The generic rigidity of triangulated spheres with blocks and holes

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal $3$-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.


Introduction
A classical result of Cauchy [1] asserts that a convex polyhedron in threedimensional Euclidean space is continuously rigid, when viewed as a bar-joint framework, if and only if the faces are triangles. Dehn [2] subsequently showed that this is also equivalent to the stronger condition of infinitesimal rigidity. If the joints of such a framework are perturbed to generic positions, with the bar lengths correspondingly adjusted, then infinitesimal rigidity may be established more directly by vertex splitting. In this case convexity is not necessary and it follows that the graphs of triangulated spheres are 3-rigid in the sense that their generic placements in R 3 provide infinitesimally rigid bar-joint frameworks. This is a theorem of Gluck [5] and in fact these graphs are minimally 3-rigid (isostatic) in view of their flexibility on the removal of any edge. The vertex splitting method was introduced into geometric rigidity theory by Whiteley [9] and it plays a key role in our arguments.
While the general problem of characterising the rigidity or minimal rigidity of generic three-dimensional bar-joint frameworks remains open, an interesting class of graphs which are derived from convex polyhedra has been considered in this regard by Whiteley [8], Finbow-Singh, Ross and Whiteley [4] and Finbow-Singh and Whiteley [3]. These graphs arise from surgery on a triangulated sphere involving the excision of the disjoint interiors of some essentially disjoint triangulated discs and the insertion of minimally rigid blocks into some of the resulting holes. Even in the case of a single block and a single hole of the same perimeter length n ≥ 4 the resulting block and hole graph need not be 3-rigid. A necessary and sufficient condition for minimal rigidity for this n-tower case, obtained in [3], requires that there exist n internally disjoint edge paths connecting the vertices of one disc boundary to the vertices of the other boundary.

The main result.
In what follows we introduce some new methods which provide, in particular, characterisations of minimal 3-rigidity for the class of block and hole graphs with a single block and finitely many holes. Such graphs may be viewed as the structure graphs of triangulated domes with windows, where the role of terra firma is played by the single block. In fact, the girth inequalities, defined in Sect. 4, provide a computable necessary and sufficient condition for 3-rigidity in terms of lower bounds on the lengths of cycles of edges around sets of windows.
T r ia n g u la te d F ix e d jo in ts F ix e d jo in ts T r ia n g u la te d The main result is as follows.
Theorem 1.1. LetĜ be a block and hole graph with a single block and finitely many holes, or, a single hole and finitely many blocks. Then the following statements are equivalent.
(iii)Ĝ is constructible from K 3 by the moves of vertex splitting and isostatic block substitution. (iv)Ĝ satisfies the girth inequalities.
Condition (ii) is a well known necessary condition for minimally 3-rigid graphs which requires the Maxwell count |E| = 3|V | − 6 together with corresponding sparsity inequalities for subgraphs (see Sect. 2). The construction scheme in (iii) involves three phases of reduction for a (3, 6)-tight block and hole graph, namely, (1) discrete homotopy reduction by (3,6)-tight preserving edge contractions, (2) graph division over critical separating cycles of edges, and, (3) admissible block-hole boundary contractions. For the single block case, the equivalence of conditions (i) − (iii) is established in Sect. 3. The girth inequalities (see Sect. 4) are a reformulation of the cut cycle inequalities of [3]. The same equivalences are obtained for the "dual" class of block and hole graphs with a single hole in Theorem 4.15. In fact, the dual of any generically isostatic block and hole graph is generically isostatic (see [4]). Theorem 1.1 confirms the single hole case and the single block case of Conjecture 5.1 in [3] (see also Remark 2.11 below). Example 4.14 shows that the conjecture is not true in general. A further corollary of Theorem 1.1 is that the following conjectures, paraphrased from [8], are true. Conjectures 4.2 and 4.3]). LetĜ be a block and hole graph with one pentagonal block and two quadrilateral holes, or, two quadrilateral blocks and one pentagonal hole. IfĜ is 5-connected then it is minimally 3-rigid.
The Appendix provides a proof of the preservation of minimal 3-rigidity under vertex splitting (established in [9]) and a simple proof of Gluck's theorem ( [5]) on the 3-rigidity of graphs of triangulated spheres.

Block and hole graphs
A cycle of edges in a simple graph is a sequence e 1 , e 2 , . . . , e r , with r ≥ 3, for which there exist vertices v 1 , v 2 , . . . , v r , such that e i = v i v i+1 for i < r and e r = v r v 1 . A cycle of edges is proper if its vertices are distinct.
2.1. Face graphs. Let S = (V, E) be the graph of a triangulated sphere, that is, S is a planar simple 3-connected graph such that each face of S is bounded by a 3-cycle. Let c be a proper cycle in S of length four or more. Then c determines two complementary planar subgraphs of S, each with a single non-triangular face bordered by the edges of c. Such a subgraph is referred to as a simplicial disc of S with boundary cycle c. The boundary cycle of a simplicial disc D is also denoted by ∂D. The edge interior of D is the set of edges in D that do not belong to ∂D. A collection of simplicial discs is internally-disjoint if their respective edge interiors are pairwise disjoint. Definition 2.1. A face graph, G, is obtained from the graph of a triangulated sphere, S, by, (1) choosing a collection of internally disjoint simplicial discs in S, (2) removing the edge interiors of each of these simplicial discs, (3) labelling the non-triangular faces of the resulting planar graph by either B or H.
A labelling of the triangular faces of G by the letter T would be redundant but nevertheless an edge of G is said to be of type BB, BH, HH, BT, HT or T T according to the labelling of its adjacent faces. A face graph is of type (m, n) if the number of B-labelled faces is m and the number of H-labelled faces is n.
Example 2.2. The complete graph K 4 is the graph of a triangulated sphere and may be expressed as the union of two simplicial discs with a common 4-cycle boundary. The edge interiors of these simplicial discs each contain a single edge. Remove these edge interiors to obtain a 4-cycle and label the two resulting faces by B and H. This is the smallest example of a face graph of type (1, 1).
If B and H are collections of simplicial discs of S then the notation G = S(B, H) indicates that the B-labelled faces of the face graph G correspond to the simplicial discs in B and the H-labelled faces of G correspond to the simplicial discs in H.

2.2.
Block and hole graphs. Let G = S(B, H) be a face graph derived from S and let B = {B 1 , B 2 , . . . , B m } be the simplicial discs in S which determine the B-labelled faces of G. Definition 2.3. A block and hole graph on G = S(B, H) is a graphĜ of the form G = G ∪B 1 ∪ · · · ∪B m where, (1)B 1 ,B 2 , . . . ,B m are minimally 3-rigid graphs which are either pairwise disjoint, or, intersect at vertices and edges of G, (2) G ∩B i = ∂B i for each i = 1, 2, . . . , m.
As in [3,4], we refer to the subgraphsB i as the blocks or isostatic blocks of G. The following isostatic block substitution principle asserts that one may substitute isostatic blocks without altering the rigidity properties ofĜ. The proof is an application of [7, Corollary 2.8].
Lemma 2.4. Let G = S(B, H) be a face graph and suppose there exists a block and hole graph on G which is simple and minimally 3-rigid. Then every simple block and hole graph on G is minimally 3-rigid.
The graph of a triangulated sphere is minimally 3-rigid ( [5]) and so such graphs provide a natural choice for the isostatic blocks in a block and hole graph.
i is referred to as a simplicial discus with poles at x i and y i . The resulting block and hole graph G ∪ B † 1 ∪ · · · ∪ B † m , denoted by G † , is referred to as the discus and hole graph for G. Note that G † is a simple graph which is uniquely determined by G. The discus and hole graphs will be used in Sect. 3 to establish a construction scheme for (3, 6)-tight block and hole graphs with a single block.
In general, a block and hole graph may not be simple. This can occur if two B-labelled faces of G share a pair of non-adjacent vertices.
Example 2.6. Let G = S(B, H) be a face graph and for each B i ∈ B construct an isostatic block B • i as follows: Define B • i to be the graph of a triangulated sphere which is obtained from the boundary cycle ∂B i by adjoining 2(|∂B i | − 3) edges so that B • i is the union of two internally-disjoint simplicial discs with common boundary cycle ∂B i . The resulting block and hole graph G ∪ B • 1 ∪ · · · ∪ B • m will be denoted G • . Note that G • is not uniquely determined and may not be simple. However, G • has the convenient property that its vertex set is that of G. This construction will be applied in Sect. 4 to characterise isostatic block and hole graphs in terms of girth inequalities.
There is a simple relationship between a face graph G and its associated block and hole graphs. It is convenient therefore to focus the reduction analysis at the level of face graphs. This perspective also underlines a duality principle of the theory under B, H transposition, a feature exposed in [4] and discussed in Sect. 4.5. Lemma 2.7. Let G, K and K ′ be graphs with the following properties, (i) K and K ′ both satisfy the Maxwell count, and, Proof. The result follows on considering the freedom numbers, A simple graph G is said to be (3,6)-sparse if f (J) ≥ 6 for any subgraph J containing at least two edges. The graph G is (3, 6)-tight if it is (3, 6)-sparse and satisfies the Maxwell count.
Lemma 2.8. Let G, K and K ′ be simple graphs with the following properties, (i) K and K ′ are both (3, 6)-tight, Proof. Suppose that G ∪ K is (3, 6)-sparse and let J be a subgraph of G ∪ K ′ which contains at least two edges. It is sufficient to consider the case where J is connected. If J is a subgraph of G then f (J) ≥ 6 since G ∪ K is (3, 6)-sparse. If J is not a subgraph of G then there are two possible cases. Case 1) Suppose that J ∩ K ′ contains exactly one edge vw and that this edge is not in G. Then, by condition (iii), either v / ∈ V (G) or w / ∈ V (G). It follows that, Case 2) Suppose that J ∩ K ′ contains two or more edges. Since K satisfies the Maxwell count, f (J ∩ K ′ ) ≥ 6 = f (K) and, since G ∪ K is (3, 6)-sparse, In each case, f (J) ≥ 6 and so G ∪ K ′ is (3, 6)-sparse. If G ∪ K is (3, 6)-tight then by the above argument, and Lemma 2.7, G ∪ K ′ is also (3, 6)-tight.
It is well-known that minimally 3-rigid graphs, and hence the isostatic blocks of a block and hole graph, are necessarily (3, 6)-tight (see for example [6]). The following corollary refers to the discus and hole graph described in Example 2.5. Proof. The statements follow by applying Lemmas 2.7 and 2.8 respectively with K and K ′ representing two different choices of isostatic block for a given B-labelled face of G. Note that in the case of (ii), if B i ∈ B then there are no edges vw of the simplicial discus B † i with v, w ∈ ∂B i other than the edges of the boundary cycle ∂B i . Thus condition (iii) of Lemma 2.8 is satisfied.

3-connectedness.
Recall that a graph is 3-connected if there exists no pair of vertices {x, y} with the property that there are two other vertices which cannot be connected by an edge path avoiding x and y. Such a pair is referred to here as a separation pair. The block and hole graphsĜ which are derived from face graphs G need not be 3-connected. However, it is shown below that in the single block case 3-connectedness is a consequence of (3, 6)-tightness. Lemma 2.10. Every (3, 6)-tight block and hole graph with a single block is 3connected.
Proof. LetĜ be a (3, 6)-tight block and hole graph with a single block and suppose thatĜ is not 3-connected. Then there exists a separation pair {x, y} with edgeconnected components K 1 , K 2 , . . . , K r . Let K 1 be the component which contains an edge ofB 1 and hence all ofB 1 . The graph K 1 and its complementary graph K ′ 1 with E(K ′ ) = E(Ĝ)\E(K) each have more than one edge and their intersection is It follows that the (3, 6)-sparse graphs K 1 and K ′ 1 are both (3, 6)-tight. In particular, K ′ 1 must be the graph of a triangulated sphere and it follows that K ′ 1 contains the edge xy. Now K 1 ∪ {xy} is a subgraph ofĜ which fails the (3, 6)-sparsity count, which is a contradiction.
Remark 2.11. The definition of a block and hole graphĜ is somewhat more liberal than the block and hole graphsP of Finbow-Singh and Whiteley [3]. A graphP is defined by considering a planar 3-connected graph P whose faces are labelled with the letters B, H and D. The B-labelled faces are replaced with isostatic block graphs and the D-labelled faces are triangulated. The resulting graphP is called a base polyhedron reflecting the fact that it is the starting point for an "expanded" graphP E . This is obtained by a further triangulation process involving adding vertices on edges of DD type, and vertices interior to triangles. In particularP andP E are also 3-connected.

Edge contraction and cycle division
For m, n nonnegative integers let G(m, n) be the set of all face graphs of type (m, n) for which the discus and hole graph G † is (3, 6)-tight. In particular, the graphs of G(0, 0) are triangulations of a triangle and the sets G(0, n) and G(m, 0) are empty for n, m ≥ 1.
3.1. T T edge contractions. The first reduction move for block and hole graphs is based on an edge contraction move for face graphs. A T T edge in a face graph G is said to be contractible if it belongs to two triangular faces and to no other 3-cycle of G. In this case the deletion of the edge and the identification of its vertices determines a graph move G → G ′ on the class of face graphs, called a TT edge contraction, which preserves the boundary cycles of the labelled faces of G.
Example 3.2. A cycle graph with length at least 4, with exterior face labelled B and interior face labelled H is evidently a terminal graph in G(1, 1). Fig. 2 shows a face graph G with a contractible T T edge which is nevertheless a terminal face graph of G(1, 5). The discus and hole graph for the contracted graph G ′ fails to be (3, 6)-tight since there is an extra edge added to the simplicial discus B † . Each block and hole graphĜ is evidently reducible by inverse Henneberg moves to a single block (i.e. by successively removing degree 3 vertices, see for example [6]). However, there is a systematic method of reduction described below in which each move is a form of edge contraction or cycle division.  Remark 3.5. The contraction of a T T edge in a graph which is both (3, 6)-tight and 3-connected may remove either one of these properties while maintaining the other. However, for a block and hole graph with a single block the situation is more straightforward since, by Lemma 2.10, 3-connectedness is a consequence of (3, 6)-tightness. In particular, if G is a terminal face graph in G(1, n), for some n ≥ 1, then the discus and hole graph G † is both (3, 6)-tight and 3-connected.

3.2.
Critical separating cycles. Let c be a proper cycle of edges in a face graph G and fix a planar realisation of G. Then c determines two new face graphs G 1 and G 2 which consist of the edges of c together with the edges and labelled faces of G which lie outside (resp. inside) c. If c is not a 3-cycle then the face in G 1 (and in G 2 ) which is bounded by c is assigned the label H. The discus and hole graph for G 1 (resp. G 2 ) will be denoted Ext(c) (resp. Int(c)). Note that Definition 3.6. A critical separating cycle for a face graph G is a proper cycle c with the property that either Ext(c) or Int(c) is (3, 6)-tight.
The boundary of a B-labelled face is always a critical separating cycle. Moreover, if G † is (3, 6)-tight then the boundary of every face of G is a critical separating cycle.
Lemma 3.7. Let G be a face graph in G(m, n). If c is a 3-cycle in G then c is a critical separating cycle for G and both Ext(c) and Int(c) are (3, 6)-tight.
For face graphs of type (1, n) a planar depiction may be chosen for which the unbounded face is B-labelled. Thus for any proper cycle, it may be assumed that Ext(c) contains the isostatic block and Int(c) is a subgraph of a triangulated sphere. Proof. If c is a 3-cycle then apply Lemma 3.7. If c is not a 3-cycle then Int(c) is a subgraph of a triangulated sphere with f (Int(c)) ≥ 6 + (|c| − 3) > 6. Proposition 3.9. Let G be a face graph of type (1, n) and suppose that there are no T T or BH edges in G.
(i) If G † satisfies the Maxwell count then G contains a proper cycle π, which is not the boundary of a face, such that Ext(π) satisfies the Maxwell count. (ii) If G ∈ G(1, n) then G contains a critical separating cycle for G which is not the boundary of a face.
Proof. Since G contains no edges of type T T or BH, every edge in the boundary cycle ∂B is of type BT (see Fig. 4) and so each vertex v in ∂B must be contained in Note that r ≤ n. Since the block and hole graphs G • satisfy the Maxwell count it follows that, This is a contradiction and so H v = H w for some distinct vertices v, w ∈ ∂B. The boundary of this common H-labelled face is composed of two edge-disjoint paths c 1 and c 2 joining v to w. The boundary cycle ∂B is also composed of two edge-disjoint paths joining v to w. Let π 1 be the path in Fig. 4 which moves anti-clockwise along ∂B from v to w and then along c 1 from w to v. Similarly, let π 2 be the path which moves clockwise along ∂B from v to w and then along c 2 from w to v. Note that π 1 and π 2 are proper cycles in G with Ext(π 1 ) ∩ Ext(π 2 ) = B † . Thus, and so, since f (G † ) = f (B † ) = 6, it follows that f (Ext(π 1 )) = f (Ext(π 2 )) = 6. Hence Ext(π 1 ) and Ext(π 1 ) both satisfy the Maxwell count. This proves (i) and now (ii) follows immediately.
3.3. Separating cycle division. The next reduction move for block and hole graphs is based on a division of the face graph with respect to a critical separating cycle of edges. The usefulness of this arises from the fact that critical separating cycles arise when there are obstructions to T T edge contraction. This division process G → {G 1 , G 2 } is referred to as a separating cycle division for the face graph G and cycle c.
Note that, under this separating cycle division, int(c) Figure 5. Separating cycle division in a face graph.
Since G 2 has no B-labelled faces it must be the graph of a triangulated sphere.
(ii) By Lemma 3.8, )-tight and Ext(c) (which intersects Int(c) in c) may be substituted by the simplicial discus B † with vertices in c to obtain G † 2 . It can happen that the only critical separating cycles in a face graph G ∈ G(m, n) are the trivial ones, that is, the boundary cycles of the faces of G. In the next section it is shown how repetition of (3, 6)-tight-preserving T T edge contractions may lead to the appearance of critical separating cycles. Through a repeated edge contraction and cycle division process a set of terminal and indivisible face graphs may be obtained. Such a face graph is illustrated in Fig. 6. 3.4. Key Lemmas. If c is a proper cycle in a face graph G, which is not the boundary of a face, then int(c) denotes the subgraph of G † obtained from Int(c) by the removal of the edges of c. The following result will be referred to as the "hole-filling" lemma.
Lemma 3.13. Let G be a face graph in G(1, n). Let K be a subgraph of G † and suppose that c is a proper cycle in G ∩ K with E(K ∩ int(c)) = ∅.
The following lemma plays a key role in the proof of the main result.
Lemma 3.14. Let G be a face graph in G(1, n) with n ≥ 1. Let e be a contractible T T edge in G with contracted face graph G ′ . Then the following statements are equivalent.
The edge e lies on a critical separating cycle of G.
Let v ′ be the vertex in G ′ obtained by the identification of u and v. Evidently, v ′ ∈ V (K ′ ) since, otherwise, G † must contain a copy of K ′ and this contradicts the (3, 6)-sparsity count for G † . There are two pairs of edges xu, xv and yu, yv in G which are identified with xv ′ and yv ′ in G ′ on contraction of e (see Fig. 7).
Case (a). Suppose that x, y ∈ V (K ′ ). Let K be the subgraph obtained from K ′ by first adjoining the edges xv ′ and yv ′ to K ′ (if necessary) and then reversing the T T edge contraction on e. Then f (K ′ ) ≥ f (K) ≥ 6 which is a contradiction.
Case (b). Suppose that x ∈ V (K ′ ) and y / ∈ V (K ′ ). Let K be the subgraph of G † obtained from K ′ by first adjoining the edge xv ′ to K ′ (if necessary) and then reversing the T T edge contraction on e. Then f (K) ≤ f (K ′ ) + 1 ≤ 6 and so f (K) = 6. In particular, K is (3, 6)-tight. Rechoose K, if necessary, to be a maximal (3, 6)-tight graph in G † which contains the edge e and does not contain the vertex y. Note that K must be connected and must contain the isostatic block in G † . Since K is maximal, by the hole-filling lemma (Lemma 3.13), K = Ext(c) for some proper cycle c in G. This cycle is a critical separating cycle for G, and so (i) implies (ii) in this case. Case (c). Suppose that x / ∈ V (K ′ ) and y / ∈ V (K ′ ). Let K be the subgraph of G † obtained from K ′ by reversing the T T edge contraction on e. Then f (K) = f (K ′ ) + 2 ≤ 7 and so f (K) ∈ {6, 7}. Once again assume that K is a maximal subgraph with this property. Then K must be connected and must contain the isostatic block in G † . By the planarity of G there are two proper cycles π 1 , π 2 of G, passing through e, with int(π 1 ) and int(π 2 ) disjoint from K and containing x and y respectively. Since K is maximal, by the hole-filling lemma (Lemma 3.13), K = Ext(π 1 ) ∩ Ext(π 2 ). Note that f (Ext(π 1 )) ≥ 6, f (Ext(π 2 )) ≥ 6 and 6 = f (G † ) = f (Ext(π 1 )) + f (Ext(π 2 )) − f (K).
For the converse, suppose that the contractible edge e lies on a critical separating cycle c. Then c is a separating cycle for a division G → {G 1 , G 2 } and G † 1 is a (3, 6)tight subgraph of G † . Since the edge e lies in exactly one triangular face of G † 1 , the graph obtained from G † 1 by contracting e is a subgraph of (G ′ ) † with freedom number 5 and so (i) does not hold. Proof. Suppose there exists a T T edge e in G. Since G is terminal, either e is not contractible or e is contractible but the graph obtained by contracting e is not in G(1, n). If e is not contractible then it must be contained in a non-facial 3-cycle c. By Lemma 3.7, c is a critical separating cycle for G. However, this contradicts the indivisibility of G. If e is contractible then by Lemma 3.14, e lies on a critical separating cycle. Again this contradicts the indivisibility of G and so the result follows.
3.5. Contracting edges of BH type. A BH edge e of a face graph G is contractible if it does not belong to any 3-cycle in G. A BH edge contraction is a graph move G → G ′ on the class of face graphs under which the vertices of a contractible BH edge of G are identified. At the level of the discus and hole graph G † , a contractible BH edge e is contained in a simplicial discus B † and is an edge of exactly two 3-cycles of G † . The contraction of e preserves the freedom number of G † and can be reversed by vertex splitting. Thus, prima facie, there is the possibility of reducing an indivisible terminal face graph with a (3, 6)-tight discus and hole graph to a smaller face graph which also has a (3, 6)-tight discus and hole graph. In the case of a block and hole graph with a single block this is always the case.
Proof. Let e = uv be the contractible BH edge in G with B 1 and H 1 the adjacent labelled faces of G and v ′ the vertex in G ′ obtained on identifying of u and v. Then e is contained in exactly two 3-cycles of G † which lie in the simplicial discus B † 1 . Clearly, (G ′ ) † satisfies the Maxwell count since f ((G ′ ) † ) = f (G † ) = 6. The BH edge contraction on e reduces the length of the boundary cycle ∂B 1 by one. If this reduction of the boundary cycle results in a 3-cycle then G ′ has no B-labelled face. Moreover, the Maxwell count for G ′ ensures that there are no H-labelled faces in G ′ . Thus G ′ ∈ G(0, 0). If G ′ has one B-labelled face then it must have either n or n − 1 H-labelled faces, depending on whether or not the BH edge contraction on e reduces the boundary cycle ∂H 1 to a 3-cycle. It remains to show that (G ′ ) † is (3, 6)-sparse in this case.
If K ′ is a subgraph of (G ′ ) † then K ′ may be obtained from a subgraph K of G † by the contraction of e. Let x and y be the polar vertices of the simplicial discus B † 1 . If K ′ contains neither of the vertices x, y then K is a subgraph of G with Suppose that K ′ contains exactly one of the polar vertices x, y. Then, assuming it is the vertex x, it follows that K is a subgraph of the triangulated sphere obtained from G by substituting the simplicial disc B 1 with the discus hemisphere for the vertex x and by inserting simplicial discs in the H-labelled faces of G. It follows that K ′ is also a subgraph of a triangulated sphere and so f (K ′ ) ≥ 6. Now suppose that K ′ contains both of the polar vertices x, y. It is sufficient to consider the case when K ′ contains the edges xv ′ and yv ′ and to assume that xu, xv, yu, yv ∈ K. Then f (K ′ ) = f (K) ≥ 6. It follows that (G ′ ) † is (3, 6)-sparse.
Example 3.17. Let G ∈ G(2, 3) be the face graph illustrated in Fig. 8. Contraction of the edge e leads to a vertex which is adjacent to four vertices in ∂B 1 and so the associated discus and hole graph is not (3, 6)-tight.
The following analogue of Lemma 3.16 applies to multi-block graphs.
Lemma 3.18. Let G be a face graph in G(m, n) with m, n ≥ 1. Let e be an edge of a path in ∂B i ∩ ∂H j which has length 3 or more and let G ′ be the face graph, of type (m ′ , n ′ ) obtained by the contraction of e. Then G ′ ∈ G(m ′ , n ′ ).
Proof. It is clear that (G ′ ) † satisfies the Maxwell count. Let K ′ be a subgraph of (G ′ ) † with decomposition We may assume that K ′ is the contraction of a subgraph K ⊆ G † containing e and that K has the corresponding decomposition K = K 1 ∪ K 2 (with K 1 ⊂ B † i and E(K 2 ) ∩ E(B † i ) = ∅). In fact we can identify the graphs K 2 and K ′ 2 . Note that in view of the path hypothesis the edge-less graph K ′ 1 ∩ K ′ 2 has the same number of vertices as K 1 ∩ K 2 .
Observe that K ′ 1 results from K 1 through the loss of a vertex v in K 1 , of degree 4 in G † . If v has degree j in K (and hence in K 1 ) with 1 ≤ j ≤ 4 then note that f (K) ≥ 6 + (4 − j) and f (K 1 ) ≥ 6 + (4 − j). Also, on contraction to K ′ 1 there is a reduction of j − 1 edges. Thus, This shows that (G ′ ) † is (3, 6)-sparse.
In the light of Lemma 3.16, the indivisible terminal face graph of Fig. 6 may be reduced by BH edge contractions and further edge contraction reductions become possible in view of the emerging edges of type T T . One can continue such reductions until termination at the terminal graph of G(0, 0) which is K 3 . In fact this kind of reduction is possible in general and forms a key part of the proof of Theorem 3.23.  Proof. Suppose there exists G ∈ G(1, n) which is terminal, indivisible and BHreduced. By Corollary 3.15, G contains no T T edges. If an edge e in G is of type BH then, since G is BH-reduced, e is not contractible and so must be contained in a non-facial 3-cycle c of G. By Lemma 3.7, c is a critical separating cycle for G. However, this contradicts the assumption that G is indivisible and so G contains no BH edges. By Proposition 3.9, G contains a critical separating cycle for G which is not the boundary of a face. However, this contradicts the indivisibility of G and so there can be no face graph in G(1, n) which is terminal, indivisible and BH-reduced.  G(1, n). Then there exists a rooted tree in which each node is labelled by a face graph such that, (i) the root node is labelled G, (ii) every node has either one child which is obtained from its parent node by a T T or BH edge contraction, or, two children which are obtained from their parent node by a critical separating cycle division, (iii) each node is either contained in G(1, m) for some m ≤ n and is not a leaf, or, is contained in G(0, 0) (in which case it is a leaf ).
Proof. The statement follows by applying Corollary 3.20 together with Lemma 3.11 and Lemma 3.16. Figure 9. Deconstructing a face graph G ∈ G(1, n). Each node is obtained from its parent by a T T or BH edge contraction, or, by a critical separating cycle division. Each leaf is contained in G(0, 0).
In the case of general block and hole graphs one can also perform division at critical cycles, and there are counterparts to Lemma 3.14 and Corollary 3.15. However, as the following example shows, there are face graphs in G(m, n), m ≥ 2, which are terminal, indivisible and BH-reduced.
Example 3.22. Fig. 10 shows a face graph G ∈ G(2, 6) which is terminal, indivisible and BH-reduced. Note that the associated block and hole graphsĜ are 3-rigid. This follows from the fact that they are constructible from K 3 by vertex splitting together with Henneberg degree 3 and degree 4 vertex extension moves.
3.6. Generic rigidity of block and hole graphs. Let J be a simple graph and let v be a vertex of J with adjacent vertices v 1 , v 2 , . . . , v n , n ≥ 2. Construct a new graphJ from J by, (1) removing the vertex v and its incident edges from J, (2) adjoining two new vertices w 1 , w 2 , (3) adjoining the edge w 1 v j or the edge w 2 v j for each j = 3, 4, . . . , n.
(4) adjoining the five edges v 1 w 1 , v 2 w 1 , v 1 w 2 , v 2 w 2 and w 1 w 2 . The graph move J →J is called vertex splitting. It is shown in [9] that if J is minimally 3-rigid then so too isJ. (See also the Appendix).  Proof. The implication (i) ⇒ (ii) is well known for general minimally 3-rigid graphs. The implication (iii) ⇒ (i) follows from the isostatic block substitution principle (Lemma 2.4) and the fact that vertex splitting preserves minimal 3-rigidity (see Appendix).
To prove (ii) ⇒ (iii), apply the following induction argument based on the number of vertices of the underlying face graph. Let P (k) be the statement that every (3, 6)-tight block and hole graphĜ with a single block and |V (G)| = k is constructible from K 3 by the moves of vertex splitting and isostatic block substitution. Note that if |V (G)| = 4 then G is a 4-cycle with one B-labelled face and one H-labelled face. In this case, every block and hole graphĜ is clearly constructible from K 3 by applying a single vertex splitting move to obtain the minimally 3-rigid graph K 4 and then substituting this K 4 with the required isostatic block forĜ. Thus the statement P (4) is true and this establishes the base of the induction. Now assume that the statement P (k) holds for all k = 4, 5, . . . , l − 1 where l ≥ 5. LetĜ be a (3, 6)-tight block and hole graph with a single block and |V (G)| = l. By Corollary 2.9, the discus and hole graph G † is also (3, 6)-tight and so G ∈ G(1, n) for some n. Thus G admits a T T edge contraction, a BH edge contraction or a critical separating cycle division as described in the reduction scheme for face graphs in G(1, n) (Corollary 3.21). In the case of a T T or BH edge contraction G → G ′ , the contracted face graph G ′ has fewer vertices than G and is contained in either G(1, m) for some m ≤ n, or, in G(0, 0). In the former case, the induction hypothesis implies that (G ′ ) † is constructible from K 3 by the moves of vertex splitting and isostatic block substitution. In the latter case, (G ′ ) † is the graph of a triangulated sphere and so is constructible from K 3 by vertex splitting alone (see Appendix). It follows that G † is itself constructible from K 3 by vertex splitting and isostatic block substitution. In the case of a critical separating cycle division G → {G 1 , G 2 }, G is obtained from two face graphs G 1 and G 2 , each with fewer vertices than G. Moreover, for each j = 1, 2 either G j ∈ G(1, m j ) for some m j ≤ n, or, G j ∈ G(0, 0). Thus it again follows that both G † 1 and G † 2 are constructible from K 3 by vertex splitting and isostatic block substitution. Note that G † 1 is minimally 3-rigid and so may be used as a substitute for the isostatic block of G † 2 . In this way G † is shown to be constructible from K 3 in the required manner. This establishes the inductive step and so the proof of the implication (ii) ⇒ (iii) is complete.

Girth inequalities
We now examine certain cycle length inequalities for block and hole graphs that were considered in Finbow-Singh and Whiteley [3]. Recall from Ex. 2.5 that G • denotes the block and whole graph obtained from a face graph G by adjoining 2(|∂B| − 3) edges to each B-labelled face so that each isostatic block B • is the graph of a triangulated sphere.  Moreover, S and G • have the same vertex set and so, The graph of a triangulated sphere S must satisfy the Maxwell count and so the result follows.  A block and hole graphĜ satisfies the girth inequalities if it is derived from a face graph G which satisfies the girth inequalities.
To obtain the reverse inequality, choose any B-labelled face B 1 in G and let C ′ = (B ∪ H)\{B 1 }. By the girth inequalities, Proof. Let S be the graph of a triangulated sphere and let c be a proper cycle of edges of length greater than 3. Then c determines two simplicial discs D 1 and D 2 with intersection equal to c. Since each simplicial disc may be completed to the graph of a triangulated sphere by the addition of |c| − 3 edges it follows that, Suppose a graph K 1 is derived from D 1 by keeping the same vertex set and subtracting and adding various edges. Then K 1 will fail the sparsity count f (K 1 ) ≥ 6 if the total change in the number of edges is an increase by more than |c| − 3 edges. Consider now the face graph G and suppose it is derived from the graph of a triangulated sphere S. Fix a planar representation of G and let c be a proper cycle in G. As in the previous paragraph, c determines two simplicial discs D 1 and D 2 in S. Without loss of generality, assume that D 1 contains the edges of S which lie inside c and D 2 contains the edges which lie outside c. Let K 1 and K 2 be the corresponding subgraphs of the block and hole graph G • . Thus K 1 and K 2 are derived from D 1 and D 2 respectively by removing edges which correspond to H-labelled faces in G and adjoining the edges of each isostatic block. (3,6)-sparse then f (K 1 ) ≥ 6. Thus the total change in the number of edges in deriving K 1 from D 1 does not exceed |c| − 3 in magnitude. This implies the inequality |c| − 3 ≥ ind(C).
(ii) Applying the argument for (i) to K 2 , f (K 2 ) ≥ 6 and so the total change in the number of edges in deriving K 2 from D 2 does not exceed |c| − 3. Thus,   4.3. One block and n holes. From the arguments of [3] it follows that a block and hole graph with a single block and a single hole is (3, 6)-tight if and only if the underlying face graph satisfies the girth inequalities. In Theorem 4.10 this equivalence is extended to the case of block and hole graphs with a single block and n holes for any n ≥ 1.   (i)Ĝ is minimally 3-rigid.
Proof. IfĜ is minimally 3-rigid then, by the isostatic block substitution principle, Lemma 2.4, G • is minimally 3-rigid for any choice of triangulated sphere B • . In particular, G • is (3, 6)-tight and so, by Proposition 4.5, G satisfies the girth inequalities.
To prove the converse, apply the following induction argument. Let P (k) be the statement that every block and hole graphĜ with a single block which satisfies the girth inequalities and has |V (G)| = k, is minimally 3-rigid. The statement P (4) is true since in this case there exists only one face graph G, namely a 4-cycle with one B-labelled face and one H-labelled face. Clearly, G satisfies the girth inequalities and has minimally 3-rigid block and hole graphs. This establishes the base of the induction.
Suppose that P (k) is true for all k = 4, 5, . . . , l − 1 and letĜ be a block and hole graph with a single block which satisfies the girth inequalities and has |V (G)| = l. Note that, by Lemma 4.4, each block and hole graph G • satisfies the Maxwell count. If G contains a critical girth cycle c, which is not the boundary of a face, then by Lemma 4.9 the face graphs G 1 and G 2 obtained by separating cycle division on c both satisfy the girth inequalities. Note that G 1 and G 2 are each either face graphs with a single B-labelled face and fewer vertices than G, or, are triangulations of a triangle. It follows that both G 1 and G 2 have minimally 3-rigid block and hole graphs. By the block substitution principle (Lemma 2.4) the isostatic block of G † 2 may be substituted with G † 1 to obtain G † . Thus G has minimally 3-rigid block and hole graphs. Now suppose that there are no critical girth cycles in G, other than the boundary cycles of faces of G. If G contains no edges of type T T or BH then, by Proposition 3.9, G contains a proper cycle π, which is not the boundary of a face, such that Ext(π) satisfies the Maxwell count. By Lemma 4.7, π is a critical girth cycle for G. This is a contradiction and so G must contain an edge of type T T or BH. Moreover, such an edge must be contractible since any non-facial 3-cycle would be a critical girth cycle for G.
Suppose a face graph G ′ is obtained from G by contracting a T T or a BH edge e. Then G ′ is either a face graph with a single B-labelled face and fewer vertices than G, or, is a triangulation of a triangle. By Lemma 4.8, G ′ satisfies the girth inequalities and so G ′ must have minimally 3-rigid block and hole graphs. Now G † may be obtained from (G ′ ) † by vertex splitting and so G also has minimally 3-rigid block and hole graphs. This establishes that the statement P (l) is true and so, by the principle of induction, the theorem is proved.
Remark 4.11. The girth inequalities give an efficient condition for the determination of generic isostaticity. Consider, for example, the structure graph of a grounded geodesic dome in which a number of edges have been removed. By such a dome we mean, informally, a triangulated bar-joint framework with a "fairly uniform" distribution of joints lying on a subset of a sphere determined by a half space and where the edges are of length less than a "small multiple" (say two) of the minimum separation distance between framework joints. Also, on the intersection of the plane and the sphere there is a cycle subgraph of "grounded" joints. If the windows arising from an edge depletion are sparsely positioned then it can be immediately evident that the girth inequalities prevail.
In [3] the following theorem is obtained. Theorem 4.12. LetĜ be a block and hole graph with one block and one hole such that |∂B| = |∂H| = r. If there exist r vertex disjoint paths in G which include the vertices of the labelled faces thenĜ is 3-rigid.
We note that this also follows from Theorem 4.10. Indeed if the disjoint path condition holds then it is evident that every cycle c associated with the single hole has length at least r since it must cross each of the r paths. Thus the girth inequalities hold. Similarly, Conjecture 1.2 follows on verifying that the 5-connectedness condition ensures that the girth inequalities hold.   (ii) If two H-labelled faces in G share more than two vertices then by the girth inequalities there exists a B-labelled face within their joint perimeter cycle. However, this implies that the block and hole graphs for G fail to be 3-connected. Similarly, if two H-labelled faces in G share two nonadjacent vertices then the block and hole graphs for G fail to be 3-connected.
The following example shows that Conjectures 5.1 and 5.2 of [3] are not true in general.
Example 4.14. Let G be the face graph of type (2, 2) with planar realisation illustrated in Fig. 11. The block and hole graph G • satisfies the separation conditions of Corollary 4.13 (and of [3]). Also, G • is (3, 6)-tight and, by Lemma 4.4, G satisfies the girth inequalities. However, G • is not minimally 3-rigid since it may be reduced to a 2-connected graph by inverse Henneberg moves on vertices of degree 3.
B H H B Figure 11. A face graph of type (2, 2) which satisfies the girth inequalities and separation conditions but does not have a 3-rigid block and hole graph.

4.5.
Block-hole transposition. We next observe that the characterisation of minimally 3-rigid block and hole graphs with a single block also provides a characterisation in the single hole case. Let G t be the face graph obtained from graph G by replacing B labels by H labels and H labels by B labels.
Corollary 4.15. LetĜ be a block and hole graph with a single hole. Then the following are equivalent.
(iii)Ĝ is constructible from K 3 by vertex splitting and isostatic block substitution.
In particular, G † is minimally 3-rigid if and only if G t † is minimally 3-rigid.
Proof. The implications (iii) =⇒ (i) ⇒ (ii) ⇒ (iv) have already been established more generally for face graphs of type (m, n). If G satisfies the girth inequalities then G t also satisfies the girth inequalities and so there exists a reduction scheme for G t as described in Corollary 3.21. This same reduction scheme may be applied to show that the block and hole graphs for G are minimally 3-rigid. Thus the equivalence of (i) − (iv) is established. The final statement follows since G satisfies the girth inequalities if and only if G t satisfies the girth inequalities.

Appendix
A bar-joint framework in R 3 consists of a simple graph G = (V, E) and a placement p : V → R 3 , such that p(v) = p(w) for each edge vw ∈ E. An infinitesimal flex of (G, p) is an assignment u : V → R 3 which satisfies the infinitesimal flex condition (u(v) − u(w)) · (p(v) − p(w)) = 0 for every edge vw ∈ E. A trivial infinitesimal flex of (G, p) is one which extends to an infinitesimal flex of any containing framework, which is to say that it is a linear combination of a translation infinitesimal flex and a rotation infinitesimal flex. The framework (G, p) is infinitesimally rigid if the only infinitesimal flexes are trivial and the graph G is 3-rigid if every generic framework (G, p) is infinitesimally rigid. See [6]. 5.1. Vertex splitting. The proof of rigidity preservation under vertex splitting indicated in Whiteley [9] is based on static self-stresses and 3-frames. For completeness we give an infinitesimal flex proof of this important result.
Let G = (V, E) with v 1 , v 2 , . . . , v r the vertices of V and v 1 v 2 , v 1 v 3 , v 1 v 4 edges in E. Let G ′ = (V ′ , E ′ ) arise from a vertex splitting move on v 1 which introduces the new vertex v 0 and the new edges v 0 v 1 , v 0 v 2 , v 0 v 3 . Some of the remaining edges v 1 v t may be replaced by the edges v 0 v t . Let p : V → R 3 be a generic realisation with p(v i ) = p i and for n = 1, 2, . . . let q (n) : V ′ → R 3 be nongeneric realisations which extend p, where q (n) (v 0 ) = q n 0 , n = 1, 2, . . . is a sequence of points on the line segment from p 1 to p 4 which converges to p 1 . Let u (n) , n = 1, 2, . . . , be infinitesimal flexes of (G ′ , q (n) ), n = 1, 2, . . . , which are of unit norm in R 3(r+1) . By taking a subsequence we may assume that u (n) converges to an infinitesimal flex u (∞) of the degenerate realisation of G ′ with q(v 0 ) = q(v 1 ) = q 1 . In view of the line segment condition we have, . Thus u (∞) restricts to an infinitesimal flex u of (G, p). Note that the norm of u is nonzero.
We now use the general construction of the limit flex in the previous paragraph to show that if G ′ is not 3-rigid then neither is G. Indeed if G ′ is not 3-rigid then there exists a sequence as above in which each flex u (n) is orthogonal in R 3(r+1) to the space of trivial infinitesimal flexes. It follows that u (∞) is similarly orthogonal and that the restriction flex u of (G, p) is orthogonal in R 3r to the space of trivial infinitesimal flexes. Since u is nonzero G is not 3-rigid, as desired.

5.2.
A proof of Gluck's theorem. In our terminology Gluck's theorem ( [5]) asserts that the (unlabelled) face graphs G of type (0, 0) are 3-rigid. For convenience we give a direct proof here. In view of 3-rigidity preservation under vertex splitting it will be enough to show that G derives from K 3 by a sequence of vertex splitting moves.
Suppose that G is not the result of a planar vertex splitting move on a face graph of type (0, 0). We show that G = K 3 . Suppose that G has the minimum number of vertices amongst all such graphs and suppose also, by way of contradiction, that G = K 3 . Every edge of G is of type T T and we may consider an edge e = uv with associated edges xu, xv and yu, yv for its adjacent faces. Since G is minimal the contraction of G under e cannot be a simple graph and so there is a nonfacial triangle with edges zu, zv and uv. But now the subgraph consisting of the 3-cycle z, u, v and its interior is of type (0, 0) and is a smaller graph than G. It is not equal to K 3 , since it contains x or y, and so it has an edge contraction. Thus G itself has an edge contraction to a simple graph, a contradiction.