H of \(\tilde{H}\) contains a spanning subgraph (V, E) satisfying the following counts:

\(\displaystyle E=\left( {\begin{array}{c}d+1\\ 2\end{array}}\right) Vd\left( {\begin{array}{c}dk\\ 2\end{array}}\right) \);

\(\displaystyle F\le \left( {\begin{array}{c}d+1\\ 2\end{array}}\right) V(F)d\left( {\begin{array}{c}dk(F)\\ 2\end{array}}\right) \) for all nonempty \(F\subseteq E\),
where k(F) is the rank of F.
Conclusion and Further Comments
Realworld structures, whether they are natural such as crystals or proteins, or manmade such as buildings or linkages, are usually nongeneric, and often exhibit nontrivial symmetries. This fact has motivated a significant amount of research in recent years on how symmetry impacts the rigidity and flexibility of frameworks (see [23], for example, for a summary of results). In Theorem 4.5, we have shown that the sufficient condition given by Tanigawa in [24] for generic global rigidity of finite frameworks can be transformed to a sufficient condition for generic global rigidity of infinite Lperiodic frameworks (under a fixed lattice L). It remains open whether this result can be extended to other types of frameworks with symmetries such as infinite periodic frameworks with (partially) flexible lattices or finite frameworks with point group symmetries. Following the proof of Theorem 5.2, such an extension would imply the characterisation of the generic global rigidity of finite bodybar frameworks with these symmetries by using the existing (local) rigidity characterisations of these frameworks by Tanigawa [25]. Furthermore, such a result would be useful for the characterisation of the generic global rigidity of bodyhinge frameworks with symmetries (where the bodies are connected in pairs by \((d2)\)dimensional hinges) such as in the (finite) generic version established by Jordán, Király, and Tanigawa [12]. However, the characterisation of generic (local) rigidity for periodic bodyhinge frameworks is still open (even for fixed lattices). For finite symmetric bodyhinge frameworks, such a characterisation is only known for groups of the form \({\mathbb {Z}}_2\times {\mathbb {Z}}_2\times \cdots \times {\mathbb {Z}}_2\) [22]. A major goal in this research area is to obtain a combinatorial characterisation of the generic global rigidity of infinite Lperiodic or finite symmetric molecular frameworks in 3space (i.e., bodyhinge frameworks in 3space with the added property that the lines of the hinges attached to each body all meet in a single point on that body), since they may be used to model crystals and protein structures. We note that for finite molecular frameworks, their generic (local) rigidity was recently characterised by the celebrated result of Katoh and Tanigawa [14]. However, their generic global rigidity has not yet been characterised, and there are also no generic local or global rigidity characterisations for infinite Lperiodic or finite symmetric molecular frameworks [18].
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Acknowledgements
Project No. NKFI128673 has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the FK_18 funding scheme (Kaszanitzky and Király). The first author was supported by the Hungarian Scientific Research Fund of the National Research, Development and Innovation Office (OTKA, Grant Number K109240 and K124171). The second author was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, by the ÚNKP17 4 New National Excellence Program of the Ministry of Human Capacities of Hungary, by the ÚNKP205 New National Excellence Program of the Ministry for Innovation and Technology, and by the Hungarian Scientific Research Fund of the National Research, Development and Innovation Office (OTKA, Grant Number K109240). The third author was supported by EPSRC First Grant EP/M013642/1.
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Kaszanitzky, V.E., Király, C. & Schulze, B. Sufficient Conditions for the Global Rigidity of Periodic Graphs. Discrete Comput Geom 67, 1–16 (2022). https://doi.org/10.1007/s00454021003469
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DOI: https://doi.org/10.1007/s00454021003469
Keywords
 Rigidity
 Global rigidity
 Bodybar framework
 Periodic framework
Mathematics Subject Classification
 52C25
 05B35
 05C10
 68R10