The Rigidity of Infinite Graphs
Abstract
A rigidity theory is developed for countably infinite simple graphs in \({\mathbb {R}}^d\). Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in \({\mathbb {R}}^2\) and Tay’s multigraph characterisation of generic infinitesimal rigidity for finite bodybar frameworks in \({\mathbb {R}}^d\). Analogous results are obtained for the classical nonEuclidean \(\ell ^q\) norms.
Keywords
Infinite graphs Infinitesimal rigidity Combinatorial rigidityMathematics Subject Classification
52C25 05C631 Introduction
In 1864 James Clerk Maxwell [25] initiated a combinatorial trend in the rigidity theory of finite barjoint frameworks in Euclidean space. In two dimensions this amounted to the observation that the underlying structure graph \(G=(V,E)\) must satisfy the simple counting rule \(E\ge 2V3\). For minimal rigidity, in which any bar removal renders the framework flexible, equality must hold together with the inequalities \(E(H)\le 2V(H)3\) for subgraphs H with at least two vertices. The fundamental result that these two necessary conditions are also sufficient for the minimal rigidity of a generic framework was obtained by Laman in 1970 and this has given impetus to the development of matroid theory techniques. While corresponding counting rules are necessary in three dimensions they fail to be sufficient and a purely combinatorial characterisation of generic rigidity is not available. On the other hand many specific families of finite graphs are known to be generically rigid, such as the edge graphs of trianglefaced convex polyhedra in three dimensions and the graphs associated with finite triangulations of general surfaces. See, for example, Alexandrov [1], Fogelsanger [9], Gluck [10], Kann [16] and Whiteley [42, 43].
A finite simple graph G is said to be generically drigid, or simply drigid, if its realisation as some generic barjoint framework in the Euclidean space \({\mathbb {R}}^d\) is infinitesimally rigid. Here generic refers to the algebraic independence of the set of coordinates of the vertices and infinitesimal rigidity in this case is equivalent to continuous (nontrivial finite motion) rigidity (Asimow and Roth [2, 3]). The rigidity analysis of barjoint frameworks and related frameworks, such as bodybar frameworks and bodyhinge frameworks, continues to be a focus of investigation, both in the generic case and in the presence of symmetry. For example Katoh and Tanigawa [18] have resolved the molecular conjecture for generic structures, while Schulze [38] has obtained variants of Laman’s theorem for semigeneric symmetric barjoint frameworks. In the case of infinite frameworks however developments have centred mainly on periodic frameworks and the infinitesimal and finite motions which retain some form of periodicity. Indeed, periodicity hypotheses lead to configuration spaces that are real algebraic varieties and so to methods from multilinear algebra and finite combinatorics. See, for example, Borcea and Streinu [4], Connelly et al. [8], Malestein and Theran [24], Owen and Power [32] and Ross et al. [37]. Periodic rigidity, broadly interpreted, is also significant in a range of applied settings, such as the mathematical analysis of rigid unit modes in crystals, as indicated in Power [34] and Wegner [41], for example.
In the development below we consider general countable simple graphs and the flexibility and rigidity of their placements in the Euclidean spaces \({\mathbb {R}}^d\) and in the nonEuclidean spaces \(({\mathbb {R}}^d, \Vert \cdot \Vert _q)\) for the classical \(\ell ^q\) norms, for \(1< q < \infty \). The constraint conditions for the nonEuclidean \(\ell ^q\) norms are no longer given by polynomial equations and so we adapt the Asimow–Roth notion of a regular framework to obtain the appropriate form of a generic framework. This strand of norm constraint rigidity theory for finite frameworks was initiated in [20] for \(\ell ^q\) norms. It was further developed in [19] for polyhedral norms and in [22] for general norms. We continue this development in Theorem 5.5 where we generalise Tay’s multigraph characterisation [40] of generically rigid finite bodybar frameworks in \({\mathbb {R}}^d\) to the nonEuclidean \(\ell ^q\) norms. As well as being a natural problem, one of the original motivations for considering rigidity with respect to a different norm was based on similarities which arose with the combinatorial methodologies used for surfaceconstrained frameworks [28, 29] and the potential for crossfertilization between these topics. Subsequently, norm based rigidity has gained interest in relation to metric embeddability [39].
Our first main result is Theorem 1.1 in which we determine the simple countable graphs which are locally generically rigid for \(({\mathbb {R}}^2,\Vert \cdot \Vert _q)\), for \(1<q<\infty \). This is a generalisation of Laman’s theorem (and its nonEuclidean analogue) to countable graphs.
Theorem 1.1
 (A)

The following statements are equivalent.
 (i)
G is rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _2)\).
 (ii)
G contains a (2, 3)tight vertexcomplete tower.
 (B)

If \(q\in (1,2)\cup (2,\infty )\) then the following statements are equivalent.
 (i)
G is rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
 (ii)
G contains a (2, 2)tight vertexcomplete tower.
We also see that these graphs are necessarily sequentially rigid in the sense of containing a spanning subgraph which is a union of finite graphs, each of which is infinitesimally rigid. This is the strongest form of infinitesimal rigidity and its equivalence with infinitesimal rigidity is particular to two dimensions; an infinite chain of double banana graphs shows that the corresponding equivalence fails to hold in higher dimensions (see Fig. 2).
Theorem 1.2
 (A)

The following statements are equivalent.
 (i)
G is rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _2)\).
 (ii)
\(G_b\) has a \(\left( \frac{d(d+1)}{2},\frac{d(d+1)}{2}\right) \)tight vertexcomplete tower.
 (B)

If \(q\in (1,2)\cup (2,\infty )\) then the following statements are equivalent.
 (i)
G is rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
 (ii)
\(G_b\) has a (d, d)tight vertexcomplete tower.
We comment on further directions and related problems at the end of Sect. 3. Accounts of the foundations of geometric rigidity theory are given in Alexandrov [1], Graver [12], Graver et al. [13] and Whiteley [45]. Also [13] has a comprehensive guide to the literature up to 1993. The influential papers of Asimow and Roth introduced regular frameworks as a more appropriate form of genericity and in Definition 2.6 we have followed Graver in requiring that all frameworks supported by vertices of G should be regular.
2 Preliminaries
In this section we state necessary definitions for finite and countably infinite graphs and we review the necessary background on the rigidity of finite graphs in \(\mathbb {R}^d\) with respect to the classical \(\ell ^q\) norms.
2.1 Continuous and Infinitesimal Rigidity
A barjoint framework in a normed vector space \((X,\Vert \cdot \Vert )\) is a pair (G, p) consisting of a simple graph \(G=(V(G),E(G))\) and a mapping \(p:V(G)\rightarrow X\), \(v\mapsto p_v\), with the property that \(p_v\not =p_w\) whenever \(vw\in E(G)\). Unless otherwise stated, the vertex set V(G) is allowed to be either finite or countably infinite. We call p a placement of G in X and the collection of all placements of G in X will be denoted by P(G, X) or simply P(G) when the context is clear. If H is a subgraph of G then the barjoint framework (H, p) obtained by restricting p to V(H) is called a subframework of (G, p).
Definition 2.1
A continuous flex is regarded as trivial if it results from a continuous isometric motion of the ambient space. Formally, a continuous rigid motion of \((X,\Vert \cdot \Vert )\) is a mapping \(\Gamma (x,t):X\times [1,1]\rightarrow X\) which is isometric in the variable x and continuous in the variable t with \(\Gamma (x,0)=x\) for all \(x\in X\). Every continuous rigid motion gives rise to a continuous flex of (G, p) by setting \(\alpha _v:[1,1]\rightarrow X\), \(t\mapsto \Gamma (p_{v},t)\), for each \(v\in V(G)\). A continuous flex of (G, p) is trivial if it can be derived from a continuous rigid motion in this way. If every continuous flex of (G, p) is trivial then we say that (G, p) is continuously rigid, otherwise we say that (G, p) is continuously flexible.
Definition 2.2
Proposition 2.3
Proof
See proof of [20, Prop. 3.1]. \(\square \)
Definition 2.4
The equivalence of continuous and infinitesimal rigidity for regular finite barjoint frameworks in Euclidean space was established by Asimow and Roth [2, 3]. In [21] this result is extended to finite barjoint frameworks in the nonEuclidean spaces \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) for \(q\in (1,\infty )\).
Theorem 2.5
We now formalise our meaning of a generic finite barjoint framework in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) for \(q\in (1,\infty )\). The complete graph on the vertices V(G) will be denoted \(K_{V(G)}\).
Definition 2.6
A finite barjoint framework (G, p) is generic in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) if \(p\in P(K_{V(G)},{\mathbb {R}}^d)\) and every subframework of \((K_{V(G)},p)\) is regular.
If (G, p) is a finite barjoint framework then p will frequently be identified with a vector \((p_{v_1}, p_{v_2},\ldots ,p_{v_{n}})\in {\mathbb {R}}^{dV(G)}\) with respect to some fixed ordering of the vertices \(V(G)=\{v_1,v_2,\ldots ,v_{n}\}\). In particular, the collection of all generic placements of G in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) is identified with a subset of \({\mathbb {R}}^{dV(G)}\).
Lemma 2.7
Let G be a finite simple graph and let \(q\in (1,\infty )\). Then the set of generic placements of G in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) is an open and dense subset of \(\mathbb {R}^{dV(G)}\).
Proof
Note that the infinitesimal flexibility dimension \(\dim _\mathrm{fl}(G,p)\) is constant on the set of generic placements of G in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\). Also, if G has a (minimally) infinitesimally rigid placement then all generic placements of G are (minimally) infinitesimally rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\).
Definition 2.8
 1.The infinitesimal flexibility dimension of G in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) iswhere p is any generic placement of G.$$\begin{aligned} \dim _{\mathrm{fl}}(G):= \dim _{d,q}(G) := \dim _\mathrm{fl}(G,p)=\dim {\mathcal {F}}_q(G,p)/ {\mathcal {T}}_q(G,p). \end{aligned}$$
 2.
G is (minimally) rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) if the generic placements of G are (minimally) infinitesimally rigid.
One can readily verify that the complete graph \(K_{d+1}\) on \(d+1\) vertices satisfies \(\dim _{d,2}(K_{d+1}) =0\) and that \(K_{d+1}\) is minimally rigid for \({\mathbb {R}}^d\) with the Euclidean norm. Also, in d dimensions we have \(\dim _{d,q}(K_{2d}) =0\), with minimal rigidity, for each of the nonEuclidean qnorms.
2.2 Sparsity and Rigidity
We recall the following classes of multigraphs.
Definition 2.9
 1.
(k, l)sparse if \(E(H)\le kV(H)l\) for each subgraph H of G which contains at least two vertices.
 2.
(k, l)tight if it is (k, l)sparse and \(E(G)=kV(G)l\).
Our main interests are in the classes of simple (2, 2)sparse and (2, 3)sparse graphs and the class of (k, k)sparse multigraphs for \(k\ge 2\).
Example 2.10
The complete graph \(K_n\) is (k, k)sparse for \(1\le n \le 2k\), (k, k)tight for \(n\in \{1,2k\}\) and fails to be (k, k)sparse for \(n>2k\). Also, \(K_2\) and \(K_3\) are (2, 3)tight while \(K_n\) fails to be (2, 3)sparse for \(n\ge 4\).
Laman’s theorem [23] provides a combinatorial characterisation of the class of finite simple graphs which are rigid in the Euclidean plane and can be restated as follows.
Theorem 2.11
 (i)
G is rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _2)\).
 (ii)
G contains a (2, 3)tight spanning subgraph.
In particular, a generic barjoint framework (G, p) is minimally infinitesimally rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _2)\) if and only if G is (2, 3)tight. In [20] the following analogue of Laman’s theorem was obtained for the nonEuclidean \(\ell ^q\) norms.
Theorem 2.12
 (i)
G is rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
 (ii)
G contains a (2, 2)tight spanning subgraph.
3 Rigidity of Countable Graphs
In this section we establish the general principle that infinitesimal rigidity is equivalent to local relative rigidity in the sense that every finite subframework is rigid relative to some finite containing superframework (Theorem 3.14). Following this we prove Theorem 1.1 which is the generalised Laman theorem. The rigidity of general infinite graphs as barjoint frameworks was considered first in Owen and Power [30, 32] and part (A) of Theorem 1.1 answers a question posed in [32].
3.1 Sparsity Lemmas
We first obtain characterisations of (k, l)tightness which are needed for the construction of inclusion chains of rigid graphs.
Lemma 3.1
 (i)
\(G'\) is (k, l)sparse, or,
 (ii)
there exists a (k, l)tight subgraph of G which contains both v and w.
Proof
If \(G'\) is not (k, l)sparse then there exists a subgraph \(H'\) of \(G'\) which fails the sparsity count. Now \(H'\backslash \{vw\}\) is a (k, l)tight subgraph of G which contains both v and w. Conversely, if H is a (k, l)tight subgraph of G which contains both v and w then \(H\cup \{vw\}\) is a subgraph of \(G'\) which fails the sparsity count. \(\square \)
Lemma 3.2
 (a)
\(k=2\), \(l=3\) and G contains at least two vertices, or,
 (b)
\(k=l\) and G contains at least 2k vertices.
Proof
Let K be the complete graph on the vertices of G. The collection of edge sets of the (k, l)sparse subgraphs of K form the independent sets of a matroid. Moreover, the edge sets of the (k, l)tight subgraphs of K are the base elements of this matroid. In case (a), this is wellknown and a consequence of Laman’s theorem, while case (b) follows from NashWilliams characterisation [26] of these graphs as those where the edge set is the disjoint union of k spanning forests. Each independent set in a matroid extends to a base element and so, in particular, the edge set of G extends to the edge set of a (k, l)tight graph \(G'\) on the same vertex set. \(\square \)
3.2 Relative Infinitesimal Rigidity
We first prove that in two dimensional \(\ell ^q\) spaces relative infinitesimal rigidity is equivalent to the existence of a rigid containing framework.
Definition 3.3
 1.
A subframework (H, p) is relatively infinitesimally rigid in (G, p) if there is no nontrivial infinitesimal flex of (H, p) which extends to an infinitesimal flex of (G, p).
 2.
A subframework (H, p) has an infinitesimally rigid container in (G, p) if there exists an infinitesimally rigid subframework of (G, p) which contains (H, p) as a subframework.
If (H, p) has an infinitesimally rigid container in (G, p) then (H, p) is relatively infinitesimally rigid in (G, p). The converse statement is not true in general as the following example shows.
Example 3.4
Figure 1 illustrates a generic barjoint framework (G, p) in \(({\mathbb {R}}^3,\Vert \cdot \Vert _2)\) with subframework (H, p) indicated by the shaded region. Note that (H, p) is relatively infinitesimally rigid in (G, p) but does not have an infinitesimally rigid container in (G, p).
In the following we will say that a finite simple graph G is independent in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) if the rigidity matrix \(R_q(G,p)\) is independent for some (and hence every) generic placement \(p:V(G)\rightarrow \mathbb {R}^d\).
Proposition 3.5
 (a)
\(q=2\), \(l=3\) and G contains at least two vertices, or,
 (b)
\(q\not =2\), \(l=2\) and G contains at least four vertices.
 (i)
G is independent in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
 (ii)
G is (2, l)sparse.
Proof
Let \(p:V(G)\rightarrow \mathbb {R}^2\) be a generic placement of G in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\). If G is independent and H is a subgraph of G then \(E(H)=\mathrm{rank}\,R_q(H,p)\le 2V(H)l\). We conclude that G is (2, l)sparse.
Conversely, if G is (2, l)sparse then, by Lemma 3.2, G is a subgraph of some (2, l)tight graph \(G'\) with \(V(G)=V(G')\). By Laman’s theorem and its analogue for the nonEuclidean case (Theorems 2.11, 2.12), \((G',p)\) is minimally infinitesimally rigid and so G is independent. \(\square \)
We now show that relative infinitesimal rigidity does imply the existence of an infinitesimally rigid container for generic barjoint frameworks in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\) for all \(q\in (1,\infty )\).
Theorem 3.6
 (a)
\(q=2\) and H contains at least two vertices, or,
 (b)
\(q\not =2\) and H contains at least four vertices.
 (i)
H is relatively rigid in G with respect to \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
 (ii)
H has a rigid container in G with respect to \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
Proof
(ii) \(\Rightarrow \) (i) Let \(H'\) be a rigid container for H in G and let \(p:V(G)\rightarrow \mathbb {R}^2\) be a generic placement of G in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\). Then no nontrivial infinitesimal flex of (H, p) can be extended to an infinitesimal flex of \((H',p)\) and so the result follows. \(\square \)
Remark 3.7
In their analysis of globally linked pairs of vertices in rigid frameworks Jackson et al. [15] remark that it follows from the characterisation of independent sets for the rigidity matroid for the Euclidean plane that linked vertices \(\{v_1,v_2\}\) must lie in the same rigid component. (See also [14].) This assertion is essentially equivalent to part (a) of Theorem 3.6. The terminology here is that a pair of vertices \(\{v_1,v_2\}\) in a graph G is linked in (G, p) if there exists an \(\epsilon >0\) such that if \(q \in P(G)\) is another placement of G with \(\Vert q_vq_w\Vert _2=\Vert p_vp_w\Vert _2\) for all \(vw\in E(G)\) and \(\Vert q_vp_v\Vert _2 <\epsilon \) for all \(v\in V(G)\) then \(\Vert q_{v_1}q_{v_2}\Vert _2=\Vert p_{v_1}p_{v_2}\Vert _2\). It can be shown that this is a generic property and that a subgraph \(H\subseteq G\) is relatively rigid in G if and only if for a generic placement (G, p) each pair of vertices in H is linked in (G, p).
3.3 Flex Cancellation and Relatively Rigid Towers
Definition 3.8
A tower of barjoint frameworks \(\{(G_k,p_k):k\in \mathbb {N}\}\) has the flex cancellation property if for each \(k\in \mathbb {N}\) and any nontrivial infinitesimal flex \(u_k\) of \((G_{k},p_k)\) there is an \(m>k\) such that \(u_k\) does not extend to an infinitesimal flex of \((G_m,p_m)\).
If a barjoint framework \((G_m,p_m)\) in a tower \(\{(G_k,p_k):k\in \mathbb {N}\}\) has a nontrivial infinitesimal flex \(u_m:V(G_m)\rightarrow X\) which can be extended to every containing framework in the tower then we call \(u_m\) an enduring infinitesimal flex for the tower.
Lemma 3.9
Let \(\{(G_k,p_k):k\in \mathbb {N}\}\) be a tower of barjoint frameworks in a finite dimensional normed space \((X,\Vert \cdot \Vert )\) and let \(u_1\) be an infinitesimal flex of \((G_1,p_1)\) which is an enduring flex for the tower. Then there exists a sequence \(\{u_k\}_{k=1}^\infty \) such that, for each \(k\in \mathbb {N}\), \(u_{k}\) is an infinitesimal flex of \((G_k,p_k)\) and \(u_{k+1}\) is an extension of \(u_k\).
Proof
Proposition 3.10
Let (G, p) be a countable barjoint framework in a finite dimensional normed space \((X,\Vert \cdot \Vert )\). If (G, p) is infinitesimally rigid then every edgecomplete tower in (G, p) has the flex cancellation property.
Proof
Suppose there exists an edgecomplete tower \(\{(G_k,p_k):k\in \mathbb {N}\}\) of finite frameworks in (G, p) which does not have the flex cancellation property. Then there exists a nontrivial infinitesimal flex of some \((G_k,p_k)\) which is an enduring flex for the tower. We may assume without loss of generality that \(k=1\). By Lemma 3.9 there is a sequence of infinitesimal flexes \(u_1,u_2,u_3,\ldots \) for the chain with each flex extending the preceding flex. The tower is edgecomplete and so this sequence defines an infinitesimal flex u for (G, p) by setting \(u(v)=u_k(v)\) for all \(v\in V(G_k)\) and all \(k\in {\mathbb {N}}\). Since \(u_1\) is a nontrivial infinitesimal flex of \((G_1,p_1)\) the flex u is a nontrivial infinitesimal flex of (G, p). \(\square \)
Remark 3.11
The key Lemma 3.9 is reminiscent of the compactness principle for locally finite structures to the effect that certain properties prevailing for all finite substructures hold also for the infinite structure. For example the kcolourability of a graph is one such property. See NashWilliams [27].
We can now establish the connection between relative rigidity, flex cancellation and infinitesimal rigidity for countable barjoint frameworks.
Definition 3.12
A tower of barjoint frameworks \(\{(G_k,p_k):k\in \mathbb {N}\}\) is relatively infinitesimally rigid if \((G_{k},p_k)\) is relatively infinitesimally rigid in \((G_{k+1},p_{k+1})\) for each \(k\in \mathbb {N}\).
Lemma 3.13
Let \(\{(G_k,p_k):k\in \mathbb {N}\}\) be a framework tower in a finite dimensional normed space \((X,\Vert \cdot \Vert )\). If \(\{(G_k,p_k):k\in \mathbb {N}\}\) has the flex cancellation property then there exists an increasing sequence \((m_k)_{k=1}^\infty \) of natural numbers such that the tower \(\{(G_{m_k},p_{m_k}):k\in \mathbb {N}\}\) is relatively infinitesimally rigid.
Proof
Let \(\mathcal {F}^{(k)}\subset \mathcal {F}(G_1,p_1)\) denote the set of all infinitesimal flexes of \((G_1,p_1)\) which extend to \((G_{k},p_{k})\) but not \((G_{k+1},p_{k+1})\). Suppose there exists an increasing sequence \((n_k)_{k=1}^\infty \) of natural numbers such that \( \mathcal {F}^{(n_k)}\not =\emptyset \) for all \(k\in \mathbb {N}\). Choose an element \(u_k\in \mathcal {F}^{(n_k)}\) for each \(k\in \mathbb {N}\) and note that \(\{u_k:k\in \mathbb {N}\}\) is a linearly independent set in \(\mathcal {F}(G_1,p_1)\). Since \(\mathcal {F}(G_1,p_1)\) is finite dimensional we have a contradiction. Thus there exists \(m_1\in \mathbb {N}\) such that \(\mathcal {F}^{(k)}=\emptyset \) for all \(k\ge m_1\) and so \((G_1,p_1)\) is relatively infinitesimally rigid in \((G_{m_1},p_{m_1})\). The result now follows by an induction argument. \(\square \)
Theorem 3.14
 (i)
(G, p) is infinitesimally rigid.
 (ii)
(G, p) contains a vertexcomplete tower which has the flex cancellation property.
 (iii)
(G, p) contains a vertexcomplete tower which is relatively infinitesimally rigid.
Proof
 1.
\(u(v)=\gamma _{n}(p(v))\) for all \(v\in V(G_{k_n})\), and,
 2.
\(u(v_{k_{n+1}})\not =\gamma _{n}(p(v_{k_{n+1}}))\) for some \(v_{k_{n+1}}\in V(G_{k_{n+1}})\).
Let \(s_n=\gamma _{n+1}\gamma _{n}\in \mathcal {T}(X)\). Then \(s_n(p(v))=0\) for all \(v\in V(G_{k_n})\) and \(s_n(p(v_{k_{n+1}}))\not =0\) for some \(v_{k_{n+1}}\in V(G_{k_{n+1}})\). Thus \(\{s_n:n\in \mathbb {N}\}\) is a linearly independent set in \(\mathcal {T}(X)\) and since \(\mathcal {T}(X)\) is finite dimensional we have a contradiction. We conclude that (G, p) is infinitesimally rigid. \(\square \)
Theorem 3.14 gives useful criteria for the determination of infinitesimal rigidity of a countable framework (G, p).
Definition 3.15
A countable barjoint framework (G, p) is sequentially infinitesimally rigid if there exists a vertexcomplete tower of barjoint frameworks \(\{(G_k,p_k):k\in \mathbb {N}\}\) in (G, p) such that \((G_k,p_k)\) is infinitesimally rigid for each \(k\in \mathbb {N}\).
Corollary 3.16
Let (G, p) be a countable barjoint framework in a finite dimensional normed space \((X,\Vert \cdot \Vert )\). If (G, p) is sequentially infinitesimally rigid then (G, p) is infinitesimally rigid.
Proof
If there exists a vertexcomplete tower \(\{(G_k,p_k):k\in \mathbb {N}\}\) in (G, p) such that \((G_k,p_k)\) is infinitesimally rigid for each \(k\in \mathbb {N}\) then this framework tower is relatively infinitesimally rigid. The result now follows from Theorem 3.14. \(\square \)
Remark 3.17
The set of placements of a countable graph with prescribed edge lengths need not be an algebraic variety even when it can be realised as a finitely parametrised set. In fact there are infinite Kempe linkages which can draw everywhere nondifferentiable curves [31, 35]. It follows that the Asimow–Roth proof [3] that infinitesimal rigidity implies continuous rigidity is not available for infinite graphs, and indeed this implication does not hold in this generality. A direct way to see this is given in Kastis and Power [17] through the construction of continuously flexible crystallographic barjoint frameworks which are infinitesimally rigid by virtue of unavoidable infinite derivatives (velocities at joints) in any continuous motion.
3.4 Generic Placements for Countable Graphs
Let G be a countably infinite simple graph and let \(q\in (1,\infty )\).
Definition 3.18
A placement \(p:V(G)\rightarrow \mathbb {R}^d\) is locally generic in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) if every finite subframework of (G, p) is generic.
A tower of graphs is a sequence of finite graphs \(\{G_k:k\in \mathbb {N}\}\) such that \(G_k\) is a subgraph of \(G_{k+1}\) for each \(k\in \mathbb {N}\). A countable graph G contains a vertexcomplete tower \(\{G_k:k\in \mathbb {N}\}\) if each \(G_k\) is a subgraph of G and \(V(G)=\bigcup _{k\in \mathbb {N}} V(G_k)\).
Proposition 3.19
Every countable simple graph G has a locally generic placement in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) for \(q\in (1,\infty )\).
Proof
We now show that infinitesimal rigidity and sequential infinitesimal rigidity are generic properties for countable barjoint frameworks in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) for all \(q\in (1,\infty )\).
Proposition 3.20
 (i)
The infinitesimal flex dimension \(\dim _\mathrm{fl}(G,p)\) is constant on the set of all locally generic placements of G.
 (ii)
If (G, p) is infinitesimally rigid then every locally generic placement of G is infinitesimally rigid.
 (iii)
If (G, p) is sequentially infinitesimally rigid then every locally generic placement of G is sequentially infinitesimally rigid.
Proof
The infinitesimal flex dimension of a countable graph and the classes of countable rigid and sequentially rigid graphs are now defined.
Definition 3.21
 (i)
G is (minimally) rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) if the locally generic placements of G are (minimally) infinitesimally rigid.
 (ii)
G is sequentially rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) if the locally generic placements of G are sequentially infinitesimally rigid.
 (iii)The infinitesimal flexibility dimension of G in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) iswhere p is any locally generic placement of G.$$\begin{aligned} \dim _\mathrm{fl}(G):= \dim _{d,q}(G) := \dim _\mathrm{fl}(G,p)=\dim {\mathcal {F}}_q(G,p)/ {\mathcal {T}}_q(G,p). \end{aligned}$$
The following example demonstrates the nonequivalence of rigidity and sequential rigidity for countable graphs. The surprising fact that these properties are in fact equivalent in two dimensions is established in Theorem 4.1 below.
Example 3.22
Figure 2 illustrates the first three graphs in a tower \(\{G_n:n\in \mathbb {N}\}\) in which \(G_n\) is constructed inductively from a double banana graph \(G_1\) by flex cancelling additions of copies of \(K_5\setminus \{e\}\) (single banana graphs). The union G of these graphs is a countable graph whose maximal rigid subgraphs are copies of \(K_5\setminus \{e\}\). Thus the locally generic placements of G are not sequentially infinitesimally rigid. However the tower is relatively rigid in \(({\mathbb {R}}^3,\Vert \cdot \Vert _2)\) and so G is rigid.
4 The Equivalence of Rigidity and Sequential Rigidity
We now prove the equivalence of rigidity and sequential rigidity for countable graphs in \(({\mathbb {R}}^2,\Vert \cdot \Vert _q)\) for \(q\in (1,\infty )\).
Theorem 4.1
 (i)
G is rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
 (ii)
G is sequentially rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
Proof
(i) \(\Rightarrow \) (ii) Suppose G is rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\) and let \(p:V(G)\rightarrow \mathbb {R}^2\) be a locally generic placement. By Theorem 3.14, (G, p) has a vertexcomplete framework tower \(\{(G_k,p):k\in \mathbb {N}\}\) which is relatively infinitesimally rigid. By Theorem 3.6, \(G_k\) has a rigid container \(H_k\) in \(G_{k+1}\) for each \(k\in \mathbb {N}\). Thus \(\{H_k:k\in \mathbb {N}\}\) is the required vertexcomplete tower of rigid subgraphs in G.
(ii) \(\Rightarrow \) (i) If \(p:V(G)\rightarrow \mathbb {R}^2\) is a locally generic placement of G in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\) then by Corollary 3.16, (G, p) is infinitesimally rigid and so G is rigid. \(\square \)
We now prove our main theorem. We use the convention that if P is a property of a graph then a Ptower is a tower for which each graph \(G_k\) has property P. Thus a (2, 3)tight tower is a nested sequence of subgraphs \(\{G_k:k\in {\mathbb {N}}\}\) each of which is (2, 3)tight.
Proof of Theorem 1
(i) \(\Rightarrow \) (ii) If G is rigid then by Theorem 4.1, G is sequentially rigid and so there exists a vertexcomplete tower \(\{G_k:k\in {\mathbb {N}}\}\) of rigid subgraphs in G. We will construct a tower \(\{H_k:k\in {\mathbb {N}}\}\) of (2, l)tight subgraphs of G satisfying \(V(H_k)=V(G_k)\) for each \(k\in {\mathbb {N}}\).
By induction there exists a vertexcomplete tower \(\{H_k:k\in {\mathbb {N}}\}\) of minimally rigid subgraphs in G. In case (A), Theorem 2.11 implies that each \(H_k\) is (2, 3)tight and in case (B) Theorem 2.12 implies that each \(H_k\) is (2, 2)tight.
(ii) \(\Rightarrow \) (i) Let \(\{G_k:k\in \mathbb {N}\}\) be a (2, l)tight vertexcomplete tower in G. By Theorems 2.11 and 2.12, each \(G_k\) is a rigid graph in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\) and so G is sequentially rigid. By Theorem 4.1, G is rigid. \(\square \)
Corollary 4.2
 (A)

The following statements are equivalent.
 (i)
G is minimally rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _2)\).
 (ii)
G contains a (2, 3)tight edgecomplete tower.
 (B)

If \(q\in (1,2)\cup (2,\infty )\) then the following statements are equivalent.
 (i)
G is minimally rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\).
 (ii)
G contains a (2, 2)tight edgecomplete tower.
Proof
(i) \(\Rightarrow \) (ii) If G is minimally rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\) then by Theorem 1.1, G contains a (2, l)tight vertexcomplete tower \(\{G_k:k\in {\mathbb {N}}\}\) and this tower must be edgecomplete.
(ii) \(\Rightarrow \) (i) If G contains a (2, l)tight edgecomplete tower \(\{G_k:k\in {\mathbb {N}}\}\) then by Theorem 1.1, G is rigid. Let \(vw\in E(G)\) and suppose \(G\backslash \{vw\}\) is rigid. By Theorem 4.1, \(G\backslash \{vw\}\) is sequentially rigid and so there exists a vertexcomplete tower \(\{H_k:k\in {\mathbb {N}}\}\) in \(G\backslash \{vw\}\) consisting of rigid subgraphs. Choose a sufficiently large k such that \(v,w\in V(H_k)\) and choose a sufficiently large n such that \(vw\in E(G_n)\) and \(H_k\) is a subgraph of \(G_n\). Then \(H_k\cup \{vw\}\) is a subgraph of \(G_n\) which fails the sparsity count for \(G_n\). We conclude that \(G\backslash \{vw\}\) is not rigid in \((\mathbb {R}^2,\Vert \cdot \Vert _q)\) for all \(vw\in E(G)\). \(\square \)
Note that it follows from this corollary that the graph of a minimally infinitesimally rigid framework may have some or all of its vertices of countable degree.
4.1 Remarks and Open Problems
One can also take a matroidal point of view for infinitesimally rigid frameworks and define the infinite matroid \({\mathcal {R}}_2^\infty \) (resp. \({\mathcal {R}}_{2,q}^\infty \)) on the set S of edges of the countable complete graph \(K_\infty \). The independent sets in this matroid are the subsets of edges of a sequential Laman graph (resp. sequentially (2, 2)tight graph). Our results show that these matroids are finitary (see Oxley [33] and Bruhn et al. [6]) and so are closely related to their finite matroid counterparts.
It is a longstanding open problem to characterise in combinatorial terms the finite simple 3rigid graphs despite progress in understanding the corresponding rigidity matroid \({\mathcal {R}}_3\). See Cheng and Sitharam [7] for example. However the absence of rotational isometries in the nonEuclidean spaces \(({\mathbb {R}}^3,\Vert \cdot \Vert _q)\) suggests that a combinatorial characterisation of finite rigid graphs might be possible in terms of (3, 3)tight graphs. If this is so then part (B) of Theorem 1.1 would extend to \(d=3\).
We note that there are a number of further directions and natural problems in which relative rigidity methods play a role.
(i) It is wellknown that generic bodybar frameworks are more tractable than barjoint frameworks and in the next section we obtain variants of Tay’s [40] celebrated combinatorial characterisation.
(ii) Finite barjoint frameworks in three dimensions whose joints are constrained to move on an algebraic surface are considered in [28, 29]. In particular the graphs for generically minimally infinitesimally rigid frameworks for the cylinder are the (2, 2)tight graphs. The methods and results above for \(({\mathbb {R}}^2,\Vert \cdot \Vert _q)\) carry over readily to the cylinder.
(iii) An important theme and proof technique in the rigidity of finite graphs and geometric systems is the use of inductive constructions, that is, the construction of all graphs in a combinatorial class through a finite number of elementary construction moves, such as Henneberg moves. In our companion paper Kitson and Power [21] we consider such constructions for infinite graphs and for infinitely faceted polytopes.
(iv) In [19] it is shown that relative infinitesimal rigidity with respect to a polyhedral norm on \({\mathbb {R}}^d\) may be determined from an edgelabelling induced by the framework placement. This provides a convenient tool which is applied to obtain an analogue of Laman’s theorem for polyhedral norms on \({\mathbb {R}}^2\). The passage to countable graphs differs from the present case in that the notion of a locally generic placement used here for \(\ell ^q\) norms is no longer appropriate in the case of a polyhedral norm. Thus an analogue of Theorem 1.1 is not available, however, Theorem 3.14 may still be applied for all polyhedral norms on \({\mathbb {R}}^d\).
(v) Globally rigid graphs are those graphs G whose generic frameworks (G, p) admit no equivalent noncongruent realisations. There have been a number of recent significant advances in the determination of such graphs [11, 14] and it would be of interest to extend such results to countable graphs.
5 Rigidity of Multibody Graphs
Tay’s theorem [40] provides a combinatorial characterisation of the finite multigraphs without reflexive edges which have infinitesimally rigid generic realisations as bodybar frameworks in Euclidean space. In this section we extend Tay’s characterisation to countable multigraphs and obtain analogues of both characterisations for the nonEuclidean \(\ell ^q\) norms for all dimensions \(d\ge 2\).
5.1 Tay’s Theorem and NonEuclidean Rigidity
We now consider barjoint frameworks in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\), where \(q\in (1,\infty )\), which arise from the following class of simple graphs.
Definition 5.1
 1.
the vertexinduced subgraph determined by \(V_k\) is a rigid graph in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\), and,
 2.
every vertex \(v\in V_k\) is adjacent to at most one vertex in \(V(G)\backslash V_k\).
The rigid vertexinduced subgraph determined by \(V_k\) is denoted \(B_k\) and is called a body of G. An edge \(vw\in E(G)\) which is incident with vertices from two distinct bodies \(B_i\) and \(B_j\) is called an interbody edge. Thus a multibody graph is composed of pairwise vertexdisjoint bodies together with interbody edges such that no pair of interbody edges of G shares a vertex.
Each multibody graph G has an associated finite or countable bodybar graph \(G_b=(V(G_b),E(G_b))\) which is the multigraph with vertex set labelled by the bodies of G and with edge set derived from the interbody edges of G.
Tay’s theorem may be restated as follows:
Theorem 5.2
 (i)
G is rigid in Euclidean space \(({\mathbb {R}}^d,\Vert \cdot \Vert _2)\).
 (ii)
\(G_b\) contains a \(\big (\frac{d(d+1)}{2},\frac{d(d+1)}{2}\big )\)tight spanning subgraph.
The following lemma shows that the bodies \(B_1,B_2,\ldots \) of a multibody graph G may be modelled in a number of different ways without altering the rigidity properties of G.
Lemma 5.3
Let G and \(G'\) be two finite multibody graphs for \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) with isomorphic bodybar graphs and \(q\in (1,\infty )\). Then \(\dim _{d,q}(G)=\dim _{d,q}(G')\).
Proof
Choose a multibody graph H with bodybar graph \(H_b\) isomorphic to \(G_b\) and \(G'_b\) such that each body of H is a complete graph with more vertices than the corresponding bodies of G and \(G'\). Then there exist natural graph homomorphisms \(\phi :G\rightarrow H\) and \(\phi ':G'\rightarrow H\). If \(p_H:V(H)\rightarrow {\mathbb {R}}^d\) is a generic placement of H then \(p:V(G)\rightarrow {\mathbb {R}}^d\) defined by \(p_v=(p_H)_{\phi (v)}\) is a generic placement of G. Now the linear mapping \(A:{\mathcal {F}}_q(H,p_H)\rightarrow {\mathcal {F}}_q(G,p)\), \(A(u)_v=u_{\phi (v)}\), is an isomorphism. Applying the same argument to \(G'\) we obtain a generic placement \(p:V(G')\rightarrow {\mathbb {R}}^d\) and a linear isomorphism \(A':{\mathcal {F}}_q(H,p_H)\rightarrow {\mathcal {F}}_q(G',p')\). The result follows. \(\square \)
Example 5.4
The complete graph \(K_{d+1}\) is \(\bigl (d,\frac{d(d+1)}{2}\bigr )\)tight and is minimally rigid for \(({\mathbb {R}}^d,\Vert \cdot \Vert _2)\). The complete graph \(K_{2d}\) is (d, d)tight and is a minimally rigid graph for \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) for each of the nonEuclidean \(\ell ^q\)norms. These sparsity and rigidity properties persist for graphs obtained from these complete graphs by a finite sequence of Henneberg vertex extension moves of degree d. Thus we may assume without loss of generality that the bodies of a finite multibody graph for \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) are \(\bigl (d,\frac{d(d+1)}{2}\bigr )\)tight in the Euclidean case and (d, d)tight in the nonEuclidean case. The convenience of modelling multibody graphs in this way is that the combinatorial and \(\ell ^q\)norm analysis of earlier sections is readytohand.
There is a natural vertexinduced surjective graph homomorphism \(\pi :G \rightarrow \overline{G}_b\) where \(\overline{G}_b\) is the multigraph obtained by contracting the bodies of G. The bodybar graph \(G_b\) is a subgraph of \(\overline{G}_b\) obtained by removing reflexive edges and \(\pi \) gives a bijection between the interbody edges of G and the edges of \(G_b\).
Theorem 5.5
 (i)
G is rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
 (ii)
\(G_b\) has a (d, d)tight spanning subgraph.
Proof
(ii) \(\Rightarrow \) (i) If \(G_b\) is (d, d)tight then it admits a partition as an edgedisjoint union of d spanning trees \(T_1,T_2,\ldots ,T_d\) (see [26]). We will construct a placement of G so that \(p_vp_w\) lies on the ith coordinate axis in \({\mathbb {R}}^d\) whenever vw is an interbody edge with \(\pi (vw)\in T_i\).
 (1)
\(p_k(V(B_k))=p_1(V(B_1))\), and,
 (2)
\(p_j(v)=p_k(w)\) whenever \(j<k\) and \(vw\in E(G)\) is an interbody edge with \(v\in V(B_j)\) and \(w\in V(B_k)\).
Suppose \(u=(u_1,\ldots ,u_n)\in \ker R_q(G,p')\). For a sufficiently small \(\epsilon \) each subframework \((B_i,p')\) is infinitesimally rigid, and so \(u_i=(a_i,\ldots ,a_i)\) for some \(a_i\in {\mathbb {R}}^d\). If vw is an interbody edge with \(\pi (vw)\in T_i\) then the corresponding row entries in \(R_q(G,p')\) are nonzero in the \(p_{v,i}\) and \(p_{w,i}\) columns only. The spanning tree property now ensures that \(a_1=\cdots =a_n\) and so the kernel of \(R_q(G,p')\) has dimension d. Thus \(p'\) is an infinitesimally rigid placement of G in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\). More generally if \(G_b\) contains a vertexcomplete (d, d)tight subgraph then by the above argument G is rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\). \(\square \)
A key feature of bodybar frameworks is the nonincidence condition for the bars. This makes available special realisations which are rigid, as we have seen in the proof of the analogue of Tay’s theorem, Theorem 5.5. Other instances of this can be seen in the matroid analysis of Whiteley [44] and in the analysis of Borcea and Streinu [5] and Ross [36] of locally finite graphs with periodically rigid periodic barjoint frameworks.
We will require the following definition and corollary to characterise the countable rigid multibody graphs for \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
Definition 5.6
A multibody graph for \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) is essentially minimally rigid if it is rigid and removing any interbody edge results in a multibody graph which is not rigid.
Corollary 5.7
 (a)
\(q=2\) and \(k=\frac{d(d+1)}{2}\), or,
 (b)
\(q\in (1,2)\cup (2,\infty )\) and \(k=d\).
 (i)
G is essentially minimally rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
 (ii)
\(G_b\) is a (k, k)tight multigraph.
5.2 Rigidity of Countable Multibody Graphs
We are now able to characterise the countable rigid multibody graphs in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) for all dimensions \(d\ge 2\) and all \(q\in (1,\infty )\). Given a finite barjoint framework (G, p) in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\) we denote by \(X_\mathrm{row}(G,p)\) the row space of the rigidity matrix \(R_q(G,p)\).
Definition 5.8
The following result is an analogue of Proposition 3.5.
Proposition 5.9
 (a)
\(q=2\), \(k=\frac{d(d+1)}{2}\) and G contains at least \(d(d+1)\) vertices, or,
 (b)
\(q\in (1,2)\cup (2,\infty )\), \(k=d\) and G contains at least 2d vertices.
 (i)
G is essentially independent with respect to \((\mathbb {R}^d,\Vert \cdot \Vert _q)\).
 (ii)
\(G_b\) is (k, k)sparse.
Proof
Conversely, if \(G_b\) is (k, k)sparse then by Lemma 3.2, \(G_b\) is a vertexcomplete subgraph of a (k, k)tight multigraph \(G'_b\) which has no reflexive edges. Let \(G'\) be a multibody graph with bodybar graph isomorphic to \(G'_b\) and which contains G as a subgraph. By Corollary 5.7, \(G'\) is essentially minimally rigid and it follows that G is essentially independent. \(\square \)
We now prove an analogue of Theorem 3.6 which shows that in the category of multibody graphs relative rigidity is equivalent to the existence of a rigid container for all dimensions d and for all \(\ell ^q\) norms.
Theorem 5.10
 (a)
\(q=2\) and H contains at least \(d(d+1)\) vertices, or,
 (b)
\(q\in (1,2)\cup (2,\infty )\) and H contains at least 2d vertices.
 (i)
H is relatively rigid in G with respect to \((\mathbb {R}^d,\Vert \cdot \Vert _q)\).
 (ii)
H has a rigid container in G with respect to \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) which is a multibody graph.
Proof
Define \(H'\) to be the union of H together with the subgraphs \(H_{v,w}\) for all such pairs \(\pi (v),\pi (w)\in V(H_b)\). Thus \(H'\) is the multibody subgraph of G with bodybar graph isomorphic to \(H'_b\). Then \(H'\) is rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) and so \(H'\) is a rigid container for H in G.
(ii) \(\Rightarrow \) (i) If H has a rigid container \(H'\) in G and \(p:V(G)\rightarrow {\mathbb {R}}^d\) is a generic placement of G then no nontrivial infinitesimal flex of (H, p) extends to \((H',p)\). The result follows. \(\square \)
We now prove the equivalence of rigidity and sequential rigidity for multibody graphs with respect to all \(\ell ^q\)norms and in all dimensions \(d\ge 2\).
Theorem 5.11
 (i)
G is rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
 (ii)
G is sequentially rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
Proof
Suppose G is rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) and let \(p:V(G)\rightarrow \mathbb {R}^d\) be a locally generic placement. By Theorem 3.14, there exists a vertexcomplete tower \(\{(G_k,p_k):k\in \mathbb {N}\}\) in (G, p) which is relatively infinitesimally rigid. Moreover, we can assume that each \(G_k\) is a multibody graph. By Proposition 5.10, \(G_k\) has a rigid container \(H_k\) in \(G_{k+1}\) for each \(k\in \mathbb {N}\). Thus the sequence \(\{H_k:k\in \mathbb {N}\}\) is a vertexcomplete tower of rigid graphs in G. For the converse apply Corollary 3.16. \(\square \)
We now prove our second main result which generalises Tay’s theorem to countable multibody graphs.
Proof of Theorem 2
(i) \(\Rightarrow \) (ii) If G is rigid then by Theorem 5.11, G is sequentially rigid. Let \(\{G_k:k\in \mathbb {N}\}\) be a vertexcomplete tower of rigid subgraphs in G and let \(B_1,B_2,\ldots \) be the bodies of G. We may assume that each \(G_k\) is a multibody graph. Applying the induction argument used in Theorem 1.1 we construct a vertexcomplete tower of essentially minimally rigid multibody subgraphs in G. To do this let \(H_1\) be the multibody graph obtained by taking all bodies which lie in \(\tilde{G}_1\) and adjoining interbody edges of \(G_1\) until an essentially minimally rigid graph is reached. The induced sequence of bodybar graphs \(\{(H_{k})_b:k\in \mathbb {N}\}\) is a vertexcomplete tower in \(G_b\). By Corollary 5.7 each bodybar graph \((H_k)_b\) is \(\bigl (\frac{d(d+1)}{2},\frac{d(d+1)}{2}\bigr )\)tight in case (A) and (d, d)tight in case (B).
(ii) \(\Rightarrow \) (i) Let \(\{G_{k,b}:k\in \mathbb {N}\}\) be a \(\bigl (\frac{d(d+1)}{2},\frac{d(d+1)}{2}\bigr )\)tight vertexcomplete tower in \(G_b\) and let \(\pi :G\rightarrow \overline{G}_b\) be the natural graph homomorphism. Define \(G_k\) to be the subgraph of G with \(V(G_k)=\pi ^{1}(V(G_{k,b}))\) such that \(G_k\) contains the body \(B_i\) whenever \(\pi (V(B_i))\in V(G_{k,b})\) and \(G_k\) contains the interbody edge vw whenever \(\pi (vw)\in E(G_{k,b})\). Then \(G_{k,b}\) is the bodybar graph for \(G_k\) and so \(G_k\) is rigid by Theorem 5.2. Thus \(\{G_k:k\in \mathbb {N}\}\) is a vertexcomplete tower of rigid subgraphs in G and so G is sequentially rigid. By Theorem 5.11, G is rigid.
To prove (B) we apply similar arguments to the above using the nonEuclidean versions of the relevant propositions and substituting Theorem 5.5 for Theorem 5.2. \(\square \)
Corollary 5.12
 (A)

The following statements are equivalent.
 (i)
G is essentially minimally rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _2)\).
 (ii)
\(G_b\) has a \(\bigl (\frac{d(d+1)}{2},\frac{d(d+1)}{2}\bigr )\)tight edgecomplete tower.
 (B)

If \(q\in (1,2)\cup (2,\infty )\) then the following statements are equivalent.
 (i)
G is essentially minimally rigid in \(({\mathbb {R}}^d,\Vert \cdot \Vert _q)\).
 (ii)
\(G_b\) has a (d, d)tight edgecomplete tower.
Proof
(i) \(\Rightarrow \) (ii) If G is essentially minimally rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) then by Theorem 1.2, \(G_b\) contains a (k, k)tight vertexcomplete tower \(\{G_k:k\in {\mathbb {N}}\}\) and this tower must be edgecomplete.
(ii) \(\Rightarrow \) (i) If \(G_b\) contains a (k, k)tight edgecomplete tower \(\{G_{k,b}:k\in {\mathbb {N}}\}\) then by Theorem 1.2, G is rigid. Let \(vw\in E(G)\) be an interbody edge and suppose \(G\backslash \{vw\}\) is rigid. By Theorem 5.11, \(G\backslash \{vw\}\) is sequentially rigid and so there exists a vertexcomplete tower \(\{H_k:k\in {\mathbb {N}}\}\) in \(G\backslash \{vw\}\) consisting of rigid subgraphs. Moreover, we can assume that each \(H_k\) is a multibody graph. Choose a sufficiently large k such that \(v,w\in V(H_k)\) and choose a sufficiently large n such that \(vw\in E(G_n)\) and \(H_k\) is a subgraph of \(G_n\). Then the bodybar graph for \(H_k\cup \{vw\}\) is a subgraph of \((G_n)_b\) which fails the sparsity count for \((G_n)_b\). We conclude that \(G\backslash \{vw\}\) is not rigid in \((\mathbb {R}^d,\Vert \cdot \Vert _q)\) for all \(vw\in E(G)\). \(\square \)
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