Global Rigidity of Periodic Graphs under Fixed-lattice Representations

In 1992, Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in $\mathbb{R}^d$. Jackson and Jordan confirmed in 2005 that these conditions are also sufficient in $\mathbb{R}^2$, giving a combinatorial characterization of graphs whose generic realizations in $\mathbb{R}^d$ are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in $\mathbb{R}^2$ in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordan.


Introduction
A bar-joint framework (or simply framework) in R d is a pair (G, p), where G = (V, E) is a graph and p : V → R d is a map. We think of a framework as a straight-line realization of G in R d in which the length of an edge uv ∈ E is given by the Euclidean distance between the points p(u) and p (v). A well-studied problem in discrete geometry is to determine the rigidity of frameworks. A framework (G, p) is called (locally) rigid if, loosely speaking, it cannot be deformed continuously into another non-congruent framework while maintaining the lengths of all edges. It is well-known that a generic framework (G, p) is rigid in R d if and only if every generic realization of G in R d is rigid [1,7]. In view of this fact, a graph G is said to be rigid in R d if some/any generic realization of G is rigid in R d . A classical theorem by Laman [19] says that G is rigid in Another central property in rigidity theory is global rigidity. A framework (G, p) is called globally rigid, if every framework (G, q) in R d with the same edge lengths as (G, p) has the same distances between all pairs of vertices as (G, p). Although deciding the global rigidity of a given framework is a difficult problem in general, the problem becomes tractable if we restrict attention to generic frameworks [8], i.e. frameworks with the property that the coordinates of all points p(v), v ∈ V , are algebraically independent over Q.
In 1992 Hendrickson [9] established the following necessary condition for a generic framework in R d to be globally rigid. Theorem 1.1 (Hendrickson [9]). If (G, p) is a generic globally rigid framework in R d , then G is a complete graph with at most d + 1 vertices, or G is (d + 1)-connected and redundantly rigid in R d , where G is called redundantly rigid if G − e is rigid for every e ∈ E(G).
Although the converse direction is false for d ≥ 3 as pointed out by Connelly, it turned out to be true for d ≤ 2.
Theorem 1.2 (Jackson and Jordán [10]). Let (G, p) be a generic framework in R 2 . Then (G, p) is globally rigid if and only if G is a complete graph with at most 3 vertices, or G is 3-connected and redundantly rigid in R 2 .
Based on the theory of stress matrices by Connelly [4,5], Gortler, Healy, and Thurston [8] gave an algebraic characterization of the global rigidity of generic frameworks. This in particular implies that all generic realizations of a given graph share the same global rigidity properties in R d , as in the case of local rigidity. However the problem of extending Theorem 1.2 to higher dimension remains unsolved.
Largely motivated by practical applications in crystallography, materials science and engineering, as well as by mathematical applications in areas such as sphere packings, the rigidity analysis of infinite periodic frameworks has seen an increased interest in recent years. Particularly relevant to our work is an extension of Laman's theorem to periodic frameworks with fixed-lattice representations by Ross [24]. In her rigidity model, a periodic framework can deform continuously under a fixed periodicity constraint (i.e., each orbit of points is fixed).
In this paper, we shall initiate the global rigidity counterpart of the rigidity theory of periodic frameworks. The global rigidity of periodic frameworks is considered at the same level as Ross's rigidity model [24,25], and we shall extend Theorem 1.1 and Theorem 1.2 to periodic frameworks. Analogous to Theorem 1.1 there are two types of necessary conditions, a graph connectivity condition (Lemma 3.1) and a redundant rigidity condition (Lemma 3.7). Our main result (Theorem 4.2) is that these necessary conditions are also sufficient in R 2 , thus giving a first combinatorial characterization of the global rigidity of generic periodic frameworks. The proof of this result is inspired by the work in [11,26]. In particular, it does not require the notion of stress matrices [4,5]. Note also that our proof does not rely on periodic global rigidity in R 2 being a generic property, meaning that all generic realizations of a periodic graph in the plane share the same global rigidity properties.
As for the rigidity of periodic frameworks, Borcea and Streinu [3] introduced a more general setting, where the underlying lattice of a framework may deform during a motion of the framework (but the framework must remain periodic), and Malestein and Theran [20] established an extension of Laman's theorem in this more general setting. Extending our global rigidity result to this flexible lattice setting is an important and challenging open problem. Important corollaries of our main theorem are toroidal and cylindrical counterparts of Theorem 1.2. Here we only give a statement for cylindrical frameworks, but the statement for toroidal frameworks can be derived in a similar fashion (see Theorem 6.1). Consider a straight-line drawing of a graph G on a flat cylinder C. We regard it as a bar-joint framework on C. Using the metric inherited from its representation as R 2 /L for a fixed one-dimensional lattice L, the local/global rigidity is defined. Ross's theorem [24] for periodic frameworks implies that a generic framework on C is rigid if and only if the underlying graph contains a spanning subgraph H with |E(H)| = 2|V (H)| − 2 such that |F | ≤ 2|V (F )| − 2 for every F ⊆ E(H) and |F | ≤ 2|V (F )| − 3 for every nonempty contractible F ⊆ E(H), where F is said to be contractible if every cycle in F is contractible on C.

Theorem 1.3. A generic framework (G, p) with |V (G)| ≥ 3 on a flat cylinder C is globally rigid if and only if it is redundantly rigid on C, 2-connected, and has no contractible subgraph H with |V (H)| ≥ 3 and |B(H)| = 2, where B(H) denotes the set of vertices in H incident to some edge not in H.
The paper is organized as follows. In Section 2, we define the concept of global rigidity for periodic frameworks with a fixed lattice representation, and then establish Hendrickson-type necessary conditions for a generic periodic framework to be globally rigid in R d in Section 3. In Section 4 we then show that for d = 2 the necessary conditions established in Section 3 are also sufficient for generic global rigidity (Theorem 4.2). Section 5 is devoted to the proofs of the combinatorial lemmas stated in Section 4.

Periodic graphs
Let Γ be a group isomorphic to Z k . In general, a pair (G, ψ) of a directed (multi-)graph G and a map ψ : E(G) → Γ is called a Γ-labeled graph.
For a given Γ-labeled graph (G, ψ), one can construct a k-periodic graphG by setting Γ is called a periodicity of G , which naturally acts on V (G) and E(G). This G is also called the covering of (G, ψ) while (G, ψ) is called the quotient Γ-labeled graph of G . See Fig. 1 for two examples of k-periodic graphs and corresponding quotient Γ-labeled graphs.
Although G is directed, its orientation is used only for the reference of the group label, and we are free to change the orientation of each edge by imposing the property that if an edge has a label γ in one direction, then it has γ −1 in the other direction. More precisely, two edges e 1 , e 2 are regarded as identical if they are parallel with the same direction and the same label, or with the opposite direction and the opposite labels. Throughout the paper, all Γ-labeled graphs are assumed to be semi-simple, that is, no identical two edges exist (although parallel edges may exist). We will assume that G is loopless. Note that a loop in G corresponds to an edge orbit in G consisting of edges that connect vertices in the same orbit in G . In our L-periodic (global) rigidity model defined in Sections 2.3 and 2.4 such edge constraints are redundant thus do not affect the (global) rigidity of the framework.
We define a walk as an alternating sequence v 1 , e 1 , v 2 . . . , e k , v k+1 of vertices and edges such that v i and v i+1 are the end vertices of e i . For a closed walk C = v 1 , e 1 , v 2 . . . , e k , v 1 in (G, ψ), let ψ(C) = k i=1 ψ(e i ) sign(e i ) , where sign(e i ) = 1 if e i has forward direction in C, and sign(e i ) = −1 otherwise. For a subgraph H of G define Γ H as the subgroup of Γ generated by the elements ψ(C), where C ranges over all closed walks in H. The rank of H is defined to be the rank of Γ H . Note that the rank of G may be less than the rank of Γ (see Fig. 1(a)). The framework in (a) has infinitely many connected components, and its quotient Z 2 -labeled graph shown in (b) is of rank one. The framework in (c) has two connected components, and its quotient Z 2labeled graph shown in (d) is of rank two. The vertices of the two components are depicted with two distinct colors in (c). We remark that neither of these frameworks is L-periodically globally rigid, by Lemma 3.1.
We say that an edge set F is balanced if the subgraph induced by F has rank zero, i.e., ψ(C) = id for every closed walk C in F .
One useful tool to compute the rank of a subgraph is the switching operation. A switching at v ∈ V (G) by γ ∈ Γ changes ψ to ψ defined by ψ (e) = γψ(e) if e is directed from v, ψ (e) = γ −1 ψ(e) if e is directed to v, and ψ (e) = ψ(e) otherwise. A switching operation preserves the rank of any subgraph (see [18] for details).
Note that the quotient Γ-labeled graph of a k-periodic graph is not unique in general. It is not difficult to see that (G = (V, E), ψ) and (G = (V, E ), ψ ) define the same covering graph G if and only if (G, ψ) can be transformed into (G , ψ ) via edge reversions (as described above) and switching operations.

Periodic frameworks
In the context of graph rigidity, a pair (G, p) of a graph G = (V, E) and a map p : V → R d is called a (bar-joint) framework in R d . A periodic framework is a special type of infinite framework defined as follows.
Let G be a k-periodic graph with periodicity Γ, and let L : Γ → R d be a nonsingular homomorphism with k ≤ d, where L is said to be nonsingular if L(Γ) has rank k. A pair (G, p) of G and p :Ṽ → R d is said to be an L-periodic framework in R d if for all γ ∈ Γ and all v ∈Ṽ .
We also say that a pair (G, p) is k-periodic in R d if it is L-periodic for some nonsingular homomorphism L : Γ → R d . Note that the rank k of the periodicity may be smaller than d. For instance, if k = 1 and d = 2, then (G, p) is an infinite strip framework in the plane. Any connected component of the framework in Fig. 1(a), for example, is such a strip framework (with Γ = (1, 1) ). An L-periodic framework (G, p) is generic if the set of coordinates is algebraically independent over the rationals modulo the ideal generated by the equations (1). For simplicity of description, throughout the paper, we shall assume that L is a rationalvalued function, 4 i.e., L : Γ → Q d .
Let (G, p) be an L-periodic framework and let (G, ψ) be the quotient Γ-labeled graph of G . Following the convention that V (G) is identified with the set {v 1 , . . . , v n } of representative vertices, one can define the quotient Γ-labeled framework as the triple (G, ψ, p) In general, a Γ-labeled framework is defined to be a triple (G, ψ, p) of a finite Γ-labeled graph (G, ψ) and a map p : V (G) → R d . The covering of (G, ψ, p) is a k-periodic framework (G, p), where G is the covering of G and p is uniquely determined from p by (1).
We say that a Γ-labeled framework (G, ψ, p) is generic if the set of coordinates in p is algebraically independent over the rationals. Note that an L-periodic framework (G, p) is generic if and only if the quotient (G, ψ, p) of (G, p) is generic.

Rigidity and global rigidity
Let G = (V, E) be a graph. Two frameworks (G, p) and (G, q) in R d are said to be equivalent if They are congruent if A framework (G, p) is called globally rigid if every framework (G, q) in R d which is equivalent to (G, p) is also congruent to (G, p). We may define the corresponding periodicity-constrained concept as follows. An L-periodic framework (G, p) in R d is L-periodically globally rigid if every L-periodic framework in R d which is equivalent to (G, p) is also congruent to (G, p). Note that if the rank of the periodicity is equal to zero, then L-periodic global rigidity coincides with the global rigidity of finite frameworks.
A key notion to analyze L-periodic global rigidity is L-periodic rigidity. By the periodicity constraint (1), the space of all L-periodic frameworks in R d for a given graph G can be identified with Euclidean space R dn so that the topology is defined. A framework (G, p) is called L-periodically rigid if there is an open neighborhood N of p in which every L-periodic framework (G, q) which is equivalent to (G, p) is also congruent to (G, p).

Characterizing L-periodic rigidity
A key tool to analyze the local or the global rigidity of finite frameworks is the lengthsquared function and its Jacobian, called the rigidity matrix. One can follow the same strategy to analyze local or global periodic rigidity.
For a Γ-labeled graph (G, ψ) and L : For a finite set V , the complete Γ-labeled graph K(V, Γ) on V is defined to be the graph on V with the edge set {(u, γv) : u, v ∈ V, γ ∈ Γ}. We simply denote f K(V,Γ),L by f V,L . By (1) we have the following fundamental fact.
Proposition 2.1. Let (G, p) be an L-periodic framework and let (G = (V, E), ψ, p) be a quotient Γ-labeled framework of (G, p). Then (G, p) is L-periodically globally (resp. locally) rigid if and only if for every q ∈ R d|V | (resp. for every q in an open neighborhood In view of this proposition, we say that a Γ-labeled framework (G, ψ, p) is Lperiodically globally (or, locally) rigid if for every q ∈ R d|V | (resp. for every q in an , and we may focus on characterizing the L-periodic global (or, local) rigidity of Γ-labeled frameworks. For The following is the first fundamental fact for analyzing periodic rigidity.
Proof. To see the necessity, assume that f V,L (p) = f V,L (q). Then p(u) − p(w) = q(u) − q(w) for every pair of elements u, w ∈ V . Since p(V ) affinely spans the whole space, this implies that there is a unique isometry h : R d → R d such that q(u) = h(p(u)) for every u ∈ V . In other words, there is an orthogonal matrix S such that affinely spans the whole space, (I − S )L(γ) = 0 for every γ. In other words, S fixes each element in L(Γ) as required.
Conversely, suppose that q is represented as q(v) = Sp(v) + t for some t ∈ R d and some orthogonal matrix S that fixes each element in L(Γ).
The following algebraic characterization is the periodic version of one of the fundamental facts in rigidity theory, and is known in the periodic case even if k = d [25].
and rank k periodicity Γ, and let L : where df G,L | p denotes the Jacobian of f G,L at p.
Proof. Since the rank of Γ is k, the set of L(Γ)-invariant isometries forms a (d By the standard argument using the inverse function theorem, it follows that (G, ψ, p) is L-periodically rigid if and only if rank df V,L | p = rank df G,L | p , implying the statement.
For d = 2 Ross [24] gave a combinatorial characterization of the rank of df G,L | p for generic (G, ψ, p), which implies the following. (Her statement is only for k = 2, but the proof can easily be adapted to the case when k = 1.) Theorem 2.4 (Ross [24]). Let (G, ψ, p) be a generic Γ-labeled framework in R 2 with rank k ≥ 1 periodicity Γ and let L : Γ → R 2 be nonsingular. Then (G, ψ, p) is L-periodically rigid if and only if (G, ψ) contains a spanning subgraph (H, ψ H ) satisfying the following count conditions: Note that if k = 0, then an L-periodic framework is simply a finite framework, and generic rigid frameworks in R 2 are characterized by the celebrated Laman theorem [19].

Necessary conditions
In this section we provide necessary conditions for L-periodic global rigidity. As in the finite case, there are two types of conditions, a connectivity condition and a redundant rigidity condition. These two conditions are stated in Lemma 3.1 and Lemma 3.7, respectively.

Necessary connectivity conditions
Let (G, ψ) be a Γ-labeled graph with Γ having rank k.
Proof. We first remark that the following holds for any Γ-labeled graph (H, ψ H ) and any v ∈ V (H): This follows from Proposition 2.2 by noting that the reflection g is L(Γ H )-invariant.
Suppose that G has an (s, t)-block H satisfying the property of the statement. If t = 0 and V (H) = V (G) (i.e., G is disconnected), then by translating p(V (H)) we obtain q with f G,L (p) = f G,L (q) and f V,L (p) = f V,L (q), contradicting the global rigidity of (G, ψ, p). Thus we have ( The affine span affP of P is a proper subspace of R d . Indeed, if t > 0, then B = B(H) and affP has dimension at most s + t − 1, which is less than d by the lemma assumption. On the other hand, if t = 0, then B = {x} and affP has dimension s, which is less than d by (3) and the lemma assumption. Thus (4) follows. By (4) we can take a hyperplane H that contains affP . Since p is generic, such a hyperplane can be taken such that H contains no point in We first show f G,L (p) = f G,L (q). Take any edge e = uv in G. If e / ∈ E(H), then To derive a contradiction we shall show f V,L (p) = f V,L (q) by splitting the proof into two cases depending on whether V (H) = V (G) or not.
Suppose that V (G) = V (H). By the definition of an (s, t)-block, we can take a v ∈ I(H).
Consider, for example, the framework shown in Fig. 2(a) and its quotient Z 2 -labeled graph (G, ψ) shown in (b). The subgraph H of (G, ψ) induced by the dotted edges is a (1, 1)-block with V (H) = V (G). Thus, by Lemma 3.1, the framework in (a) is not globally L-periodically rigid. Here aff(P ) is one of the thin black lines in (a) connecting the copies of the black vertices, and g is the reflection in aff(P ). The framework (G, q) is obtained from (G, p) by reflecting each connected component of the framework with dotted edges in the corresponding parallel copy of aff(P ) containing the black vertices of the component. Fig. 2(c) shows another example of a framework which is not globally L-periodically rigid by Lemma 3.1. Consider the corresponding quotient Z 2 -labeled graph (G, ψ) shown in Fig. 2(d). The subgraph H induced by the dotted edges is a (0, 2)-block of G with V (H) = V (G) but 0 = s < k = 2. Here aff(P ) is one of the lines in (c) indicated by thin black line segments connecting pairs of black vertices, and g is again the reflection in aff(P ). The framework (G, q) is obtained from (G, p) as described in the previous case.
Sometimes Lemma 3.1 can be strengthened by decomposing graphs. Consider for example the case d = 2. Suppose that (G 1 , ψ 1 , p 1 ) is redundantly L-periodically rigid but contains a (1, 1)-block H. Then by Lemma 3.1 (G 1 , ψ 1 , p 1 ) is not L-periodically globally rigid. Now consider attaching a new L-periodically globally rigid framework (G 2 , ψ 2 , p 2 ) at a vertex in I(H) with |V (G 1 ) ∩ V (G 2 )| = 1. Then the resulting framework is clearly not L-periodically globally rigid but H is no longer a (1, 1)-block and the resulting graph may satisfy the cut condition (and may also be redundantly L-periodically rigid). In general, if a framework has a cut vertex, then we should look at each 2-connected component individually based on the following fact.
Lemma 3.2. Let (G, ψ, p) be a Γ-labeled framework with rank d periodicity Γ, and suppose that it can be decomposed into two frameworks is L-periodically globally rigid and of rank d.
Proof. Note that, if the underlying periodicity group Γ has rank d, then every L(Γ)invariant isometry is a translation. Hence the claim follows from Proposition 2.2.
A similar statement to Lemma 3.2 holds if we assume that the intersection of the two frameworks forms an L-periodically globally rigid subframework. Extending it to a more general gluing scenario (and sharpening the necessary condition for global periodic rigidity) is left as an open problem.

The necessity of redundant L-periodic rigidity
Let f : R d → R k be a smooth map. Then x ∈ R d is said to be a regular point of f if the Jacobian df | x has maximum rank, and is a critical point of f otherwise. Also We use the following lemmas.
Lemma 3.3. (See, e.g., [11]) Let f : R d → R k be a polynomial map with rational coefficients and p be a generic point in For a vector p in R d , let Q(p) be the field generated by the entries of p and the rationals. For a field F and an extension K, let td[K : F ] denote the transcendence degree of the extension. For a field K, let K be the algebraic closure of K. We also need the following lemma which will be used in the proof of Lemma 4.5. (See, e.g., [12,Proposition 13] for the proof.)

a polynomial map with rational coefficients and p be a generic point in
We now return to our discussion of L-periodically globally rigid frameworks. Let Γ be a group isomorphic to Z k , and (G, ψ) be a Γ-labeled graph with |V | ≥ t, where t = max{d − k, 1}. Let L : Γ → R d be a nonsingular homomorphism, and for simplicity we suppose that the linear span of L(Γ) is {0} d−k × R k , the linear subspace spanned by the last k coordinates. We pick any t vertices v 1 , . . . , v t , and define the augmented is a rational polynomial map given by . Augmenting f G,L by appending g corresponds to "pinning down" some coordinates to eliminate trivial continuous motions. Proposition 3.6. Let (G, ψ, p) be a Γ-labeled framework in R d with rank k periodicity and Proof. We consider the system (df G,L | p )ṗ = 0 of linear equations with variables ṗ ∈ R d|V | . By regarding ṗ ∈ R d|V | as a map ṗ : V → R d , this system can be described as By isometry is a composition of a rotation fixing the last k coordinates and a translation. Hence a map ṗ : V → R d is a solution of the linear system (5) if it is of the forṁ for some x ∈ R d and some skew-symmetric matrix S ∈ R d×d such that only the top-left (d − k) × (d − k) block of S may be nonzero and the remaining entries are zero. Such ṗ is called a trivial infinitesimal motion. The set of trivial infinitesimal motions forms a linear space of dimension d + t 2 , and hence it suffices to prove that no trivial infinitesimal motion is in the kernel of df G,L | p .
Let us take any trivial infinitesimal motion ṗ described by S and x as above, and suppose that ṗ is in the kernel of df G,L | p . Note that df G,L | p is obtained from df G,L | p by augmenting dg| p .
By the first d rows in dg| p , we have x = 0. Similarly, if we denote the i-th row of S by s i , then by the remaining t 2 rows of dg| p , we get . . .
Since p is generic and S is a skew-symmetric matrix such that only the top-left (d − k) × (d − k) block of S may be nonzero, we have s i = 0 for every i, implying S = 0 and ṗ = 0.
We say that (G, ψ, p) is redundantly L-periodically rigid if (G −e, ψ, p) is L-periodically rigid for every e ∈ E(G). (For simplicity, we slightly abuse notation here and denote the restriction of ψ to E(G) − e also by ψ.) Proof. The proof strategy is analogous to the one for Theorem 8.2 in [14]. Suppose for a contradiction that (G − e, ψ, p) is not L-periodically rigid for some e ∈ E. Since (G, ψ, p) is L-periodically globally rigid, it is L-periodically rigid. Hence by Proposition 2.3 and Proposition 3.
Hence its preimage is a 1dimensional smooth manifold (see, e.g., [23]). Since this manifold is bounded (due to the "pinning" of some of the vertices) and closed, it is compact, and it consists of a disjoint union of cycles by the classification of 1-dimensional manifolds. Let O be the component that contains p.
Consider f e,L : [13,Lemma 3.4] for the proof of the first equation). Hence df e,L | p is nonzero (i.e., p is not a critical point of f e,L ), and so the intermediate value theorem implies that there is a q ∈ O with f e,L (q) = f e,L (p) and q = p. We can assign an orientation to O and we may assume that q is chosen as close to p as possible in the forward direction. Take a path γ : If γ travels in the same direction as γ within O then we can assume that f e,L increases as we pass through p in the forward direction. Then f e,L has to increase as we pass through q. Thus there are values . Using the intermediate value theorem, we then get a contradiction, because there exists a point p with f e,L (p ) = f e,L (p) between p 0 and p 1 .
If γ travels in the direction opposite to γ within O, then there exists a t ∈ [0, 1] such that γ(t) = γ (t). At this t, we have p t (u) = h(p t (u)) for every u ∈ V . In other words, p t (V ) is contained in the invariant subspace H of h, which is a proper affine subspace of R d as p = q. Let d (< d) be the affine dimension of H. Since H contains L(Γ) whose basis is rational, H is determined by (d + 1)d parameters, at most (d + 1 − k)d of which are independent over Q.
The last term is at most d if k ≥ 1 and at most d + 1 if k = 0, which contradicts the assumption of the statement.

Main theorems
In this section we characterize periodic global rigidity in the plane based on the necessary conditions given in Section 3. We need to introduce one more term to describe the main theorem combinatorially. Given a Γ-labeled graph (G, ψ), Proposition 2.3 implies that (G, ψ, p) is L-periodically rigid for some generic p if and only if (G, ψ, p) is L-periodically rigid for every generic p. Moreover, Theorem 2.4 says that the choice of L is not important as long as L is nonsingular. In view of these facts, we say that We are now ready to state our main theorems.
is redundantly periodically rigid in R 2 , 2-connected, and has no (0, 2)-block.    Proof. Let V (G) = {u, v}, and suppose that every edge is oriented to v.
If k = 1, then there exist two edges from u to v with distinct labels. By switching, we may assume that these labels are id and γ. Given q(v), there are only two possible positions for q(u) if (G, ψ, p) is equivalent to (G, ψ, p). In both cases, the resulting framework is congruent to (G, ψ, p), and hence (G, ψ, p) is L-periodically globally rigid.
If k = 2, then there exists a third edge with label γ not spanned by γ. The given distance between q(u) and γ q(v) then uniquely determines the position of q(u). Thus, (G, ψ, q) is again congruent to (G, ψ, p), and hence (G, ψ, p) is L-periodically globally rigid.
We remark here that there is an efficient algorithm to check whether (G, ψ) satisfies the combinatorial conditions of the main theorem. After finding the 2-connected components of G the method given in [18] can be used here as well to check their redundant periodic rigidity. Their rank can also be checked easily by finding an equivalent gain function by switchings (see, e.g., [18]). Finally, for each pair of vertices we can check whether they are the boundary of a (0,2)-block.
The proofs of Theorem 4.1 and Theorem 4.2 are almost identical and consist of two parts, an algebraic part and a combinatorial part. The algebraic part is solved in Lemma 4.5, and the combinatorial part is solved in Lemmas 4.6 and 4.7.

Algebraic part
Let (G, ψ, p) be a Γ-labeled framework. We say that a vertex v is nondegenerate in Note that this is a generic property, that is, v is nondegenerate in a generic realization of (G, ψ) if and only if it is nondegenerate in any generic realization of (G, ψ). Thus a vertex v in (G, ψ) is said to be d-nondegenerate if v is nondegenerate in a generic realization of (G, ψ).
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, ψ), let e 1 · e 2 be the edge from u to w with label ψ(vu) −1 ψ(vw). We define (G v , ψ v ) to be the Γ-labeled graph obtained from (G, ψ) by removing v and inserting e 1 · e 2 for every pair of nonparallel edges e 1 , e 2 incident to v (unless an edge identical to e 1 · e 2 is already present in (G, ψ)). The following is the periodic generalization of an observation given in [11,26].
Proof. Pin the framework (G, ψ, p) (as done in Section 3.2) and take any q , v is different from the vertices selected when augmenting f G,L to f G,L ). Our goal is to show that p = q.
Let p and q be the restrictions of p and q to Thus we can take a spanning subgraph H of G − v such that df H,L | p has linearly independent rows, i.e. is nonsingular. As q ∈f −1 H,L (f H,L (p )), it follows from Lemma 3.5 that Q(p ) = Q(q ). This in turn implies that q is generic.
We may assume that all the edges incident to v are directed from v.
and let P and Q be the d × d-matrices whose i-th column is x i and y i , respectively. Note that since v is d-nondegenerate and p , q are generic, x 1 , . . . , x d and y 1 , . . . , y d are, respectively, linearly independent, and hence P and Q are both nonsingular. Let . We then have x v = y v since G has the edge vv 0 with ψ(vv 0 ) = id. Due to the existence of the edge e i we also have where we used x v = y v . Denoting by δ the d-dimensional vector whose i-th coordinate is equal to x i 2 − y i 2 , the above d equations can be summarized as which is equivalent to where I d denotes the d × d identity matrix. Note that each entry of P is contained in Q(p ), and each entry of Q is contained in Q(q ). Since Q(p ) = Q(q ), this implies that each entry of P Q −1 is contained in 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 Q(p ). Therefore, by regarding the left-hand side of (7) as a polynomial in x v , the polynomial must be identically zero. In particular, we get Thus, P Q −1 is orthogonal. In other words, there is some orthogonal matrix S such that P = SQ, and we get p Since Thus we obtain p = q.

Combinatorial part
The combinatorial part consists of the following two lemmas whose proof will be given in the next sections separately. Lemma 4.6. Let (G, ψ) be a Γ-labeled graph with rank k ≥ 1 periodicity Γ. Suppose that (G, ψ) is 2-connected, redundantly periodically rigid in R 2 , and has no (0, 2)-block. Then at least one of the following holds: redundantly periodically rigid, and has no (0, 2)-block. (ii) G has a vertex of degree three.
Suppose that (G, ψ) is 2-connected, redundantly periodically rigid in R 2 , and has no (0, 2)block. Then the minimum degree of G is at least three and the following hold for every vertex v of degree three.
• v is 2-nondegenerate. The proof of the sufficiency is done by induction on the lexicographic order of the list (V (G), E(G)). Suppose that |V (G)| = 3. By the 2-connectivity, G contains a triangle and the minimum degree in G is at least three. Let G be an inclusionwise minimal spanning subgraph that is 2-connected and redundantly periodically rigid. Then it consists of five edges, two parallel classes and one simple edge. Since G has a vertex of degree three, we can use Lemmas 4.5 and 4.4 to deduce that (G , ψ) (and hence (G, ψ)) is L-periodically globally rigid.
Assume that |V (G)| ≥ 4. Suppose that G has a vertex of degree three. By Lemma 4.7, (G v , ψ v ) is 2-connected, redundantly periodically rigid, and has no (0, 2)-block. Hence, by induction, (G v , ψ v , p) is L-periodically globally rigid. Therefore it follows from Lemma 4.7 and Lemma 4.5 that (G, ψ, p) is L-periodically globally rigid.
Thus we may assume that G has no vertex of degree three. Then by Lemma 4.6 there is an edge e such that (G − e, ψ) is 2-connected, redundantly periodically rigid, and has no (0, 2)-block. By induction, (G −e, ψ, p) (and hence (G, ψ, p)) is L-periodically globally rigid.
Proof of Theorem 4.2. By Lemma 3.2 we may assume that G is 2-connected. Then the necessity of the conditions again follows from Lemmas 3.1 and 3.7.
The proof of the sufficiency is similar to that of Theorem 4.1, and it is done by induction on the lexicographic order of the list (V (G), E(G)). The case when |V (G)| = 3 is exactly the same as that for Theorem 4.1. Hence we assume |V (G)| ≥ 4.
Suppose that G has a vertex of degree three. By Lemma 4.7, (G v , ψ v ) is 2-connected, redundantly periodically rigid, and has no (0, 2)-block. Also, by the definition of ( Hence, by induction, (G v , ψ v , p) is L-periodically globally rigid.
Therefore it follows from Lemma 4.7 and Lemma 4.5 that (G, ψ, p) is L-periodically globally rigid.
Thus we may assume that G has no vertex of degree three. Then by Lemma 4.6 there is an edge e such that (G − e, ψ) is 2-connected, redundantly periodically rigid, and has no (0, 2)-block. If the rank of Γ G−e is equal to two, then we can apply the induction hypothesis, meaning that (G − e, ψ, p) (and hence (G, ψ, p)) is L-periodically globally rigid. Therefore, we assume that the rank of Γ G−e is smaller than two.
Since Γ G has rank 2, the rank of Γ G−e is nonzero. Thus the rank of Γ G−e is one. By switching, we may suppose that every label in E(G − e) is in Γ G−e and consider the restriction L : Γ G−e → R 2 of L. By Theorem 4.1, (G − e, ψ, p) is L -periodically globally rigid. To show that (G, ψ, p) is L-periodically globally rigid, take any q : , ψ, p) is L -periodically globally rigid, Proposition 2.2 implies that q can be written as q = h • p for some L(Γ G−e )-invariant isometry h. Since the rank of Γ G−e is one and the dimension of the ambient space is two, h is either the identity or the reflection along the line whose direction is in the span of L(Γ G−e ).
Let Due to Theorem 2.4, we may work in a purely combinatorial world in order to prove Lemma 4.6 and Lemma 4.7.
Since all the graphs we treat in the following discussions are Γ-labeled graphs, we shall use the following convention. Throughout the section, we omit the labeling function ψ to denote a Γ-labeled graph (G, ψ). The underlying graph of each Γ-labeled graph is directed, but the direction is used only to refer to the group labeling. So the underlying graph is treated as an undirected graph if we are interested in its graph-theoretical properties, such as the connectivity, vertex degree, and so on.
Let G = (V, E) be a Γ-labeled graph. For disjoint sets X, Y ⊆ V , d G (X, Y ) denotes the number of edges between X and Y , and let d G (X) : Given two Γ-labeled graphs G 1 = (V 1 , E 1 ) and . Note that G 1 ∪ G 2 and G 1 ∩ G 2 are Γ-labeled graphs whose labeling are inherited from those of G 1 and G 2 . Note also that, when taking the union or the intersection, the labels are taken into account and two edges are recognized as the same edge if and only if they are identical. Hence G 1 ∪ G 2 may contain parallel edges even if G 1 and G 2 are simple.
Several terms defined for edge sets will be used for graphs G by implicitly referring to E(G). If there is no confusion, terms for graphs will be conversely used for edge sets E by referring to the graph (V (E), E).
We use the following terminology from matroid theory. Let M = (E, I) be a matroid, where E is the ground set and I is the family of independent sets. A circuit in M is a minimal dependent subset of E, i.e. a dependent set whose proper subsets are all independent. We define a relation on E by saying that e, f ∈ E are related if e = f or if there is a circuit C in M with e, f ∈ C. It is well-known that this is an equivalence relation. The equivalence classes are called the components of M. If M has at least two elements and only one component then M is said to be connected. If M has components

Count matroids
Let V be a finite set and Γ be a group isomorphic to Z k . We say that an edge set E is independent if |F | ≤ 2|V (F )| − 3 for every nonempty balanced F ⊆ E and |F | ≤ 2|V (F )| − 2 for every nonempty F ⊆ E. Proposition 2.3 and Theorem 2.4 imply that the family of all independent edge sets forms the family of independent sets of a matroid on K(V, Γ), which is denoted by R 2 (V, Γ) or simply by R 2 (V ). (This can also be checked in a purely combinatorial fashion, see, e.g., [18].) Let r 2 be the rank function and cl 2 be the closure operator of R 2 (V ).
We say that G is a circuit if E(G) is a circuit in R 2 (V (G)). An example of a balanced circuit is the simple complete graph on four vertices with identity label on each edge. An example of an unbalanced circuit is a Γ-labeled graph with two vertices and three parallel edges such that no two parallel edges form a balanced set.
We say that G is M -connected if the restriction of R 2 (V (G)) to E(G) is connected. For a Γ-labeled graph G on V , denote r 2 (G) = r 2 (E(G)). By Theorem 2.4, G is periodically rigid in R 2 if and only if r 2 (G) = 2|V (G)| − 2.
Note that if G is balanced, then the restriction of R 2 (V ) to E(G) is isomorphic to the 2-dimensional generic rigidity matroid. Hence we say that G is rigid if G is balanced and r 2 (G) = 2|V (G)| − 3.
We can use known properties of the 2-dimensional generic rigidity matroids for balanced sets. The following statements can be found in [10] (with a slightly different terminology).

Lemma 5.1. If a Γ-labeled graph G is balanced and M -connected, then G is rigid.
Lemma 5.2. Let G 1 and G 2 be two rigid graphs. Suppose that |V (G 1 ) ∩ V (G 2 )| ≥ 2 and The following simple property was first observed in [24]. Lemma 5.4. Let G 1 and G 2 be Γ-labeled graphs.

Lemma 5.3 ([24]). Let v be a vertex of degree two in
To see the first claim, suppose that G 1 is periodically rigid and G 2 is rigid. Since G 2 is rigid, G 2 is balanced, and we may suppose, by switching (see [ denotes the set of edges on V 2 whose labels are the identity. Since |V 1 ∩V 2 | ≥ 2, there are two distinct vertices u and v in V 1 ∩V 2 . Let F be the edge set of the complete bipartite subgraph of K 0 (V 2 ) whose one partite set is {u, v} and whose other partite set is V 2 \V 1 . By Lemma 5.3, we have r 2 (E 1 ∪F ) = r 2 (E 1 ) +2|V 2 \V 1 |.
Thus G 1 ∪ G 2 is periodically rigid. We can use the same argument to prove the second statement. In this case we have K 0 (V 2 ) ⊂ K(V 2 , Γ) ⊆ cl 2 (E 2 ) and we can again use Lemma 5.3 to deduce that r 2 (E 1 ∪ E 2 ) = 2|V 1 ∪ V 2 | − 2, which completes the proof. Lemma 5.1 implies that a balanced circuit is always rigid. By Theorem 2.4, the following periodic version can be easily proved.
Lemma 5.5. If G is an unbalanced circuit, then it is periodically rigid.

M -connectivity and ear decomposition
Jackson and Jordán [10] used ear decompositions of connected rigidity matroids as a key tool in their proof of Theorem 1.2. Let C 1 , C 2 , . . . , C t be a non-empty sequence of circuits of the matroid M. For 1 ≤ j ≤ t, we denote D j = C 1 ∪ C 2 ∪ · · · ∪ C j and define the lobeC j of C j by C j = C j \ D j−1 . We say that C 1 , C 2 , . . . , C t is a partial ear decomposition of M if for every 2 ≤ i ≤ t the following properties hold: An ear decomposition of M is a partial ear decomposition with D t = E. We need the following facts about ear decompositions.
For an M -connected graph G = (V, E), an ear decomposition of E means an ear decomposition of the restriction of R 2 (V ) to E.
When the matroid M is specialized to the generic 2-dimensional rigidity matroid, several properties of ear-decompositions were proved in [10,17]. We will use the following (with some of the terminology adjusted to fit our context).
Then the following hold.
(a) Either Y = ∅ and |C t | = 1, or Y = ∅ and every edge e ∈C t is incident to Y .
In [17,Lemma 3.4.3], Jordán pointed out that Lemma 5.7 immediately implies the existence of a degree three vertex in a minimally M -connected graph. His proof actually gives a slightly stronger statement as follows.
Lemma 5.8. Let G be balanced and M -connected, and let C 1 , . . . , C t be an ear decompo- Then Y contains a vertex which has degree three in G. Moreover, if |Y | ≥ 2, then Y contains at least two vertices of degree three in G.
Proof. It is a well known fact that a circuit in the ordinary 2-dimensional generic rigidity matroid has at least four vertices of degree three. Hence the claim follows if t = 1. Suppose that t ≥ 2. Lemma 5.7(a) says that, if |C t | > 1, then Y = ∅ and every edge in C t is incident to Y . Moreover, Lemma 5.7(b) says that Hence if |Y | = 1 then the vertex in Y has degree three. If |Y | ≥ 2, then the edge set of G induced by Y is a proper subset of C t , and hence i G (Y ) ≤ 2|Y | − 3. Combining this with (9), we have that the total degree of the vertices Thus Y contains at least two vertices that have degree three in G.
It is well known that every circuit in the generic rigidity matroid (i.e., every balanced circuit) is 2-connected and 3-edge-connected. The next lemma shows that even if G is unbalanced or M -connected, we can guarantee the same connectivity. Proof. The statement is known if G is balanced (see, e.g., [10]). Hence we assume that G is unbalanced.

Redundant rigidity and M-connectivity
Based on the ear decomposition, in this section we shall reveal a connection between redundant rigidity and M -connectivity. In particular, we show that, under a certain connectivity condition, these properties are equivalent. This is an analogue of the fact that M -connectivity is equivalent to redundant rigidity under (near) 3-connectivity in the plane [10, Theorem 3.2].
In Lemma 5.11 we first give the unbalanced version of Lemma 5.1. For the proof we need the following technical lemma.
Lemma 5.10. Let E 1 and E 2 be balanced edge sets in a Γ-labeled graph G. Suppose that Proof. This is a special case of [18,Lemma 2.4]. Since the proof is easy, we include it for completeness. Suppose that (V (E 1 ) ∩ V (E 2 ), E 1 ∩ E 2 ) is connected. Then one can take a spanning tree T in (V (E 1 ∪ E 2 ), E 1 ∪ E 2 ) such that T ∩ E i is a spanning tree in (V (E i ), E i ) for i = 1, 2. By switching, we may assume that ψ(e) = id for every e ∈ T (see [18,Proposition 2.3]). Since E i is balanced, we get ψ(e) = id for every e ∈ E 1 ∪ E 2 , contradicting that E 1 ∪ E 2 is unbalanced.
Lemma 5.11. If G is unbalanced and M-connected, then G is periodically rigid.
Proof. Since G is M-connected, E(G) has an ear decomposition C 1 , . . . , C t . It follows from |D i ∩ C i+1 | ≥ 1 that there are at least two vertices in V (D i ) ∩ V (C i+1 ). We will prove by induction on i that D i is rigid if D i is balanced, and D i is periodically rigid otherwise.
This follows from Lemma 5.1 and Lemma 5.5 if i = 1. Hence, we assume i ≥ 2.
If D i−1 is unbalanced, then by induction D i−1 is periodically rigid. By Lemma 5.4, D i−1 ∪ C i is periodically rigid, implying (10).
If D i−1 is balanced, then by induction it is rigid. If C i is unbalanced, then by Lemma 5.4(i) and Lemma 5.5, D i−1 ∪ C i is periodically rigid. If C i is balanced and D i−1 ∪ C i is balanced, then by Lemma 5.1 and Lemma 5.
To see this, observe first that s ≥ 0 holds since C i ∩ D i−1 is balanced and independent. Also (11) trivially holds if s ≥ 1. Hence suppose s = 0. Then , implying (11) even for s = 0.
Since D i−1 is rigid and C i is a balanced circuit, Lemma 5.6(c) implies where the last inequality follows from (11).
Since E(G) = D t and G is unbalanced, we conclude that G is periodically rigid by (10).
The following is an analogue of [10,Theorem 3.2]. Using the lemmas collected so far, its proof is now a direct adaptation of that of [10, Theorem 3.2].

Theorem 5.12. Let G be unbalanced and suppose that it has no (0, 2)-block. Then G is 2-connected and redundantly periodically rigid if and only if G is M-connected.
Proof. Suppose first that G is M-connected. By Lemma 5.9 G is 2-connected. Also G is periodically rigid by Lemma 5.11. Since every edge is included in a circuit, we have r 2 (G − e) = r 2 (G) for every edge e ∈ E(G), meaning that G − e is periodically rigid. Thus G is redundantly periodically rigid.
Conversely, suppose that G is 2-connected and redundantly periodically rigid. Suppose further for a contradiction that G is not M -connected. Then E(G) can be decomposed into M-connected components E 1 , . . . , E t with t ≥ 2. By Lemmas 5.11 and 5.1, we have Since G is redundantly periodically rigid, every M -connected component contains a circuit. Hence, for each balanced E i , we have |V (E i )| ≥ 4 (since the smallest balanced circuit is K 4 ). This means that, if a i ≤ 2 for a balanced E i , then G[E i ] is a (0, 2)-block in G. Hence we have a i ≥ 3 for each balanced E i . On the other hand, for each unbalanced E i , we have a i ≥ 2 since G is 2-connected. Therefore the last two terms in (12) are nonnegative. Observe finally that k i=1 (2|V (E i )| − a i ) ≥ 2|V (G)|, since each vertex of G contained in a unique E i is counted twice while the rest of the vertices are counted at least twice. Therefore (12) is at least 2|V (G)|, which is a contradiction.

Proof of Lemma 4.7
We first solve the following main case of the proof of Lemma 4.7.
Lemma 5.13. Let G be a 2-connected and redundantly periodically rigid graph that has no (0, 2)-block. Then for any vertex v with degree 3 in G, G v is 2-connected and redundantly periodically rigid and has no (0, 2)-block.
Proof. The 2-connectivity of G v follows from the 2-connectivity of G. Let E v be the set of edges incident with v in G, and assume that those edges are directed from v. Let F be the set of edges e 1 · e 2 over all pairs of nonparallel edges e 1 , e 2 in E v . Recall that G v is obtained from G by removing v and adding F , where the label of e 1 · e 2 is defined to be ψ(e 1 ) −1 ψ(e 2 ). Thus, we have that We first show that G v has no (0, 2)-block. Suppose to the contrary that G v contains a (0, 2)-block H. Take a maximal such H. We split the proof into four cases depending on how H intersects with F as follows. To see that G v is redundantly periodically rigid, we first check that G v is periodically rigid. Indeed, since G is redundantly periodically rigid, G − e is periodically rigid for an edge e incident to v. Then v has degree two in G − e , and by Lemma 5 Now take any edge e ∈ E(G v ). Our goal is to show that G v − e is periodically rigid. We take an edge f from E v . Suppose first that there is a circuit C in G + F with f ∈ C and e / ∈ C. Since C ⊆ E(G) + F − e, we have cl 2 (E(G) + F − e) = cl 2 (E(G) + F − e − f ). On the other hand, since G is redundantly periodically rigid, cl 2 (E(G) + F ) = cl 2 (E(G) + F − e). Thus, cl 2 (E(G) + F ) = cl 2 (E(G) + F − e − f ), and we get So we may suppose that every circuit in G + F that contains f also contains e. Then by (13), e ∈ F . Suppose that G v − e is not periodically rigid.
Proof. We split the proof into two cases depending on the size of N G (v). We first show that D 1 = D 2 . To see this, suppose to the contrary that D 1 = D 2 . Then we have e ∈ cl 2 (D 1 ). Indeed, if D 1 is unbalanced, then cl 2 (D 1 ) contains every edge on V (D 1 ) by Lemma 5.11, implying e ∈ cl 2 (D 1 ). On the other hand, if D 1 is balanced, then by Lemma 5.1 cl 2 (D 1 ) contains every edge e on V (D 1 ) for which D 1 +e is balanced, and D 1 + e is indeed balanced by the balancedness of {e, e 1 , e 2 } and Lemma 5.10. Hence e ∈ cl 2 (D 1 ). This implies that cl 2 (E(G v )) = cl 2 (E(G v − e)) and hence G v is not periodically rigid as G v − e is not. This contradiction implies D 1 = D 2 .
Suppose, contrary to the claim, that D 1 and D 2 are all the M -connected components in G v − e. By Lemma 5.1 and Lemma 5.11, The first equation implies that v 3 is a cut vertex in G v − e. The second inequality implies that at least D 1 or D 2 is balanced. Without loss of generality we assume that D 2 is balanced. If |V (D 2 )| ≥ 3, then D 2 is a (0, 2)-block in G v since {v 2 , v 3 } forms a cut, which contradicts that G v has no (0, 2)-block. On the other hand, if |V (D 2 )| = 2, then D 2 = {e 2 }, and the edges e and e 2 induce a (0, 2)-block in G v , which is again a contradiction. This contradiction completes the proof for Case 1.
Case 2: |N G (v)| = 2. Let e 1 be the edge in F different from e. Let D 1 be the M -connected component in G v − e that contains e 1 . If D 1 is unbalanced, then D 1 is periodically rigid by Lemma 5.11, and hence e ∈ cl 2 (D 1 ). This however implies that G v is not periodically rigid, which is a contradiction. Thus D 1 is balanced.
If Let D be an M -component in G v − e satisfying the condition of Claim 5.14. Let g ∈ D be arbitrary. By Theorem 5.12 G is M -connected, and hence G + F is Mconnected. Hence G + F contains a circuit C 1 with e, g ∈ C 1 . Recall that E v ∪ F is a circuit by (13). Since e ∈ F and g / ∈ E v + F , by the circuit elimination, we have a circuit To see this recall that every circuit in G + F containing f also contains e. As e / ∈ C 2 , we get f / ∈ C 2 . Moreover, since v has degree three in (14).
By (14), C 2 is a circuit in G v −e. However, according to the construction of C 2 , we have C 2 ∩ (E v + F ) = ∅, which means that C 2 intersects F − e by (14). Since D ∩ (F − e) = ∅ and g ∈ C 2 ∩ D , C 2 intersects more than one M -connected component in G v − e, which is a contradiction.
Proof of Lemma 4.7. By Lemma 5.3, the minimum degree of G is at least three. Let v be a vertex of degree three, and suppose without loss of generality that all the edges (ψ(e)) : e = vv i ∈ E(G)} is affinely independent for any generic p : V (G) → R 2 . If v has three distinct neighbors, then this is clearly true. If v has exactly two neighbors, any parallel edges are not identical, and hence the parallel edges form an unbalanced cycle. Thus the statement is again true. Hence by 2-connectivity of G we conclude that v is 2-nondegenerate.
As shown in the proof of Lemma 5.13, G − v is periodically rigid. Moreover, the statement for G v has also already been proved in Lemma 5.13.

Proof of Lemma 4.6
The proof of Lemma 4.6 is rather involved, and it consists of three major parts. In Lemma 5.15 we first show the existence of a degree three vertex in a minimally M -connected graph. By this result, we may focus on the case when G is not minimally Mconnected, i.e., E = {e ∈ E(G) : G − e is M -connected} is nonempty. By Theorem 5.12, G − e is 2-connected and redundantly periodically rigid for every e ∈ E for which G − e has no (0, 2)-block. Thus, what remains to show is that there is an edge e ∈ E such that G − e has no (0, 2)-block. We prove this by contradiction. A key lemma for this will be Lemma 5.21, which shows that, for any (0, 2)-block H e in G − e for e ∈ E , H e contains an edge f in E . The proof is completed by first taking e ∈ E such that |V (H e )| is as small as possible and then showing that G − f has no (0, 2)-block for f ∈ E(H e ) ∩ E .
We start with the following lemma, which is an unbalanced version of Lemma 5.8. The proof strategy is similar, but its proof is technically more involved.
Lemma 5.15. If G is unbalanced and minimally M-connected, then G has a vertex of degree three.
Proof. Let C 1 , . . . , C t be an ear decomposition of E(G). The statement trivially follows from the edge count of G if t = 1. Hence we assume t ≥ 2. Let D i = i j=1 C j . We will use the notation V = V (G) and V = V (D t−1 ). Our goal is to prove that the average degree of V \ V (denoted by d avg (V \ V )) is less than 4, implying that V \ V must contain a vertex of degree three. The proof is split into two cases depending on whether D t−1 is balanced or not. Case 1: Suppose that D t−1 is unbalanced. Then it follows from Lemma 5.11 that D t−1 and D t−1 ∪ C t are periodically rigid. Using Lemma 5.6(c) we have Since D t−1 is periodically rigid, K(V , Γ) ⊆ cl 2 (D t−1 ). Therefore, since G is minimally M -connected, no edge in C t \ D t−1 is induced by V . This implies V \ V = ∅, and by Lemma 5.9 we further have Combining (15) and (16), we get Case 2: Suppose that D t−1 is balanced. We may assume that C t is balanced (otherwise, by Lemma 5.6, we can start the ear decomposition with the unbalanced circuit C t , and hence we are back in Case 1). We prove that V \ V = ∅.
Since D t−1 is balanced, we may assume ψ(e) = id for every e ∈ D t−1 .
] − X must be disconnected. This in turn implies |X| ≥ 3 by Lemma 5.9, and hence On the other hand, by Lemma 5.11 D t−1 ∪ C t is periodically rigid, and it follows from Lemma 5.1 that D t−1 is rigid since it is balanced and M-connected. Thus, by Lemma 5.6(c) we have Combining (17) and (18), we get V \ V = ∅. By Lemma 5.9 we again have (16). Let F be the set of edges in C t \ D t−1 induced by V . If F = ∅, then we have where the third equation follows from (18) and the last inequality follows from (16) and F = 0. Hence we suppose F = ∅ and every edge in Hence Lemma 5.9 implies |V \ V | ≥ 2. Since C t is balanced, the number of edges of C t induced by V \ V is at most 2|V \ V | − 3, and we obtain This completes the proof.
The next target is to prove Lemma 5.19 and Lemma 5.20, which will be needed for the proof of Lemma 5.21. Lemmas 5.19 and 5.20 may be considered as periodic versions of the 2-sum lemma and the cleaving lemma given in [10]. For the proofs we first need three technical lemmas.
Lemma 5.16. Let F 1 and F 2 be edge sets in a Γ-labeled graph G. Suppose that F 1 is balanced, |V (F 1 ) ∩ V (F 2 )| = 2, and F 1 ∩ F 2 has exactly one edge, denoted by f .
Proof. (i) Since F 1 is balanced, we may assume that the label of each edge in F 1 is identity. If F 2 is balanced, then F 1 ∪F 2 is balanced by Lemma 5.10, and hence F 1 ∪F 2 −f is balanced as well. If F 2 is unbalanced, then it has an unbalanced cycle C.
then the sum of the labels in C − f is non-identity as f has the identity label. Concatenating C − f with a path in F 1 − f , we get an unbalanced cycle in F 1 ∪ F 2 − f .
Suppose that F is not a circuit. Then there exists a proper edge subset F of F with |F | ≥ 2|V (F )| − 2 + δ , where δ = 0 if F is balanced, and otherwise δ = 1. Let F i = F ∩ F i . Then F is the disjoint union of F 1 and F 2 . Also since each F i is a circuit, each F i is nonempty.
Since F 1 is balanced and independent (as Moreover, since F is a proper edge subset of F , the equations do not hold simultaneously. (If both equations hold, then we would have Therefore, every proper edge subset F of F satisfies |F | ≤ 2|V (F )| − 3 + δ , and F is a circuit.
Since our matroid becomes an ordinary generic rigidity matroid if the graph is balanced, by adapting notations, we have the following from Lemma 4.2 in [2] (see also Lemma 2.18 in [10]). We prove the counterpart for unbalanced circuits. Since H is balanced, there is exactly one edge f 0 in K({a, b}, Γ) whose addition to H keeps the balancedness. By Lemma 5.16 If H + f 0 is not a circuit, then H has a proper edge subset F with |F | = 2|V (F )| − 3 and {a, b} ⊆ V (F ). Then |F ∪E(G 1 )| = 2|V (F )| −3 +2|V (G 1 )| −2 = 2|V (F ) ∪V (G 1 )| −1, which contradicts that G is a circuit. Therefore H + f 0 is a circuit.
If G 1 + f 0 is not a circuit, then G 1 has a proper edge subset F such that F + f 0 is a circuit. Since H +f 0 is a circuit, by Lemma 5.16(ii), E(H) ∪F would be a circuit properly contained in G, a contradiction. Therefore We now extend Lemma 5.18 to general M -connected graphs.
Proof. Let G 1 = G − I(H). By Lemma 5.19, H + f 0 is balanced and Take any edge e ∈ E(H −f ) and any edge e ∈ E(G 1 ). Since H +f 0 −f (resp., G 1 +f 0 ) is M -connected, it contains a circuit C (resp., C ) with e, f 0 ∈ C (resp., e , f 0 ∈ C ). By Lemma 5.16(ii), We are now ready to prove the following key lemma. Proof. Let u and v denote the end vertices of e. Since H e is balanced and G has no (0, 2)-block, we may suppose that the label of each edge in H e is the identity, and that e has a non-identity label if Let {a, b} = B(H e ) and let G = H e + f 0 be the cleavage graph for H e . At least one end vertex of e is contained in I(H e ), since otherwise H e would be a (0, 2)-block in G. Hence we may assume u ∈ I(H e ). As G is (balanced) M -connected by Lemma 5.6 and Lemma 5.19, we can take an ear decomposition C 1 , . . . , C t of E(G ) such that f 0 ∈ C 1 and u ∈ V (C 1 ).
We first solve the case when an end vertex of e is not in I(H e ). We first remark that t > 1. Otherwise C 1 contains at least four vertices of degree three in C 1 , one of which is also a degree three vertex in G since an end vertex of e is not in I(H e ). This contradicts that the minimum degree of G is at least four. Thus t > 1. By Lemma 5.8 and the minimum degree condition for G, we have |C t | = 1. Let f be the edge in C t . Then Then it is not (balanced) rigid, since every rigid graph is 2-connected. This in turn implies that H e − f + f 0 is not M -connected by Lemma 5.1, a contradiction. This completes the proof in the case where an end vertex of e is not in I(H e ).
The difficult case is when both end vertices of e are contained in I(H e ). We may assume that |C t | > 1, for otherwise the edge in C t has the desired property by Lemma 5.20.
Proof. If t = 1, then there are at least four vertices of degree three in G since E(G ) is a balanced circuit. Since the minimum degree of G is at least four, we have (i).
Suppose t > 1, and suppose also that (ii) does not hold. Then by Lemma 5 , w is distinct from them. Hence w has degree three even in G, contradicting the minimum degree condition of G.
Since v ∈ I(H e ), we can take an edge f in G incident to v. We claim that G − f has a circuit C * with e ∈ C * and f / ∈ C * . Indeed, since G − e is M -connected, G − e has a circuit C with f ∈ C, and G has a circuit C with e ∈ C and f ∈ C . By the circuit elimination, we get C * ⊆ C ∪ C − f with e ∈ C * . By (20) we have Proof. If C * ⊆ E(H e ) + e, then we would have r(C * ) = r(C * − e) + 1 by e ∈ C * and (19), which contradicts that C * is a circuit. Hence C * must contain at least one edge from E(G) − E(H e + e). Thus the 2-connectivity of C * implies {a, b} ⊆ V (C * ). Suppose that C * is balanced. Then every path between u and v in C * − e passes through a, since the concatenation of e and a simple path between u and v avoiding a is unbalanced by (19). Hence a is a cut vertex in C * − e, contradicting the rigidity of G[C * ]. (Note that u and v are distinct from a by u, v ∈ I(H e ).) By Claim 5.23 and Lemma 5.5, C * is periodically rigid with {a, b} ⊆ V (C * ), and f 0 ∈ cl 2 (C * ). Hence C * + f 0 contains a circuit C * 0 with f 0 ∈ C * 0 . Note that C * 0 − f 0 ⊂ C * . For any e 1 , e 2 ∈ E(G − f ), we denote e 1 ∼ e 2 if G − f has a circuit that contains e 1 and e 2 . In order to show the M -connectivity of G − f , we show that e 1 ∼ e 2 for any e 1 , e 2 ∈ E(G − f ). Since ∼ is an equivalence relation, we just need to show e ∼ e for any e ∈ E(G − f ), and this follows by showing that G − f has a circuit intersecting e and C * for each e ∈ E(G − f ). For e ∈ E(G − f ) \ E(H e − f 0 ) this can be rapidly shown as follows. Since G − I(H e ) + f 0 is M -connected by Lemma 5.19, it contains a circuit C e with e , f 0 ∈ C e . If C e = C * 0 , then we have e ∈ C * by C * 0 − f 0 ⊂ C * . If C e = C * 0 , then by the circuit elimination C e ∪ C * 0 contains a circuit intersecting e and avoiding f 0 . This circuit has the desired property since C * 0 − f 0 ⊂ C * and f / ∈ C e ∪ C * 0 . In the following discussion, we prove that for each e ∈ E(H e − f ) there is a circuit intersecting e and C * . The proof consists of two cases, depending on whether t = 1 or t > 1. The case when t = 1 is easily solved by the following claim.
Also let k be the number of edges between a and b in G. Note that C 1 = E(H e ) + f 0 by t = 1. (Recall that C 1 is the initial circuit in the ear decomposition C 1 , . . . , C t .) By Claim 5.23 and e ∈ C * , {u, v, a, b} ∩ Y = ∅. Hence the edge set of G induced by Y is a proper subset of C 1 . Hence if |Y | ≥ 2. On the other hand, for X, we claim To see this, recall first that C * is an unbalanced circuit by Claim 5.23. Also, since V (C * ) = {a, b}, C * can contain at most two edges between a and b.
Let C * 1 be the set of edges in C * induced by V (H e ), and C * 2 be the set of edges in C * induced by V (G) − I(H e ). Then Finally we claim To see this, recall that E(H e ) +f 0 is a balanced circuit. Also X ∪Y induces E(H e ) +e and the edges on {a, b}. Thus, if f 0 ∈ E(G) then we have i G (X∪Y ) = (2|X∪Y | −2) +1 +(k−1), and otherwise we have Combining (22)(23)(24), we get Those imply that, if Y = ∅, then G has a vertex of degree three in Y . By the minimum degree condition, we conclude that Y = ∅.
Suppose that t = 1. Since C * is periodically rigid, e ∈ cl 2 (C * ) holds for any e ∈ E(H e − f ). Hence C * + e contains a circuit C e intersecting e and C * , which has the desired property.
Suppose that t > 1. By Claim 5.22(ii) and (21), every edge in C t − f is included in Otherwise, since f 0 ∈ C 1 from the definition of C 1 , by the circuit elimination there is a circuit C e with e ∈ C e ⊆ C 1 ∩ C * 0 − f 0 . This circuit is contained in G − f and intersects e and C * (by C * 0 − f 0 ⊂ C * ). This solves the base case. For the general case, let e ∈ C i − f 0 . If f 0 ∈ C i , one can apply exactly the same argument as in the base case. Otherwise C i is contained in G − f by f ∈C t . Moreover by the definition of ear decomposition C i contains an edge e in j<i C j . By induction e ∼ e, and we get e ∼ e ∼ e. This completes the proof.
The following lemma lists properties of (0, 2)-blocks which we will use frequently in the following. Most of them follow directly from the definition.   We are now ready to prove Lemma 4.6.
Proof of Lemma 4.6. We show that (i) holds if (ii) does not. Hence suppose that the minimum degree of G is at least four. For any edge e ∈ E(G), G − e is unbalanced since otherwise r 2 (G) = 2|V (G)| − 2 > 2|V (G)| − 3 = r 2 (G − e) would hold, contradicting the M -connectivity of G. Hence, by Theorem 5.12 it suffices to show that G has an edge e such that • G − e is M -connected, and • G − e has no (0, 2)-block. By Lemma 5.15, G is not minimally M -connected. In other words, E = {e ∈ E : G − e is M-connected} is not empty. We show that G − e has no (0, 2)-block for some e ∈ E .
Suppose that G − e has a (0, 2)-block for every e ∈ E . Let H e be an inclusionwise minimal (0, 2)-block in G − e for each e ∈ E , and take e ∈ E such that |V (H e )| is as small as possible. Since the minimum degree of G is at least four, Lemma 5.21 implies that H e contains an edge f ∈ E , i.e., G − f is M -connected. Lemma I(H e )). Then G 1 , G 2 and G 3 are balanced, and each G i has at least two vertices. Thus, r 2 (G −e −f ) ≤  I(H f )). Note that G − e − f = 4 i=1 G i . Also, by a ∈ V (H f ), a ∈ G 1 and a ∈ G 3 . As {a, b} (resp., {x, y}) is the boundary of H e − f (resp., We have two subcases. Case 3-1: Suppose that G 1 does not contain a path between a and b. If G 1 has no path between a and x, then b is a cut vertex in G 1 ∪ G 3 (= H f − e). Then b(= y) is still a cut vertex in H f − e + f xy , and H f + f xy cannot be 3-connected, a contradiction. Therefore G 1 has a path between a and x (and has no path between b and x). This in turn implies that x is a cut vertex in G 1 ∪ G 2 . Also a is a cut vertex in G 1 ∪ G 3 (= H f − e). Hence, for the 3-connectivity of H f + f xy at least one end vertex of e is not in I(H e ) (as e should be incident to a vertex in V (G 3 )). Thus, by (26), H e − f − f ab is 2-connected. However, as G 1 ∪ G 2 = H e − f , this contradicts the fact that x is a cut vertex in G 1 ∪ G 2 .  (25). This vertex also belongs to I(H f ), since H e has a path between the end vertices of f internally avoiding {x, y} by the 3-connectivity of H e + f ab and x, y / ∈ I(H e ). Thus we can take v * ∈ I(H e ) ∩ I(H f ). We next prove that H e ∩ H f contains a path between a and b. Since H e + f ab is 3-connected, H e contains a path from v * to a (resp., b) avoiding f and b (resp., a). By x, y / ∈ I(H e ), such a path goes through the interior of H f (and the last vertex a (resp., b) may be the boundary). Hence such a path is in H e ∩ H f , and the concatenation of those paths would be a desired path between a and b in H e ∩ H f . Therefore, by Claim 5.27, H e ∪ H f is balanced, and we may suppose that the label of each edge in H e ∪ H f is identity. . As H f + f xy is 3-connected, we should have e ∈ E(H f ). This however implies that H e ∪H f is a balanced graph whose boundary in G is {x, y}(= {a, b}), i.e., a (0, 2)-block in G. This contradicts that G has no (0, 2)-block, and the proof is complete.

Global rigidity of cylindrical/toroidal frameworks
Given ∈ R 2 , we consider the following equivalence relation of R 2 : for a, b ∈ R 2 , a ∼ 1 b if and only if a = b + z for some z ∈ Z.
A flat cylinder C = C is obtained by factoring out R 2 with the relation ∼ 1 . Similarly, given a pair of vectors 1 2 of R 2 , consider an equivalence relation of R 2 by: for a, b ∈ R 2 , a ∼ 2 b if and only if a = b + z 1 1 + z 2 2 for some pair z 1 , z 2 ∈ Z.
A flat torus T 1 , 2 = T is obtained by factoring out R 2 with the relation ∼ 2 .
Given a straight-line drawing of an undirected graph on C (resp. on T ), we regard it as a bar-joint framework. Note that such frameworks on C (resp. on T ) have a oneto-one correspondence with 1-periodic (resp. 2-periodic) frameworks in R 2 through ∼ 1 (resp. ∼ 2 ). Hence our combinatorial characterization of the global rigidity of periodic frameworks immediately implies a characterization of the global rigidity of cylindrical/toroidal frameworks.
The underlying combinatorics of a drawing on C (resp. on T ) is captured by using a Z-labeled graph (G, ψ) (resp. Z 2 -labeled graph), where each label determines the geodesic between two end vertices. A cylindrical framework (resp. a toroidal framework) is defined as a pair (G, ψ, p) of a Z-labeled graph (G, ψ) (resp. Z 2 -labeled graph) and p : V (G) → C (resp. p : V (G) → T ). Note that a subgraph H is balanced if and only if it is contractible on the surface. Hence Theorem 4.1 can be translated to Theorem 1.3 for cylindrical frameworks. For toroidal frameworks the statement becomes as follows.

Open problems
There are quite a few remaining questions. As mentioned in the introduction, an important challenging problem is to extend our results to more general settings of global rigidity of periodic frameworks, as was done for local rigidity in [20,21].
As mentioned above, Theorems 4.1 and 4.2 may be viewed as a characterization of generic globally rigid bar-joint frameworks on a flat cylinder or a flat torus, respectively. A natural open problem is to establish counterparts of Theorem 1.2 in other flat Riemannian manifolds. (Note that for frameworks on the cylinder and other surfaces with edge lengths measured by the standard Euclidean distance in R 3 , generic global rigidity has been investigated in [14][15][16].) A similar question would be about the global rigidity of frameworks on a flat cone. Since a flat cone with cone angle 2π/n is the quotient of R 2 by an n-fold rotation, the global rigidity of such frameworks can be understood by the global rigidity of frameworks in R 2 with n-fold rotational symmetry (under the given symmetry constraints). The corresponding local rigidity question has been studied in [18,22] and an extension of Laman's theorem is known. Since the space of trivial motions is only of dimension one in this case (only rotations are trivial), an unbalanced rigidity circuit G satisfies the count |E(G)| = 2|V (G)|. Therefore, we cannot guarantee the existence of a vertex of degree three in G. This constitutes the key obstacle in applying our current proof method to this problem.
We may also ask about the global rigidity of finite frameworks with other point group symmetries. However, for the same reason, it also remains open to characterize the symmetry-forced global rigidity of finite bar-joint frameworks in R 2 that are generic modulo reflection or dihedral symmetry. In fact, it is currently not even known whether symmetry-forced global rigidity is a generic property for any point group in dimension 2. Necessary redundant rigidity and connectivity conditions, however, may be obtained in a similar fashion as described in Section 3.