Rigid cylindrical frameworks with two coincident points

We develop a rigidity theory for frameworks in $\mathbb{R}^3$ which have two coincident points but are otherwise generic and only infinitesimal motions which are tangential to a family of cylinders induced by the realisation are considered. We then apply our results to show that vertex splitting, under the additional assumption that the new edge is redundant, preserves the property of being generically globally rigid on families of concentric cylinders.


Introduction
A framework (G, p) in R d is the combination of a finite, simple graph G = (V, E) and a map p : V → R d . It is rigid if every edge-length preserving continuous motion of the vertices arises as a congruence of R d (see, for example, [12] for basic definitions and background). The theory of generic rigidity aims to characterise the graphs G for which (G, p) is rigid for all generic choices of p. This was accomplished by Laman [6] for d = 2, but is a long-standing open problem for d ≥ 3.
We are interested in frameworks in R 3 whose motions are restricted such that all vertices are realised on a fixed surface and only those continuous motions that keep (G, p) on the surface are considered. Generic rigidity in this context has been characterised for various surfaces [8,9]. In this paper we consider frameworks on concentric cylinders in which two of the vertices are mapped to the same point, but are otherwise generic. For such frameworks we give a combinatorial characterisation of rigidity.
Frameworks in R 2 with two coincident points were studied in [3] where the following deletion-contraction characterisation of rigidity was proved. A graph G is uv-rigid in R 2 if there exists a realisation p of G in R 2 such that p(u) = p(v), p| V −v is generic and the framework (G, p) is rigid. We use G − uv to denote the graph formed from G by deleting the edge uv if it exists and G/uv to denote the graph which arises from G by contracting the vertices u and v. Theorem 1.1. [3] Let G = (V, E) be a graph and let u, v ∈ V be distinct vertices. Then G is uv-rigid in R 2 if and only if G − uv and G/uv are both rigid in R 2 .
Our main result is an analogue of this theorem for frameworks on concentric cylinders. One motivation for studying frameworks on cylinders with coincident points is an ongoing work to understand global rigidity for frameworks on concentric cylinders [4,5]. In particular we utilise our main result to prove that vertex splitting, under the additional assumption that the new edge is redundant, preserves generic global rigidity for families of concentric cylinders.
In Section 2 we provide background on frameworks on concentric cylinders. In Section 3 we define a count matroid M uv (G) on a graph G with two distinguished vertices u and v. In Section 4 we derive an inductive construction for graphs whose edge set is independent in Date: March 8, 2022. M uv (G). We then use this construction to prove our characterisation of rigidity on families of concentric cylinders for frameworks in which u and v are coincident but are otherwise generic. In Section 5 we discuss global rigidity and apply our coincident point result to prove that the vertex splitting operation preserves generic global rigidity for frameworks on families of concentric cylinders when the new edge is redundant. Finally, in Section 6 we comment on extensions to other surfaces.
In this paper we consider simple graphs only as parallel edges correspond to the same distance constraint and thus one of them is always redundant.

Frameworks on concentric cylinders
Let G = (V, E) where V = {v 1 , . . . , v n }. We will consider realisations of G on a family of concentric cylinders Y = Y 1 ∪ Y 2 ∪ · · · ∪ Y k where Y i = {(x, y, z) ∈ R 3 : x 2 + y 2 = r i } and r = (r 1 , . . . , r k ) is a vector of positive real numbers. A framework (G, p) on Y is an ordered pair consisting of a graph G and a realisation p such that p(v i ) ∈ Y for all v i ∈ V .
Two frameworks (G, p) and (G, The framework (G, p) is rigid on Y if there exists an > 0 such that every framework (G, q) on Y which is equivalent to (G, p), and has p(v i ) − q(v i ) < for all 1 ≤ i ≤ n, is congruent to (G, p). Moreover (G, p) is minimally rigid if (G, p) is rigid but (G − e, p) is not for any e ∈ E. The framework (G, p) is generic on Y if td[Q(r, p) : Q(r)] = 2n.
It was shown in [8] that a generic framework (G, p) on any family of concentric cylinders is rigid if and only if it is infinitesimally rigid in the following sense. An infinitesimal flex s of (G, p) on Y is a map s : V → R 3 such that s(v i ) is tangential to Y at p(v i ) for all v i ∈ V and (p(v j ) − p(v i )) · (s(v j ) − s(v i )) = 0 for all v j v i ∈ E. The framework (G, p) is infinitesimally rigid on Y if every infinitesimal flex is an infinitesimal isometry of R 3 .
The rigidity matrix R Y (G, p) is the (|E| + |V |) × 3|V | matrix R Y (G, p) = R 3 (G, p) S(G, p) where: R 3 (G, p) has rows indexed by E and 3-tuples of columns indexed by V in which, for e = v i v j ∈ E, the submatrices in row e and columns v i and v j are p(v i ) − p(v j ) and p(v j ) − p(v i ), respectively, and all other entries are zero; S(G, p) has rows indexed by V and 3-tuples of columns indexed by V in which, for v i ∈ V , the submatrix in row v i and column v i isp(v i ) = (x i , y i , 0) when p(v i ) = (x i , y i , z i ). The rigidity matroid R Y (G) is the row matroid of R Y (G, p) for any generic p.
The following characterisation of generic rigidity on Y was proved in [8].
Theorem 2.1. Let (G, p) be a generic framework on a union of concentric cylinders Y. Then (G, p) is minimally rigid if and only if G is a complete graph on at most 3 vertices or G is (2, 2)-tight and simple.
2.1. Coincident realisations on concentric cylinders. Let G = (V, E) be a graph and We denote the uv-coincident cylinder rigidity matroid by R Y uv (G) (this is the row matroid of R Y (G, p) for any generic uv-coincident realisation (G, p)). Note that the matroid depends on G but not on the choice of generic uv-coincident realisation. That is, for any two generic uv-coincident realisations (G, p) and (G, p ) on Y, we get the same matroid. We also use r uv (G) to denote the rank of R Y uv (G). We say that G is uv-rigid if r uv (G) = 2|V |−2 and that G is minimally uv-rigid if G is uv-rigid and |E| = 2|V | − 2.
Note that the term uv-rigid and the notation r uv (G) refer to generic realisations on a fixed family of concentric cylinders Y, and hence appear to depend on Y. We will see, however, that this is not the case since our characterisation of R Y uv (G) depends only on the graph G.

A count matroid
In this section we define a count matroid M uv (G) on the edge set of a graph G with two distinguished vertices u and v. We will show that M uv (G) is equal to R Y uv (G) in Section 4.
We may omit the subscripts referring to G if the graph is clear from the context. Figure 1 for an example. We define the value of subsets of V and of uv-compatible families as follows. For a nonempty subset We will show that the edge sets of the uv-sparse subgraphs of G form the independent sets of a matroid that we will denote by M uv (G).
The next lemmas will enable us to 'uncross' tight sets and tight uv-compatible families in a sparse graph. The first result follows immediately from the definition of the i-and valfunctions.
Then there is a uv-compatible family H with cov(H) ⊆ cov(H ) and val(H ) < val(H).
Proof. Using Lemma 3.1(b) and the facts that  Proof. We have This implies that t X∩Y = 2 and equality holds throughout. Thus X ∪ Y and X ∩ Y are both tight and either |X ∩ Y | ≥ 4 or |X ∩ Y | = 1.
This gives where the first inequality follows from the fact that edges spanned by H or Y are spanned by H and if some edge is spanned by both H and Y then it is spanned by Y ∩ H i for some i. The equality holds because H and Y are tight, and the second inequality holds by our calculations above. The last inequality holds because G is uv-sparse. Hence equality must hold everywhere, which implies that H is tight and that Y ∩ H i is also tight for all j ≤ i ≤ k.
Proof. We may assume that i = k − 1 and j = k.
Using Lemma 3.1(b) twice and the fact that with equality only if |H k−1 | = |H k | = 3. Thus val(H ) ≤ val(H) + val(Y ) as claimed. Now suppose that G is uv-sparse. and H and Y are both tight. Then we have where the last inequality follows since |Y ∩ H k−1 | = |Y ∩ H k | = 1 and |Y ∩ H i | ≤ 1 for all 1 ≤ i ≤ k. Hence equality must hold throughout. Thus H is tight and |H k−1 | = |H k | = 3.
We define two uv-compatible families as follows: let It is easy to see that H ∪ and H ∩ are both uv-compatible. For convenience we rename the families as It follows from their construction that |A i | = 3 for all 3 ≤ i ≤ p and |B j | = 3 for all 2 ≤ j ≤ q and also at least one of |A 2 | = 3, |B 1 | = 3 holds. It can be seen easily that p + q = k + l. We also have We will show that equality occurs at both ends of the above inequality.
Hence equality must hold throughout the displayed inequality. In particular, H ∪ and H ∩ are both tight. Since cov(H 1 ) ∪ cov(H 2 ) ⊆ cov(H ∪ ), the maximality of the covers implies that cov(H 1 ) = cov(H 2 ) which in turn gives H 1 = H 2 .
We have now shown that H 1 = H max is unique and that properties (a) and . This would contradict the maximality of H max and hence {u, v} ⊆ Y . If |Y ∩ X i | ≥ 2 or |Y ∩ X i | = 1 and |X i | ≥ 4 then Lemma 3.5 would imply that there exists a uv-compatible family H with cov(H max ) ∪ cov(Y ) ⊆ cov(H ). Hence |Y ∩ X i | ≤ 1 and |X i | = 3. This tells us that |Y ∩ X j | ≤ 1 for all j and hence cov( 3.1. The matroid and its rank function. We first remind the reader of the simple (2, 2)sparse matroid and its rank function. Given a graph G = (V, E), a set F ⊆ E is independent if and only if it is simple and induces a (2, 2)-sparse subgraph Now we define the count matroid M uv (G). Let G = (V, E) be a graph and u, v ∈ V be distinct vertices of G. We will prove that the family of sets The value of the system K is given by val(K) = val(H) + k i=1 val(H i ). We will show that the rank of an arbitrary subset F ⊆ E in M uv (G) is given by Let G = (V, E) be the graph shown in Figure 1. It is not difficult to see that G is (2,2)sparse and simple, and hence E is independent in the simple (2,2)-sparse matroid. We will show that E is not independent in M uv (G). Consider the following sets: K is either a thin cover or a thin uv-cover of E }.
We shall prove that there is a thin (uv-)cover K of E with |F | = val(K), from which the theorem will follow.
Let J = (V, F ) denote the subgraph defined by the edge set F . First suppose that there is no tight uv-compatible family in J and consider the following cover of F : where H 1 , H 2 , . . . , H t are the maximal tight sets with size at least four in J for some t ≤ k and H t+1 , . . . , H k are the pairs of end vertices of edges in follows. We claim that K 1 is a cover of E . To see this consider an edge ab = e ∈ E − F . Since F is a maximal subset of E in I we have F + e ∈ I. By our assumption there is no tight uv-compatible family in J, and hence there must be a tight set X in J with a, b ∈ X. Hence X ⊆ H i for some 1 ≤ i ≤ k which implies that K 1 covers e.
Next suppose that there is a tight uv-compatible family in J and consider the following uv-cover of F : Then K 2 is a uv-cover of F . By Lemmas 3.4 and 3.7, the uv-cover K 2 is thin, and hence We claim that K 2 is a uv-cover of E . As above, let ab = e ∈ E − F be an edge. By the maximality of F we have F + e ∈ I. Thus either there is a tight set In the latter case Lemma 3.7 implies that cov(H ) ⊆ cov(H max ) and hence e is covered by K 2 . In the former case, when a, b ∈ X for some tight set X in J, we have |X| ≥ 5 since if |X| = 2, 3 or 4 then X induces a complete graph in J and e = ab would be an edge of F .
and hence X ⊆ H i for some 1 ≤ i ≤ k, since every edge of J induces a tight set and every tight set is contained in a maximal tight set. Thus e is covered by K 2 , as claimed.

Characterisation of the uv-coincident cylinder rigidity matroid
Our aim is to show that the uv-coincident cylinder rigidity matroid R Y uv (G) of a graph G = (V, E) is equal to the count matroid M uv (G). To simplify terminology we will say We first show that independence in R Y uv implies independence in M uv . Let G/uv denote the graph obtained from G by contracting the vertex pair u, v into a new vertex z uv (and deleting the resulting loops and parallel copies of edges). Given a uv-coincident realisation (G, p) of G on Y we obtain a realisation (G/uv, p uv ) of G/uv on Y by putting p uv (z uv ) = p(u) = p(v) and p uv (x) = p(x) for all x ∈ V − {u, v}. Furthermore, each vector in the kernel of R Y (G/uv, p uv ) determines a vector in the kernel of R Y (G, p) in a natural way. It follows that We can use this fact to prove that independence in R Y uv implies independence in M uv . Lemma 4.1. Let G = (V, E) be a graph and let u, v ∈ V be distinct vertices. If G is independent in R Y uv then G is independent in M uv . Proof. Let (G, p) be an independent uv-coincident realisation of G. Independence implies that i(X) ≤ val(X) holds for all X ⊆ V by Theorem 2.1. Since p(u) = p(v), uv / ∈ E follows.
Let H = {X 1 , . . . , X k } be a uv-compatible family and consider the subgraph . By contracting the vertex pair u, v in F we obtain the graph F/uv, in which H uv = {X 1 /uv, . . . , X k /uv} is a cover where X i /uv denotes the set that we get from X i by identifying u and v. Thus we get r( (3.2). This bound and (4. 2)). Since (G, p) is an independent uv-coincident realisation of G, we have Thus G is independent in M uv , as claimed.
We next define operations on uv-sparse graphs and use them to show that independence in M uv implies independence in R Y uv . The (two-dimensional versions of) the well-known Henneberg operations are as follows. Let G = (V, E) be a graph. The 0-extension operation (on a pair of distinct vertices a, b ∈ V ) adds a new vertex z and two edges za, zb to G. The 1-extension operation (on edge ab ∈ E and vertex c ∈ V − {a, b}) deletes the edge ab, adds a new vertex z and edges za, zb, zc.
We shall need the following specialized versions. Let u, v ∈ V be two distinct vertices. The 0-uv-extension operation is a 0-extension on a pair a, b with {a, b} = {u, v}. The 1-uvextension operation is a 1-extension on some edge ab and vertex c for which {u, v} is not a subset of {a, b, c}. The inverse operations are called 0-uv-reduction and 1-uv-reduction, respectively.
We will also need two further moves. The vertex-to-K 4 move deletes a vertex w and substitutes in a copy of K 4 with V (K 4 ) ∩ V (G) = {w} and with an arbitrary replacement of edges xw by edges xy with y ∈ V (K 4 ). The inverse operation is known as a K 4 -contraction.
A vertex-to-4-cycle move takes a vertex w with neighbours v 1 , v 2 , . . . , v k for any k ≥ 2, splits w into two new vertices w, w with w / ∈ V (G), adds edges wv 1 , w v 1 , wv 2 , w v 2 and then arbitrarily replaces edges xw with edges xy where x ∈ {v 3 , . . . , v k } and y ∈ {w, w }. The inverse move is known as a 4-cycle-contraction. The only difference in the specialised versions of these moves are that we require |V (K 4 ) ∩ {u, v}| ≤ 1 in a uv-K 4 -contraction and similarly |V (C 4 ) ∩ {u, v}| ≤ 1 in a uv-4-cycle-contraction.
We first consider the 0-extension and 1-extension operations. It was shown in [8] that these operations preserve independence in R Y . The same arguments can be used to verify analogous results for R Y uv .
Lemma 4.2. Let G = (V, E) be independent in R Y uv and suppose that G is obtained from G by a 0-uv-extension or a 1-uv-extension Then G is independent in R Y uv .
In the case of 0-extensions we will also need the following result. Proof. The 0-extension adds 3 rows and 3 columns to R Y (G, p), the 3 columns being 0 everywhere except the 3 new rows. The genericness of p and the fact that uv / ∈ E implies the new 3 × 3 block is invertible. Hence R Y (G , p ) has linearly independent rows so G is independent in R Y uv . We next consider the vertex-to-4-cycle operation. It was shown in [9] that this operation preserves independence in R Y . A similar argument would yield the analogous result for R Y uv but we will need a stronger result that a vertex-to-4-cycle move which creates two coincident vertices preserves independence in R Y .
andp(w) are linearly independent wherep(w) is the projection of p(w) onto the plane z = 0. Let G be obtained by performing a vertex-to-4-cycle operation at w in G such that v 1 and v 2 are both adjacent to w and w in G . Put p (a) = p(a) for all a ∈ V − w and put p (w) = p (w ) = p(w). Then R Y (G , p ) has linearly independent rows.
Proof. We will construct R Y (G , p ) from R Y (G, p) by a series of simple matrix operations that preserve the independence of the rows.
We first add three zero columns corresponding to w . We then add three rows corresponding to the edges w v 1 , w v 2 and the vertex w . Adding these rows increases the rank by 3 since p(w) − p(v 1 ), p(w) − p(v 2 ) andp(w) are linearly independent so the 3 × 3 matrix formed by the entries in the columns corresponding to w and the rows corresponding to w v 1 , w v 2 , w is non-singular and the rest of the entries in these columns are zero. The matrix M we obtain by this modification has the following form: To obtain R Y (G , p ) from M we need to modify some of the rows in M corresponding to edges (wv i ) into the form of rows corresponding to edges (w v i ), i.e. we need to move the entries in the columns of w to the columns of w and replace them with zeros. We will do this one by one.
Since (p(w)−p(v 1 )), (p(w)−p(v 2 )) andp(w) are linearly independent, for every 3 ≤ i ≤ k there exist unique values α, β, γ such that α(p(w) − p(v 1 )) + β(p(w) − p(v 2 )) + γp(w) = (p(w) − p(v i )). Now subtract the row of (wv 1 ) multiplied by α, the row of (wv 2 ) multiplied by β and the row of w multiplied by γ from the row of (wv i ) in M . Then add the row of (w v 1 ) multiplied by α, the row of (w v 2 ) multiplied by β and the row of w multiplied by γ to the same row (and change its label from (wv i ) to (w v i )) for every neighbour v i of w in G to obtain R Y (G , p ). These operations also preserve independence, thus we conclude that the rows of R Y (G , p ) are independent.
Corollary 4.5. Let G be independent in R Y uv and suppose that G is obtained from G by a vertex-to-4-cycle operation. Then G is independent in R Y uv .
Proof. We choose a generic uv-coincident realisation (G, p). Then (G, p) satisfies the hypotheses of Lemma 4.4. Hence G has a uv-coincident realisation (G , p ) such that R Y (G , p ) has linearly independent rows. It follows that every generic uv-coincident realisation is independent.
We next consider a generalisation of the vertex-to-K 4 operation. It was shown in [8] that this operation preserves independence in R Y . We will need an analogous result for uv-coincident realisations. Proof. (a) Let |V | = n, |V (H)| = r and consider R Y (G, p) where (G, p) is a generic uv-coincident framework on Y and p = (p(v 1 ), p(v 2 ), . . . , p(v n )). By reordering rows and columns if necessary we can write R Y (G, p) in the form where M 2 (p) is a square matrix with 3(n − r) rows. Suppose, for a contradiction, that G is not uv-rigid. Then there exists a vector m ∈ ker R Y (G, p) which is not an infinitesimal isometry of Y. Since (H, p| H ) is uv-rigid we may suppose that m = (0, . . . , 0, m r+1 , . . . , m n ). Consider the realisation (G, p ) where p = (p(v r ), p(v r ), . . . , p(v r ), p(v r+1 ), . . . , p(v n )) and define the realisation (G/H, p * ) by setting p * = (p(v r ), p(v r+1 ), . . . , p(v n )). Since p * is generic, (G/H, p * ) is infinitesimally rigid on Y by assumption. Now, M 2 (p) is square with the nonzero vector (m r+1 , . . . , m n ) ∈ ker M 2 (p). Hence rank M 2 (p) < 3(n − r). Since p is generic, we also have rank M 2 (p ) < 3(n − r) and hence there exists a nonzero vector m ∈ ker M 2 (p ). Therefore we have contradicting the infinitesimal rigidity of (G/H, p * ). (b) A similar proof holds. We choose a generic uv-coincident framework (G, p), a vector m ∈ ker R Y (G, p) which is not an infinitesimal isometry of R 3 , and uv-coincident realisations (G, p ) and (G/H, p * ) as above. We then use the facts that H is rigid on Y and G/H is uv-rigid on Y to obtain a contradiction.
We next consider the 0-uv-reduction, 1-uv-reduction, uv-K 4 -contraction and uv-4-cycle contraction operations.   Proof. Suppose (a) does not occur. Then, for all 1 ≤ i < j ≤ 3, either v i v j ∈ E, or there exists a tight uv-compatible family H ij in G − z with v i v j ∈ cov(H ij ) or there exists a tight set X ij in G − z with {v i , v j } ⊂ X ij and {u, v} ⊂ X ij . If the second alternative occurs we may assume that H ij has been chosen to be the unique tight uv-compatible family in G − z with maximal cover. If G[v 1 , v 2 , v 3 ] ∼ = K 3 then (b) occurs. So we may assume that v 1 v 2 / ∈ E. We first show that v i v j / ∈ E and that H ij exists for some 1 ≤ i < j ≤ 3. Suppose H 12 does not exist. Then X 12 exists. If v 3 ∈ X 12 then X 12 +z contradicts the independence of G in M uv .
If X 13 exists, then X 12 ∪ X 13 ∪ {z} contradicts the independence of G in M uv . Hence H 13 exists. Relabeling if necessary we assume that If v 3 ∈ X i then |X i | ≥ 4, since |N (z) ∩ {u, v}| ≤ 1, and the uv-compatible family obtained from H 12 by replacing X i by X i + z will contradict the independence of G in M uv . Hence v 3 ∈ X i .
Suppose that {v 1 , v 2 } ∩ {u, v} = ∅. Then |X i | ≥ 4 and neither v 1 v 3 nor v 2 v 3 are covered by H 12 . The maximality of cov(H 12 ) now implies that H 13 and H 23 do not exist. If v 1 v 3 , v 2 v 3 ∈ E, then the uv-compatible family obtained from H 12 by replacing X i by X i +v 3 will contradict the maximality of cov(H 12 ). Relabeling if necessary, we may suppose that v 1 v 3 / ∈ E, and hence X 13 exists. Then X i ∩ X 13 = ∅, |X i | ≥ 4, |X 13 | ≥ 4 and v 1 v 3 ∈ cov(X 13 )\cov(H 12 ). This contradicts the final part of Lemma 3.7. Hence {v 1 , v 2 }∩{u, v} = ∅ and we may assume, without loss of generality, that u = v 1 .
If v 3 ∈ V (H 12 ), then Lemma 3.7(c) implies that v 1 v 3 ∈ E and hence X 13 exists. This contradicts the final part of Lemma 3.7 since u ∈ X 13 ∩ X i . Hence v 3 ∈ X j for some X j ∈ H 12 − X i . The final part of Lemma 3.7 now implies that X 23 does not exist and hence Then the facts that G is independent in M uv and H 12 is tight imply that |X i | = 3 = |X j | (so X = N (z) ∪ {u, v, z}), and that H is a tight uv-compatible family in G with i(X) ≥ 2|X| − 4.
Lemma 4.10. Let G = (V, E) be a graph and let u, v ∈ V be distinct vertices. Suppose that G is independent in M uv , H = {X 1 , X 2 , . . . , X k } is a tight uv-compatible family in G and that H − X i is not tight for all 1 ≤ i ≤ k. Then either: (a) k = 1 and X 1 is tight; The fact that G is independent in M uv and H is tight now imply that k = 1 or 2 and that the sets in H satisfy the assertions in the lemma.
Note that if alternative (d) holds then X 1 ∪ X 2 is tight so we can reduce to alternative (a). Proof. Let G be the result of a K 4 contraction applied to H with w being the contracted vertex. Suppose that (a), (b) and (e) fail. It is easy to check that G is (2, 2)-sparse. Since (a) fails, G is simple. Since (b) fails, uv / ∈ E(G ). Since (e), there is a uv-compatible family H = {X 1 , X 2 , . . . , X k } for which val(H) < i(H) and w ∈ V (H). Without loss of generality we may assume w ∈ X 1 . If |X 1 | ≥ 4 then we get a contradiction as the uv-compatible  Proof. Suppose G is not independent in M uv . Since G = G − z + v 1 x and xv 1 / ∈ E, there exists either a tight uv-compatible family H in G − z with xv 1 ∈ cov(H), or a tight set Suppose X exists. Then X ∪ Y and X ∩ Y are tight by Lemma 3.4. Since {v 1 , x} ⊆ X ∩ Y and no proper subset of Y containing v 1 and x is tight, we have X ∩ Y = Y . This implies that z ∈ X contradicting the choice of X. Hence H = {X 1 , X 2 , . . . , X k } exists.
Since xv 1 ∈ cov(H), we may assume, without loss of generality, that x, v 1 ∈ X 1 . Then x, v 1 ∈ X 1 ∩ Y . Since |{u, v} ∩ Y | ≤ 1 by the hypotheses of the lemma, Lemma 3.5 implies that X 1 ∩ Y is tight. Since no proper subset of Y containing v 1 and x is tight we have X 1 ∩ Y = Y . This implies that z ∈ X 1 and contradicts the choice of H.
Theorem 4.13. Let G = (V, E) be a graph and let u, v ∈ V be distinct vertices. Then G is independent in R Y uv if and only if G is independent in M uv . Proof. Necessity follows from Lemma 4.1. Now suppose that G is independent in M uv . We prove that G is independent in R Y uv by induction on |V |. It is straightforward to check that G is independent in R Y uv when |V | ≤ 4. Hence we may assume that |V | ≥ 5. By extending |E| to a base of M uv (K |V | ) if necessary, we may also assume that |E| = 2|V | − 2. Case 1. G contains a vertex of degree 2. First suppose that u has degree 2. Then G − u is (2, 2)-sparse. Hence, by Theorem 2.1, R Y (G − u, p) has linearly independent rows for any generic p. We can now use Lemma 4.3 a contradiction. Hence {a, b} = {u, v}, which implies that the 0-uv-reduction operation can be applied at w to obtain a graph G = (V − w, E ) that is independent in M uv and satisfies |E | = 2|V − w| − 2. By induction, G is independent in R Y uv . Now Lemma 4.2 implies that G is independent in R Y uv . Case 2. There is a 4-cycle in G containing u and v. By Lemma 4.8, we may apply a uv-4-cycle-contraction (contracting u and v) to obtain a graph H which is simple and (2, 2)-sparse. Theorem 2.1 implies that any generic realisation (H, p) on Y is infinitesimally rigid. Now we can use Lemma 4.4 to show that G is independent in R Y uv . Henceforth we assume that Cases 1 and 2 do not occur. Case 3. There is a proper tight set X containing u and v. Since Case 1 does not occur, we may suppose X is a maximal proper tight set (where proper means X = V and maximal means there is no vertex w ∈ V − X with more than one neighbour in X). Now by the maximality of X, G/X is simple and |V − X| ≥ 3. Hence G/X is (2, 2)-tight. Theorem 2.1 implies that any generic framework (G/X 1 , p) on Y is infinitesimally rigid. We may now apply Lemma 4.6(a) to show that G is independent in R Y uv . Henceforth we may assume that Case 3 does not occur.  Lemma 4.9 and i(X) ≥ 2|X| − 3, or v 1 belongs to a subgraph H ∼ = K 4 satisfying alternative (b) of Lemma 4.11. If the first alternative occurs then we may use the facts that G is independent in M uv and Case 2 does not occur to deduce that i(X) = 2|X| − 3 = 7. It follows that, in both cases, we may relabel the vertices of such that H is one of the graphs shown in Figure 2. The fact that G is (2, 2)-sparse implies that, in both cases, there exists a (necessarily bad) degree three vertex v 4 ∈ V \ V (H). Since Case 2 does not occur, v 4 is not adjacent to both u and v. Hence v 4 also belongs to a subgraph H which is isomorphic to one of the graphs shown in Figure 2. Since Case 2 does not occur, v 4 ∈ V (H ). Since G is (2, 2)-sparse, V (H) ∩ V (H ) = {u, v, v 3 }. Now H ∪ H is one of the graphs shown in Figure 3. Since all three graphs are tight, we may use the fact that Case 3 does not occur to deduce that G = H ∪ H . The fact that Case 1 does not occur now tells us that G is not the graph in Figure 3(a). The graph in Figure 3(b) cannot be equal to G since X = N (v 1 ) ∪ {u, v, v 1 } does not belong to a tight uv-compatible family (so v 1 is not bad). Hence G is as shown in Figure 3(c).
(c) Figure 3. The three alternatives for G.
We will complete the discussion of this case by showing that G is minimally uv-rigid on Y. Let (G, p) be a generic uv-coincident realisation of G on Y and m be an infinitesimal motion of (G, p) with m(u) = 0. Since K 4 is rigid, m(w) = 0 for all w ∈ V (H) − v. In particular m(v 3 ) = 0 and hence m(w) = 0 for all w ∈ V .
Case 6. None of the previous cases occur. Let z 1 , z 2 , . . . , z k be the good degree three vertices in G. If the edge set of some 1-reduction of G at z i is independent in M uv then we may apply induction to the reduced graph and then apply Lemma 4.2 to deduce that G is independent in R Y uv . Hence we may assume that alternative (b) or (c) of Lemma 4.9 holds for z i . Similarly, if alternative (b) of Lemma 4.9 holds for z i and z i is contained in a K 4 -subgraph whose contraction results in a graph which is independent in M uv , then we may apply induction to the reduced graph and then apply Lemma 4.6(b) to deduce that G is independent in R Y uv . Since Case 4 does not occur, it follows that, for every good degree 3 vertex z i , there exists a tight uv-compatible family H i as described in alternative (c) or (d) of Lemma 4.11, or alternative (c) of Lemma 4.9. We may assume that the first alternatives holds for all 1 ≤ i ≤ k and that the second alternative holds for l + 1 ≤ i ≤ k. Choose a subgraph H i ∼ = K 4 that contains z i and satisfies alternative (c) or (d) of Lemma 4.11 for each 1 ≤ i ≤ k and let X i be the element of contradicting the fact that no subset of X s+1 that contains u, v (in each of the three possibilities for X s+1 ) satisfies this inequality.
We may apply Lemma 4.10 to a minimal tight uv-compatible subfamily of H i for all 1 ≤ i ≤ k, and use the facts that Cases 2 and 3 do not occur to deduce that alternative (c) of Lemma 4.10 must hold. Hence there exist sets Y i and {u, v, The fact that we are not in Case 2 also implies that y i = y j = y, say, for all Then Y ∩ X = {u, v} and y ∈ Y . We can now use Lemma 3.1(a) and the fact that Case 3 does not occur to prove inductively that i(Y ) = 2|Y | − 3.
Let W = V \X. Since i(W ) ≤ 2|W |−2 there is an integer t for which i(W ) = 2|W |−2−t. Since i(Y ) = 2|Y | − 3 and G is (2, 2)-sparse, there are at least 3 edges from Y \ {u, v} to {u, v}. Since Y \ {u, v} ⊆ W , y ∈ W \ Y and there are two edges from y to {u, v}, we have at least five edges between {u, v} and W . Note that the definition of X tells us that all degree 3 vertices in W are bad.
Suppose that no (bad) degree three vertex z ∈ W is contained in a set X ⊆ V which satisfies alternative (c) of Lemma 4.9 and has i(X) ≥ 2|X| − 3, or a subgraph H ∼ = K 4 that satisfies alternative (b) of Lemma 4.11. Then every (bad) degree three vertex in W is adjacent to both u and v. Since Case 2 does not occur we have at most one degree three vertex in W . Since i(Y ) ≥ 2|Y | − 4, we have |E| − |E(Y )| − |E(W )| ≤ 4 + t. The total degree of the vertices in W is at most 2(2|W | − 2 − t) + 4 + t = 4|W | − t. Since there is at most one degree three vertex in W , t ≤ 1. If t = 0, then W is tight and W + u + v violates sparsity. Hence t = 1 and W + u + v is a proper tight set which contradicts the fact that Case 3 does not occur. Now consider the case when there is a (bad) degree three vertex z ∈ W which is contained in a set X ⊆ V which satisfies alternative (c) of Lemma 4.9 and has i(X) ≥ 2|X| − 3, or a subgraph H 1 ∼ = K 4 that satisfies alternative (b) of Lemma 4.11. Then H = G[N (z) ∪ {u, v, z}] is isomorphic to one of the graphs shown in Figure 2  Proof. Necessity follows from the fact that an infinitesimally rigid uv-coincident realisation of G is an infinitesimally rigid realisation of G − uv, and also gives rise to an infinitesimally rigid realisation of G/uv by (4.1).
To prove sufficiency, suppose, for a contradiction, that G − uv and G/uv are both rigid but G is not uv-rigid. By Theorems 3.8 and 4.13 this implies that there is a thin cover K of G − uv with val(K) ≤ 2|V | − 3. If K consists of subsets of V only, then r(G − uv) ≤ 2|V | − 3 follows, which contradicts the fact that G − uv is rigid. Hence . . , X l } is a uv-compatible family. Contract the vertex pair u, v in G into a new vertex z uv . This gives rise to a cover which implies that G/uv is not rigid, a contradiction. This completes the proof.
A similar proof can be used to verify the following more general result: Theorem 4.15. Let G = (V, E) be a graph and let u, v ∈ V be distinct vertices. Then r uv (G) = min{r(G − uv), r(G/uv) + 2}. Theorems 4.14 and 4.15 show that the polynomial-time algorithms for computing the rank of a count matroid (see e.g. [1,7]) can be used to test whether G is uv-rigid, or more generally, to compute r uv (G). v n } and (G, p) is a realisation of G on a family of (not necessarily distinct) concentric cylinders Y = Y 1 ∪ Y 2 ∪ . . . ∪ Y n such that p(v i ) ∈ Y i for 1 ≤ i ≤ n. We say that (G, p) is globally rigid if every equivalent framework (G, q) on Y, with q(v i ) ∈ Y i for all 1 ≤ i ≤ n, is congruent to (G, p).

Vertex splitting and global rigidity
Let G = (V, E) be a graph and v 1 be a vertex of G with neighbours v 2 , v 3 , . . . , v t . A vertex split of G at v 1 is a graphĜ which is obtained from G by deleting the edges v 1 v 2 , v 1 v 3 , . . . , v 1 v k and adding a new vertex v 0 and new edges v 0 v 1 , v 0 v 2 , . . . , v 0 v k , for some 2 ≤ k ≤ t. We will refer to the new edge v 0 v 1 as the bridging edge of the vertex split. We will show in this section that a vertex splitting operation, in which the bridging edge is redundant, preserves generic global rigidity on the cylinder. Given a map p : V → R 3n , there is a unique family of concentric cylinders Y with p(v i ) ∈ Y i for all 1 ≤ i ≤ n as long as p(v i ) does not lie on the z-axis for all 1 ≤ i ≤ n. We will refer to Y as the family of concentric cylinders induced by p and denote it by Y p . We shall need the following analogue of [2, Theorem 13].
Lemma 5.1. If (G, p) is infinitesimally rigid and globally rigid on Y, then there exists an open neighbourhood N p of p on Y such that for any q ∈ N p the framework (G, q) is infinitesimally rigid and globally rigid on Y.
Proof. Suppose |V | ≥ 5 and that for any open neighbourhood N p , there is a p * ∈ N p such that the framework (G, p * ) is not globally rigid on Y. Then there is a convergent sequence (G, p k ) of non-globally rigid frameworks converging to (G, p). For each framework (G, p k ), let (G, q k ) be an equivalent but non-congruent realisation on Y. We may assume that (G, p k ) and (G, q k ) are in standard position (that is p k (v 1 ) = q k (v 1 ) = (0, 1, 0) assuming, without loss of generality, that r 1 = 1). By the compactness of R 3|V | , there is a convergent subsequence (G, q m ) converging to a limiting framework (G, q). As the limits of the respective sequences, (G, q) must be equivalent to (G, p).
If (G, q) is not congruent to (G, p) then we contradict the global rigidity of (G, p). So (G, p) and (G, q) are congruent, i.e. we can transform q to p by a reflection in the plane x = 0, a reflection in the plane z = 0 or a combination of the two. We apply this same congruence to all the (G, q m ) to obtain a sequence (G, r m ) converging to (G, p) with (G, r m ) being equivalent but not congruent to (G, p m ) for each m. Since p m and r m are not congruent, p m − r m is a nontrivial infinitesimal motion. This means that the rank of the rigidity matrix for each framework (G, p m +r m 2 ) is less than maximal. Since both p m and r m converge to p, so does p m +r m 2 . Thus (G, p) is a limit of a sequence of infinitesimally flexible frameworks and hence itself is infinitesimally flexible, a contradiction. (The fact that (G, p) is infinitesimally rigid implies that the rank of R Y q (G, p) is maximum for all q ∈ R 3|V | sufficiently close to p.) We can use this lemma and our main result to show that vertex splitting preserves global rigidity on Y under the additional assumption that the new edge is redundant.
Theorem 5.2. Let (G, p) be a generic globally rigid framework on a family of concentric cylinders Y. LetĜ be a vertex split of G at the vertex v 1 with new vertex v 0 and suppose thatĜ − v 0 v 1 is rigid on Y. Letp(v) = p(v) for all v = v 0 andp(v 0 ) = p(v 1 ). Then for any q on Y which is sufficiently close top, (Ĝ, q) is globally rigid on Y.
Proof. Since (Ĝ/v 0 v 1 , p) = (G, p) is globally rigid on Y and p is generic,Ĝ/v 0 v 1 is rigid on Y. Since G − v 0 v 1 is also rigid on Y, Theorem 4.14 implies thatĜ has a v 0 v 1 -coincident generic rigid realisation (Ĝ,p), wherep(v) = p(v) for all v = v 0 andp(v 0 ) = p(v 1 ). Since (G, p) is globally rigid on Y, (Ĝ,p) is also globally rigid on Y. We can now use Lemma 5.1 to deduce that (Ĝ, q) is globally rigid on Y for all q sufficiently close top.

Concluding remarks
Similarly to our definition of a framework (G, p) on Y we can define a framework on a family of concentric spheres S = S 1 ∪ S 2 ∪ · · · ∪ S k where S i = {(x, y, z) ∈ R 3 : x 2 + y 2 + z 2 = r i } and r = (r 1 , . . . , r k ) is a vector of positive real numbers. We can map a framework on S to a framework in the union of parallel (affine) planes P 1 ∪ P 2 ∪ · · · ∪ P k , where P i is the plane z = r i in R 3 , by central projection. In [10,11] this process was shown to preserve infinitesimal rigidity for any framework on S. Since the projection also preserves the property that u an v are coincident, the problem of characterising generic rigidity for frameworks with two coincident points on concentric spheres is equivalent to the problem of characterising generic rigidity for frameworks with two coincident points on parallel planes. This latter problem can be characterised using the proof technique of Theorem 1.1. This gives us the following result. We can also replace Y with other surfaces. In particular if we choose a surface with 1 ambient rigid motion (such as the cone, hyperboloid or torus) then the analogue of Theorem 2.1 requires the graph to be (2, 1)-tight [9]. In the uv-coincident case we would define the value as val(H) = 2|H| − t H where t H = 3 if |H| ∈ {2, 3} and H = {u, v}, t H = 2 if |H| ∈ {0, 4} or H = {u, v} and t H = 1 if |H| ≥ 5. We expect that, using similar techniques to Section 3, the appropriate count matroid can be established. However we do not know how to prove an analogue of Theorem 4.13. To make a start on this problem would require dealing with the case when the only vertices of degree less than 4 are u and v.