The rigidity of a partially triangulated torus

A simple graph is 3‐rigid if its generic embeddings in R3 are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal 3‐rigidity of a simple graph obtained from a triangulated torus by the deletion of edges interior to an embedded triangulated disc.


Introduction
The graph G = (V, E) of a triangulated sphere is generically 3-rigid in the sense that any generic placement of the vertices in three-dimensional Euclidean space determines a bar-joint framework which is continuously rigid. This generic version of Cauchy's rigidity theorem for convex polyhedra follows from Dehn's determination [5] of the infinitesimal rigidity of convex triangulated polyhedra. See also Gluck [7] for an alternative proof and Cauchy [1,2]. Moreover, the graphs of triangulated spheres are minimally 3-rigid since Maxwell's necessary condition |E| 3|V | − 6 holds with equality. Although this fact is prominent in combinatorial rigidity, little seems to be known about the minimal rigidity of general triangulated surfaces and manifolds.
We note that Fogelsanger [6] has shown that a finite simple graph given by a triangulated compact surface without boundary is 3-rigid. The proof is an extended argument employing combinatorial edge contraction methods to reduce to the case of a complete graph, which is evidently 3-rigid. The methods also extend to higher dimensions. However, with the exception of the sphere, the triangulated surface graphs without boundary are over-constrained, in the sense that |E| > 3|V | − 6. In what follows we begin a systematic analysis of minimal 3-rigidity for the graphs of triangulations of surfaces with boundaries by considering certain partially triangulated tori that satisfy equality in the Maxwell condition. Our main result is a necessary and sufficient condition for minimal 3-rigidity in the case of a partially triangulated torus graph which has edges removed from the interior of an embedded triangulated disc. A precise definition of these graphs, together with their inherited facial structure and integral homology, is given in Section 2. Moreover, we show that (i) there are 17 distinct forms of boundary graph for this class, (ii) there are two irreducible graphs, and (iii) the graphs are generated from these two irreducibles by vertex splitting.
There is currently considerable interest in determining conditions for the rigidity of nongeneric bar-joint frameworks, and particularly those with prescribed symmetry groups. See, for example, Connelly et al. [3] and Schulze and Tanigawa [10] where symmetric variants of Maxwell counting conditions play a key role. We shall be concerned entirely with generic frameworks but the results and methods may nevertheless give insights into the rigidity of symmetric toroidal structures.
The main theorem may be stated as follows.
Theorem 1.1. Let G be a torus with hole graph. Then, the following conditions are equivalent.
(iii) G is constructible from K 3 by vertex splitting.
Although (3,6)-tightness is a well-known necessary condition (see Section 3 for the definition) its sufficiency here is a more subtle issue than in the case of the generic Cauchy theorem. For example, we note in Section 3 that the substitution of a triangulated subdisc by a triangulated disc with the same boundary need not preserve (3,6)-tightness. Also we see that there are torus graphs with two holes which are (3,6)-tight and yet are generically flexible.
The development is structured as follows. In Section 2, a torus with hole graph is formally defined. In Section 3, we consider the subfamily T of (3,6)-tight graphs G of this type and we give the 17 possible forms for the boundary graphs. In Section 4, we define critical separating cycles and associated fission moves within the class T . Exploiting the toroidal facial structure of the graphs in T we obtain a key lemma, Lemma 4.4, which shows that if the contraction of an edge e preserves the simplicity of the graph but violates the (3,6)-tight sparsity count, then there exists a critical separating cycle through e. In this case, an associated fission move G → {G 1 , G 2 } is possible, which provides a pair of strictly smaller graphs in T if |V (G)| 10. It follows that there is a contraction-fission reduction scheme to a certain family of small graphs in T with no more than nine vertices.
In Section 5, we prove that the graphs in T with no more than eight vertices are 3-rigid and that the 9-vertex graphs in T are contractible in T . The inverse move for edge contraction is a vertex splitting move, which is known to preserve 3-rigidity (Whiteley [11]). Also the inverse fission moves, or fusion moves, are rigid subgraph substitution moves preserving 3-rigidity, and so the equivalence of (i) and (ii) follows. We note that critical separating cycle methods have also proven useful recently in a simpler reduction scheme for modified triangulated spheres (Cruickshank, Kitson and Power [4]) and we expect this methodology to be a key tool in the analysis of higher genus surface graphs.
In Section 6, we give an alternative proof of the equivalence of (i) and (ii) which is more direct. The proof is based on (a), a nested application of the key lemma, in order to identify a contractible edge whose contraction preserves membership in T , and (b) an analysis of the graphs of T which are not contractible in this manner. This leads to the fact that there are two such irreducible graphs (see Theorem 6.2). Each of these irreducible graphs is constructible from K 3 by vertex splitting and so the equivalence of (i), (ii) and (iii) follows.

Surface graphs in the torus
Let M be a classical compact surface of finite genus, with or without a boundary. In this case, we may define a surface graph for M to be the graph determined by the 1-skeleton of a simplicial complex with geometric realisation M, when this 1-skeleton graph is simple. Equivalently, M is the graph determined by a triangulation of M in the usual sense, which requires simplicity [9]. However, the 1-skeleton perspective is the most useful for us since we are interested in graphs associated with modified compact surfaces which have superficial holes, obtained by the removal of embedded open discs. These spaces, as is evident in the figures below for example, are more general than classical surfaces with boundaries.
We refer to a surface graph for the torus as a torus graph while a surface graph for the disc is referred to as a triangulated disc. Definition 2.1. Let M be a simplicial complex for the torus whose 1-skeleton is a simple graph T . Let D be the simplicial complex of a triangulated disc, and let ι be an injective map from the set of 2-simplexes of D to the set of 2-simplexes of M which respects the adjacency relation between 2-simplexes of D. Finally, let G be the subgraph of T obtained by deleting the edges associated with the 1-simplexes which are images, under the map induced by ι, of the interior 1-simplexes of D. Then, G is said to be a torus with hole graph.
We also view a torus with hole graph G as being endowed with the facial structure inherited from M . This consists of the set of 3-cycles for the faces that are not in the range of ι. Thus, G has well-defined simplicial integral homology groups.
A torus with hole graph G is determined by a triple (M, D, ι) where ι : D → M is a simplicial map from a simplicial complex D of a triangulated disc to a simplicial complex M , of the torus, which is injective on 2-simplexes. It is also convenient to abuse notation and let D, T and ∂D denote the graphs of the simplicial complexes D, M and ∂D. Also we write i for the associated simple graph homomorphism i : D → T . We refer to a subgraph of G as an embedded triangulated disc graph if it has the form j(H), where H is a triangulated disc and j : H → T is graph homomorphism which is injective on facial 3-cycles.
The boundary graph ∂G of a torus with hole graph G is the subgraph whose edges do not lie in two facial 3-cycles of G. Thus, ∂G is the graph i(∂D).
A torus with hole graph G is also endowed with a closed path of edges that cover the edges of ∂G, perhaps with repetition. This path is the image under ι of the boundary cycle of the graph D. We also refer to the graph homomorphism as the detachment map of G. This map has the form α : C r → ∂G where C r is the r-cycle graph, and is uniquely associated with G.
We now note some examples of torus with hole graphs. Let M be the compact surface with boundary derived from S 1 × S 1 by the removal of the interior of a closed topological disc D, where the topological boundary ∂D is a simple closed curve. Then, a surface graph for M is a torus with hole graph. These graphs correspond to the injectivity of the detachment map. Figure 1 illustrates such a graph.
On the other hand, Figure 2 indicates a torus with hole graph for which α is not injective and for which Recall that a simple graph G is 3-connected if there exists no pair of vertices x, y which separates the graph in the sense that there are vertices v, w such that each path from v to w contains one of the vertices in the pair. We note that a torus with hole graph may fail to be 3-connected, as indicated in Figure 3. However, we shall see that the combinatorial condition   of (3,6)-tightness, defined in the next section, limits the possible forms of non-injectivity of the detachment map. In particular, these graphs are necessarily 3-connected. Figure 4 gives a perspective view of two torus with hole graphs with non-injective detachment map which can arise as (3,6)-tight graphs. Here we may regard the detached disc interior (the hole) as being wrapped around the torus. The graph of the second figure has an exposed edge, that is, one which is incident to no face. This arises when the detachment map α is non-injective on edges.
Finally, we remark that a torus with hole graph G may take an extreme form with ∂G = G. In particular, G may consist of two cycles of edges joined at a single vertex. Evidently such graphs are not (3,6)-tight.

Planar representations
We find it convenient to illustrate torus with hole graphs by rectangular face graph representations. A rectangular face graph for a torus graph may be formally defined as a planar triangulated disc R whose outer boundary path ∂R, as a directed cycle of edges, is a concatenation ∂R = π 1 π 2 π 3 π 4 where π 1 and π 3 are paths of length r, π 2 and π 4 are paths of length s and where these paths are appropriately identified. This identification corresponds to bijections V (π 1 ) → V (π 3 ) and V (π 2 ) → V (π 4 ) which are order reversing. The associated identification graph R/ ∼ is a torus graph. If D is a triangulated disc in R and R is obtained from R by the removal of edges interior to D, then the identification graph G = R / ∼ is a torus with hole graph. The class of graphs in T with no more than nine vertices is surprisingly rich and in the Appendix we use face graph representations extensively for their case-by-case analysis.
Additionally, it is useful to consider torus with hole graphs as embedded graphs on the topological torus, and we do this in the proofs of Lemmas 4.5, 5.3 and 5.7 for example. Thus, a torus graph has an embedded graph representation in [0, 1] 2 / ∼ and the hole for a torus with hole graph can be viewed as an embedded open disc in the open subset (0, 1) 2 .

(3,6)-tight torus with hole graphs
We now consider torus with hole graphs G which are (3,6)-tight, and we give the 17 possible forms for a detachment map α : Recall that the freedom number for a finite simple undirected graph G = (V, E) is f (G) = 3|V | − |E|, and that if f (G) = 6, then G is said to satisfy the Maxwell count. Proof. From Euler's formula it follows that f (T ) = 0 if T is a surface graph for the torus. If ∂D has length r, then G can be completed to a torus graph by the addition of r − 3 edges, and so f (G) = 6 if and only if r = 9.
A simple graph G is (3,6)-tight if f (G) = 6 and f (K) 6 for every subgraph K with at least three vertices. We write T for the class of torus with hole graphs which are (3,6)-tight. Figure 5 indicates a rectangular face graph representation for a graph H 1 in T for which the boundary graph is a 9-cycle.
It is elementary to verify that H 1 is (3,6)-tight by means of the following principle. If a graph H a arises from a (3,6)-tight graph H b by vertex splitting, in the sense of the following definition, then H a is also (3,6)-tight. In this way, it also readily follows that the graphs H 2 , . . . , H 17 given below are graphs in T .
Note that if H 1 is obtained from H 1 by retriangulation, then H 1 need not be (3,6)-tight. This is the case, for example, if H 1 has a path of edges around the hole which has length less that 9. In this case, the path defines a torus with hole graph with freedom number less than 6. In general, replacing a triangulated disc by another with the same boundary need not preserve the (3,6)-tightness of torus with hole graphs.
Let  The proof of rigidity preservation under vertex splitting is due to Whiteley [11]. A different proof, together with a proof of Gluck's theorem, is given in Cruickshank, Kitson and Power [4].

Graphs in T with non-injective detachment map
When the detachment map α is not injective the closed path i(∂D) can be regarded as a 9-cycle which has been pinched together in some manner, with several self-contact points. The simplest form of this occurs when |V (∂G)| = 8 and we note that ∂G then takes one of two forms, which we denote as v3v6 and v4v5. Figure 6 indicates the graphs H 2 and H 3 which are of these types.
In general, a detachment map α : C 9 → ∂G determines a vertex word of length 9 whose letters are the vertices of ∂G, with repetitions, ordered in correspondence with the 9-cycle of C 9 . This word is uniquely determined up to cyclic permutation and order reversal. However, we shall employ the economy of writing the short form v3v6 for the full form v 1 The short form should be read as 'v followed by three distinct edges to v, followed by six further distinct edges' (terminating in the first vertex v). An example of a cyclic word for a detachment map with two repeated vertices (and no repeated edges) is v1w2v2w4. The numbers represent the nine distinct edges in i(∂D) in this example. We use the letters v, w and also x to denote repeated vertices. When edges are repeated we adopt a more economical notation. For example, e3e4 indicates that edge e is repeated and that e is followed by three distinct edges then followed by e (even though traversed in a different order) which is then followed by four distinct edges to complete the cycle. We use the letters e, f and g to denote repeated edges. In every cyclic word for the detachment map type, the number of edge letters (e, f or g), counted with multiplicities, together with the total of the numerals, is equal to 9.
The next lemma and the table in Figure 7 show that even the small graphs in T , with nine or fewer vertices, form a surprisingly varied class. The proof of the lemma is a straightforward case-by-case analysis and is given in the Appendix. (i) If G ∈ T with detachment map α, then there is a unique graph H i and a graph isomorphism φ : We remark that the graphs H 16 and H 17 are uncontractible torus with hole graphs G in the sense that there are no edges belonging to two faces in G whose contraction yields a simple graph (and hence a torus with hole graph). On the other hand, H 1 , . . . , H 15 do have such edges, referred to as F F edges in the next section.

Torus graphs with several holes
There is an evident modification of Definition 2.1 which defines a torus graph with several (superficial) holes. We note the following two examples which are also (3,6)-tight. Figure 8 shows a rectangular face graph R 1 for a torus graph G 1 with two holes. Two triples of edges (dashed) have been deleted from the interiors of two triangulated discs in R 1 . The graph G 1 is (3,6)-tight and has a separating pair of vertices v, w. In particular, it is not 3-rigid.  Figure 9 indicates a rectangular face graph R 2 for a torus graph G 2 with six holes. The graph G 2 may be obtained from G 1 by the addition of two degree 3 vertices and so G 2 is also (3,6)-tight and fails to be 3-rigid.

Contraction moves in T and critical separating cycles
Let G be a torus with hole graph. An edge of G is of type F F if it is contained in two facial 3-cycles, and an F F edge is contractible if it is not contained in any non-facial 3-cycle. For such an edge, there is a natural contraction move on the graph G which creates another torus with hole graph G . To define this formally, let e = vw be a contractible F F edge in G and let avw and bvw be the two facial 3-cycles which contain e. A simple graph G is said to be obtained from G by an edge contraction on e = vw if G is obtained by (i) deleting the edges aw and bw, (ii) replacing all remaining edges of the form xw with xv, (iii) deleting the edge e and the vertex w. Note that G is again a torus with hole graph with the natural facial structure inherited from G. Also G can be recovered from G by applying a vertex splitting move at v which introduces the new vertex w.
The contraction move for a contractible F F edge need not preserve (3,6)-tightness, and therefore membership in the class T . However, we obtain in this section a key lemma which shows that when this occurs there exists a critical separating cycle, in the sense below, and an associated graph division Definition 4.1. An annulus graph A is a planar subgraph of the graph of a triangulated disc D which is obtained by the removal of the interior edges of a triangulated subdisc D 1 ⊆ D. Such a graph has an inherited facial structure and the cycles ∂D and ∂D 1 are the boundary cycles of A.
With this terminology, and in analogy with the notion of an embedded triangulated disc graph i(D) in a torus graph G, we define an embedded triangulated annulus graph in G to be one of the form i(A), where A is an annulus graph and i is injective on facial 3-cycles. We refer to such a subgraph as an annular subgraph. In particular, we shall see that G • 2 is an annular subgraph.

Critical separating cycles
Let G be a torus with hole graph with triple (T, D, i), and let c be a closed path of edges in G. Then c is said to be a hole separating cycle or simply a separating cycle if the context is clear, if there is a subgraph G 1 ⊆ G which is a torus with hole graph with triple (T, D 1 , i 1 ), where D ⊆ D 1 , and c is equal to i 1 (∂D 1 ) viewed as a path of edges. In the case of a non-injective detachment map with repeated edges, the path c may be viewed more properly as a directed cycle in which, for a chosen orientation, no directed edges are repeated. A separating cycle c gives a division move G → {G 1 , G • 2 } where G 1 is the torus with hole graph defined by a triple (T, D 1 , i 1 ), and G • 2 is the annular subgraph determined by c and the faces of G which are not faces of G 1 .
We shall see, in Lemma 4.5, that in all cases the graph G 2 is simple and (3,6)-tight and so the resulting graphs of a fission move both lie in T .
Note that critical separating cycles have length 9 and the annular graph G • 2 has two boundary cycles of length 9. Also we note in Figure 10 that G • 2 need not be a planar graph. The following 'filling in' lemma will be useful for the analysis of critical separating cycles. It may be paraphrased as the assertion that a (3,6)-tight subgraph of a graph G ∈ T contains no holes on its surface, bounded by four or more edges, which do not contain the superficial hole of G. Lemma 4.3. Let G ∈ T and let H be an embedded triangulated disc graph in G.
Proof. (i) The embedded assumption means that H is determined by a simplicial map from a triangulated disc to the simplicial complex for G with the property of being injective on 2-simplexes. Let us write H c for the subgraph of G which contains the edges of ∂H and the edges of G not contained in H.
Let H = i(D), where D is a triangulated disc, and i is the embedding map. Since i is injective on the set of interior vertices of D and the set of interior edges of D, it follows that We deduce that f (D) − f (∂D) = 0. Since D is a triangulated disc this is only possibly if ∂D is a 3-cycle. Thus, ∂H must also be a 3-cycle graph.
(ii) The argument above leads to −1 f (H) − f (∂H) and hence to the inequality −1 f (D) − f (∂D). This implies that ∂H is either a 3-cycle or 4-cycle graph.

Contraction and fission
We now give a key lemma, Lemma 4.4, which will be used for the deconstruction and construction of graphs in T . The following terminology will be useful.
Two facial 3-cycles of a torus graph are adjacent if they have a common edge. An edge of a subgraph K of a torus graph is a boundary edge if at most one of its facial 3-cycles is a subgraph of K, and the boundary ∂K of K is the graph determined by its boundary edges. Proof. Let (T, D, i) be a defining triple for G and suppose G / ∈ T . Note that the Maxwell count is preserved on contraction of the edge e and so G must fail the (3,6)-sparsity count. Thus, there exists a subgraph K of G containing e for which the edge contraction results in a graph K satisfying f (K ) < 6.
Let e = vw and let c and d be the facial 3-cycles which contain e. Note that if both c and d are subgraphs of K, then f (K) = f (K ) < 6, which contradicts the sparsity count for G. Thus, K must contain either one or neither of these facial 3-cycles.
Suppose first that K is a maximal subgraph among all subgraphs of G which contains the cycle c, does not contain d and for which contraction of e results in a graph K which fails the (3,6)-sparsity count. Note that f (K) = f (K ) + 1 which implies f (K) = 6 and f (K ) = 5. In particular, K is (3,6)-tight, and is a connected graph.
Consider now an embedding of T on a topological torus. This provides a topological embedding of K determining a closed set X(K). Also K has faces inherited from the facial structure of G, and we letX(K) be the union of X(K) and the embeddings of these faces of K. Let U 1 , . . . , U k be the maximal connected open sets of the complement ofX(K) in the topological torus. Note that one of these open sets, U 1 say, contains the topological hole of G which, more precisely, is the complement of the closed setX(G). There are two possible cases. For the first case, each U i is homeomorphic to an open unit disc. In the second case, some U i contains a curve which is not homotopic on the torus to a point. However, in the second case it follows that K is a planar graph, since it can be embedded in the complement of U i . This is then a contradiction, since the edge contraction of a contractible F F edge in a planar triangulated graph preserves (3,6)-sparsity. Thus, each set U i is the interior of the closed set determined by an embedded triangulated disc graph, say H(U i ). (Indeed, the facial 3-cycles defining H(U i ) are those whose torus embedding have interior set contained in U i .) It follows that we may apply the filling in lemma, Lemma 4.3, to see that ∂H(U i ) is a 3-cycle for i > 1. By the maximality of K, we have k = 1 (since adding the edges and vertices of T interior to these non-facial 3-cycles gives a subgraph of G with the same freedom count). Thus, K is a subgraph of G determined by a hole separating cycle, namely the boundary cycle for ∂H(U 1 ). This cycle is a critical separating cycle for G and so the proof is complete in this case.
It remains to consider the case for which K contains neither of the facial 3-cycles which contain e. Thus, f (K) = f (K ) + 2 and f (K) ∈ {6, 7}. Once again we assume that K is a maximal subgraph of G with respect to these properties and consider the complementary components U 1 , . . . , U k . As before each set U i is homeomorphic to a disc and determines an embedded triangulated disc graph H(U i ), one of which, say H(U 1 ), contains the graph i(D).
The filling in lemma and maximality now implies that each boundary of H(U i ), for i > 1, is a 4-cycle. By the maximality of K, we see once again that k = 1 (since adding the missing edge for such a 4-cycle gives a subgraph of G with a lower freedom count) and the proof is completed as before.
We now show that the move G → {G 1 , G 2 }, associated with a critical separating cycle, is indeed a fission move in the class T .
be a division move associated with a critical separating cycle whose detachment map has associated graph H i . Let 2 be the torus with hole graph obtained by substituting H i for G 1 in the graph G. Then, G 2 is simple and (3,6)-tight.
Proof. Let A be the annular graph G • 2 . Then, the graph intersections G 1 ∩ A and H i ∩ A coincide. Also we have It remains to show that G 2 is simple and (3,6)-sparse. If G 2 is not simple then, since A and H i are simple there is an edge e of H i with the same vertices as an edge f of A. Also these vertices belong to the critical separating cycle. But then G 1 ∪ {f } is a subgraph of G with freedom count 5 which is a contradiction.
To determine the sparsity of G 2 let K be a subgraph of the graph G 2 = H i ∪ A with at least three vertices. Then, (3,6)-sparse and f (G 1 ) = 6. On the other hand Thus, f (K) 6 (as desired) except possibly in the case that K ∩ H i consists of a single edge. If this edge is an edge of the boundary cycle, then K is a subgraph of G and it follows immediately that f (K) 6. So the final case to consider is the case K = K 1 + e where K 1 is a (3,6)-tight subgraph of A which meets H i at two vertices, being the vertices of a non-boundary edge e of H i . We next show that this does not occur by making use of the filling in lemma.
As in the previous proof we consider representations of torus with hole graphs on a topological torus, [0, 1] 2 / ∼, given through a representation on the closed unit square [0, 1] 2 . Also we will consider subsets of this torus determined by the given facial structure on the graphs.
First, represent G on [0, 1] 2 / ∼ so that the critical separating cycle, c say, which is associated with the division G → {G 1 , G • 2 }, is represented by a simple closed curve γ in [0, 1] 2 , marked with the vertices of c. (As usual the structure of c is reflected in the way in which γ meets the boundary of [0, 1] 2 and some vertices may be marked on opposite points of this boundary.) It follows that the annular graph A = G • 2 is represented by the embedding in [0, 1] 2 of the triangulated annulus graph associated with A.
The graph G 2 is defined through a substitution of G 1 by the graph H i and it too has a division G 2 → {H i , A} determined by c. Also G 2 has a representation on [0, 1] 2 / ∼ which agrees with the representation of G on the common subgraph A. Consider now the subgraph K 1 ⊂ A. We denote the vertices of the edge e, as x and y.
The edges of K 1 determine a closed subset X(K 1 ) of the torus and the edges and faces (that is, facial 3-cycles) of K 1 determine the closed setX(K 1 ). The open subset of the torus determined by the intersection of the interior of γ and the complement ofX(K 1 ) contains at least two connected component regions (such as the regions A and B in Figure 11). One of these corresponds to or contains the hole of G. (The hole of G is represented in the torus as the interior of the setX(D) for the triangulated disc D associated with G.) Figure 11 depicts an example of the representation γ in [0, 1] 2 and subsets which correspond toX(K 1 ) and these regions (and labelled accordingly). The simple curve representing the edge e is not depicted. In general, this will be a curve exterior to γ with initial and final points carrying the labels x and y (for which there may be choices), and the curve may have several components corresponding to transits through opposing boundary points.
Since K 1 and G 1 are (3,6)-tight the subgraph G = G 1 ∪ K 1 of G is (3,6)-tight and the intersection of K 1 and G 1 consists of the two vertices of e, denoted x and y. The filling in lemma, Lemma 4.3, applies to G and we deduce that the boundary curve of each of the non-hole regions of the torus (such as B, C and E, etc.) represents a 3-cycle subgraph of G. (Specifically, the role of the subgraph K in that lemma is played by G and the role of the embedded triangulated disc graph H is played by the subgraph corresponding to one of the non-hole containing regions of the torus.) Consider the curve π from y to x which lies in the boundary ofX(K 1 ) and the boundary of one of the regions indicated above on the torus. If this region is not the hole-containing region (for the hole region associated with G), then it corresponds to a subpath p of a 3-cycle. If p has length 1, then adding the corresponding edge to G 1 gives a subgraph of G with freedom count 5, which is a contradiction. On the other hand, if the length of p is 2, then the edge e lies in c and adding this edge to K 1 gives a subgraph of G with freedom count 5, and so this again is a contradiction. Since there is at least one region that does not contain the hole region for G the proof is complete.
We now deduce that there is a contraction fission sequence for any graph G ∈ T as described in the next corollary.
Note that the boundary cycle i(D) determined by the triple (T, D, i) for G is a critical separating cycle which is improper in the sense that G 1 = G and G • 2 is equal to c as a graph. We say that c is proper if it is not improper. Also we note in Figure 10 a proper critical separating cycle c 1 need not provide a fission move with the property, which we call the reducing property, that G 2 is smaller than G in the sense of having fewer faces. However, we now show that if there is a proper critical separating cycle, then there is a contraction move in the class T or there is a critical separating cycle with the reducing property. The corollary then follows readily from this fact and the key lemma.
Suppose then that c is a proper critical separating cycle for G ∈ T . Then G • 2 contains a face of G with an edge xy on c and third vertex z ∈ G • 2 . Also z has degree 3 or more, since G is (3,6)-tight. Since G • 2 is an annular graph it follows that there is an F F edge e in G • 2 with both faces in G • 2 . If contraction of e yields a graph in T , then this edge contraction may be used for the reduction scheme in the corollary. So we may assume that this is not the case. By the key lemma, either there is a critical separating cycle c though e or the edge e lies in a non-facial 3-cycle of G • 2 . In the latter case, G • 2 contains the graph of a triangulate disc (all of whose faces are faces of G) whose outer boundary is the non-facial 3-cycle. It is well known that such a graph (a triangulated sphere) has an edge f of F F type whose contraction is simple graph (see [4, Section 5.2] for example) and so f can play the role of e in this case. We can also assume that c lies in G • 2 . To see this, note first that if c, c are oriented critical cycles for G with source vertex v and the same orientation (relative to the hole), and if π, π are subpaths with initial vertex v and common final vertex w, then necessarily π, π have the same length. (If not then a substitution between π and π creates a critical separating directed cycle of length less than 9.) In view of this principle, if c, c have common vertices, then we may substitute such a subpath of π containing e for its corresponding subpath π to obtain a critical cycle in G • 2 . (If c, c do not have a common vertex, then c already lies entirely in G • 2 .) It now follows that the division move for c produces an annular graph with fewer faces than G • 2 .
Definition 4.6. A torus with hole graph is uncontractible if it has no F F edges or if every edge of type F F lies on a non-facial 3-cycle.
Corollary 4.7. Let G be a torus with hole graph in T . Then there exists a finite rooted tree in which each node is labelled by an element of T such that, (i) the root node is labelled G, (ii) every node has either one child which is obtained from its parent node by an F F edge contraction, or, two children which are obtained from their parent node by a fission move for a critical separating cycle, (iii) each leaf is an uncontractible graph.
The inverse move for edge contraction in the class T is a vertex splitting move, which as we have noted, preserves 3-rigidity. Also the inverse of a fission move, which we refer to as a fusion move, corresponds to substitution of the subgraph H i of G 2 by the graph G 1 . It is immediate from the definition of infinitesimal rigidity that if G 1 and G 2 are 3-rigid, then so too is G. A proof of the equivalence of (i) and (ii) in the main theorem can therefore be completed by showing, as we do in the next section, that the uncontractible graphs in T , which appear in the terminal nodes of Figure 12, are 3-rigid.

The rigidity of uncontractible graphs
We now show that the uncontractible graphs of T are 3-rigid. This completes our first proof of the equivalence of (i) and (ii) in the main theorem. This will be achieved in two steps. The first step shows that uncontractible graphs G in T have no interior vertex. This means that each vertex of G lies on ∂G, and so, in particular, |V (G)| 9. This leads quickly to the fact that the uncontractible graphs with at most eight vertices are 3-rigid. In the second step, we show that the graphs with nine vertices and no interior vertices are contractible.
Lemma 5.1. Let G ∈ T be an uncontractible graph. Then, the interior vertices of G have degree at least 6.

Proof. A vertex v is an interior vertex if and only if all edges incident to v are of F F type.
Since G is (3,6)-tight it contains no vertices of degree 1 or 2. If deg(v) = 3, then it follows from the simplicity of G that each of the three edges incident to v does not lie on a non-facial 3-cycle. This contradicts the uncontractibility of G. If v has degree 4, then the induced subgraph X(v) for v and its four neighbours has at least ten edges. These are the eight edges for the faces incident to v and at least two further edges to fulfil the uncontractibility condition. Thus, f (X(v)) 5 which is contrary to G being (3,6)-tight. One can similarly check that f (X(v)) 5 if the degree of v is 5.
The proof of the next lemma exploits the topological nature of the torus. For this and subsequent arguments it is convenient to define the homology class of an F F edge e in the case that e = xy with x, y ∈ ∂G and e / ∈ ∂G. We refer to such an edge as a crossover edge. The homology class of a crossover edge e is well defined and not equal to the trivial homology class. For simplicity, we write [ẽ] for the homology class of e. In Figure 13, we indicate the partitioning of the crossover edges of the graph H 1 according to homology class.
The limited possibilities for the classes [ẽ] become apparent on considering G as an embedded graph on the topological torus. In the case of boundary type 9v the boundary graph edges determine a simple closed curve, γ say. If there are no interior vertices, then the curves for the remaining edges are disjoint except possibly at their endpoints on γ. In view of this disjointness it follows that there can be at most three distinct homology classes for such edges. Indeed, Figure 14 indicates three embedded crossover edges. (With respect to a natural identification of the homology group with Z 2 these have homology classes (1,0), (0,1) and (1,1), up to sign.) Note that no further homology class is possible for any additional embedded F F edge. The figure illustrates the embedding of a 9v type graph but in fact the argument is the same in general, when γ may have points of self-contact.
Suppose first that the degree of z is 6. Then, the subgraph X(z) induced by z and its neighbours includes the six edges incident to z, the six perimeter edges v i v i+1 and additional edges between non-adjacent perimeter vertices v 1 , . . . , v 6 . There are at least three such edges, and since f (X(z)) 6 it follows that there are exactly 3, say e 1 , e 2 , e 3 . Letẽ 1 ,ẽ 2 ,ẽ 3 be choices of three non-facial 3-cycles for the edges e 1 , e 2 , e 3 , and let [ẽ 1 ], [ẽ 2 ], [ẽ 3 ] be the associated homology classes in H 1 (T, Z).   In the example of Figure 16, with homology triple (1, 0), (0, 1), (1, 1) there are two nonfacial regions each of which is bounded by an embedded 6-cycle. Thus, in both cases that the non-facial regions are bounded by 4, 6 or 8 edges.
The remaining case in which the homology classes [ẽ 1 ], [ẽ 2 ], [ẽ 3 ] coincide cannot occur since G is a simple graph without loops and the embedded edges for e 1 , e 2 , e 3 are disjoint except at their endpoints.
Consider now the embedded graph for the containing torus graph T for G. Note that each of the non-facial regions R defines a subgraph G R of T which is a torus with hole graph whose boundary corresponds to the boundary of the region. In view of the observations on boundary lengths, each graph G R has freedom number f (G R ) < 6. However, one of these graphs is a subgraph of G, and this contradicts the (3,6)-sparsity count for G. So we conclude that G has no interior vertices of degree 6.
Suppose now that deg(z) = 7. We may view X(z) in this case as arising from the deg(z) = 6 case by the addition of a new vertex v * between v i and v i+1 , with the edge v i v i+1 replaced by two edges v i v * , v * v i+1 and with the edge zv * added. Moreover an embedded graph in the degree 6 case can be augmented in a corresponding way to provide the embedded graph for the degree 7 case. By the uncontractibility of G, there is an edge v * v j for some j which is embedded and we see that its embedding divides an r-cycle region (for the degree 6 case) into two regions. Moreover, it follows that the boundary cycles for these regions do not use more edges. Thus, we obtain a contradiction as before. By induction the same conclusion holds in general and so G has no interior vertices of degree n for all n 6. In view of Lemma 5.1, there can be no interior vertices of any degree.
Recall that the double banana graph, G DB say, is formed by joining two copies of the (single banana) graph K 5 \e at their degree 3 vertices. Evidently this graph is (3,6)-tight and fails to be 3-rigid and it is well known that this is the only graph with eight or fewer vertices with this property. The next lemma combined with the previous lemma shows that the uncontractible graphs with eight or fewer vertices are 3-rigid.
Lemma 5.4. Let G ∈ T be a graph in T with no interior vertices and no more than eighy vertices. Then G is 3-rigid.
Proof. It suffices to show that if G has eight vertices, then it is not equal to G DB . Since G and ∂G have the same vertex set it follows that the boundary graph is of type v3v6 or v4v5.
Considering an embedded graph representation of G in a torus R/ ∼, with the detached disc represented by an open subset of R, it follows that in fact any torus with hole graph with this form of boundary is 3-connected. Since the double banana graph is not 3-connected the proof is complete.
We now embark on showing the remaining case (step 2) that the graphs G in T with boundary type 9v and no interior vertices are contractible. It is possible to give a rather extended ad hoc embedded graph argument to show this, and in fact this method is employed in Section 5 for a range of small graph types. However, we now give a more sophisticated proof, exploiting once again the homology classes of edges, and which provides some general methods and other insights.
The main idea may be illustrated by considering again the nine vertex graph H 1 , labelled as in Figure 13. The edges vw and v w are F F edges of the same homology class, as are the three 'intermediate v to w edges'. We note that the 'interior edge' v 1 w 1 (in contrast to v w 1 ) does not lie on a non-facial three cycle. Also it can be shown that it does not lie on a critical separating cycle, and so the graph is reducible in T by edge contraction. In general, we will identify such edges within subgraphs (called panel subgraphs) determined by crossover edges of the same homology class.
First we note two general lemmas that ensure contractibility. (These lemmas also play a role in the ad hoc arguments in Section 5.) Lemma 5.5. Let G be a graph of T with a degree 3 vertex on the boundary graph ∂G which is incident to an F F edge e. Then, the graph obtained from the contraction of e is in T .
Proof. That the contracted graph is (3,6)-tight follows from the fact that subtracting v and its three incident edges from a subgraph does not change the freedom number. Also it is elementary to check that the contracted graph is a torus with hole graph.
Lemma 5.6. Let G ∈ T with V (G) = V (∂G) and let c be a critical cycle of edges which is not the detachment map cycle for G. Then, there is an edge contraction of G to a graph in T .
Proof. Let π be the directed 9-cycle for the detachment map of G. Then c and π form the boundary of an annular subgraph H of G and we assume that they have the same orientation. Let x be a vertex such that the edges e = xy, f = xw are edges of π and c which start at x, with e = f , and such that there are subpaths π 1 , c 1 of the critical cycles from x to a common vertex z. We may also assume that z is the first such vertex. Thus, the subpaths form the  boundary of either a facially triangulated disc, if z = x, or, if z = x, a triangulated disc with two boundary vertices identified. We denote this subgraph of H as H 1 . See Figure 17. The subpaths π 1 , c 1 are of the same lengths, say r, since π and c are critical separating cycles, and the triangulation is formed by the addition of edges only. It follows by elementary graph theory that there is a degree 3 vertex u strictly between x and z on the subpath π 1 . Indeed, the graph H 1 has exactly 2r − 2 bounded faces from the triangulation by facial 3-cycles of G. If the degrees of the intermediate vertices u are greater than 3, then there would be at least 2(r − 1) + 1 faces incident to these vertices.
Since the vertex u is incident to an F F edge the previous lemma applies to complete the proof.
We now define panel subgraphs of a graph G ∈ T with |V (G)| = |V (∂G)| = 9, and show that their strongly interior edges (defined below), should they exist, allow contraction to a graph in T .
Consider two distinct crossover edges e = vw, e = v w which have the same homology class. They determine a triangulated disc subgraph of G, and its containing torus graph T , which may be visualised as a planar triangulated panel of G. The vertex set consists of the vertices of e, e , of which there are three or four in number, and vertices on the hole boundary lying on two paths, between v and v and between w and w . Figure 18 indicates such a panel subgraph with five faces.
Considering embedded graphs in the torus it follows readily that every panel graph is contained in a unique maximal panel subgraph.
Note that the interior vertices of the panel, that is, those without incidence with e or e , are only incident to edges of the panel subgraph. It follows that any strongly interior edge v 1 w 1 of such a panel, that is, one with both v 1 and w 1 interior vertices, does not lie on a non-facial 3-cycle.  Note that Lemma 5.6 implies that if G is not contractible to a graph in G, then there can be no critical separating cycles. Thus, in view of the previous paragraph, there can be no strongly interior edges of the panel if G is not contractible to a graph in T .
Lemma 5.7. Let G be a graph in T with |V (G)| = |V (∂G)| = 9. Then, G may be contracted to a graph in T .
Proof. By the previous discussion we may assume that the maximal panel subgraphs do not have a strongly interior edges. Also, by Lemma 5.5, we can assume that there are no vertices in G of degree 3. It follows that each such subgraph has at most four crossover edges. Since there are 12 crossover edges, by (3,6)-tightness, and at most three crossover edge homology classes, it follows that there are three panels, each with four crossover edges. Thus, the panel subgraphs have the form indicated in Figure 19.
By Lemma 5.6, we may assume that every crossover edge is not on a critical cycle. However, we now show that this is not possible, completing the proof.
Consider a single pair of interior edges, v 1 w, v 1 w on one of the panel graphs. These edges and their panel subgraph are illustrated in the embedded graph diagram of Figure 20. If v 1 w lies on a non-facial 3-cycle, then this is achieved through edge v 1 v of the panel and one of the eight remaining crossover edges with a different homology class. This crossover edge from v to w is indicated in the figure with label g. Note however, that if g is another embedded edge of the same homology type as g then, from the disjointness requirement for embedded edges, g has the form v x (or the form yw), where x (respectively, y) lies on the hole boundary arc from v to w (respectively, v to w ) as indicated in the figure. It follows that the form of the panel for this homology type cannot hold, and this contradiction completes the proof.

Vertex splitting construction
We now obtain an alternative proof of the equivalence of (i) and (ii) in the main theorem. Also we determine exactly the uncontractible graphs in T and with this we complete the proof of the equivalence of (i), (ii) and (iii).
Lemma 6.1. Every graph in T admits a contraction sequence in T to an uncontractible graph.
Proof. Let G ∈ T and suppose that G is a contractible torus with hole graph. We show that there exists a contractible edge for which the contraction yields a graph in T .
Since G is contractible there exists an F F edge which is not on a non-facial 3-cycle. If the contracted graph is in T we may continue the argument with a smaller graph. So we may assume, by the critical cycle lemma, Lemma 4.4, and the discussion preceding Corollary 4.7, that there exists a proper critical cycle c 1 and an associated division G → {G 1 , G • 2 } where G 1 is in T and both G 1 and the annular graph G • 2 have fewer vertices than G. Since G • 2 contains a face of G it follows, again by the discussion preceding Corollary 4.7, that the annular graph contains an edge e of F F type.
We may assume, moreover, that the edge e does not lie on a non-facial 3-cycle. To see this we note the two possibilities that hold if e does lie on a non-facial 3-cycle. Either the 3-cycle is triangulated by faces of G, or the triangulation of the 3-cycle in the containing torus graph for G contains the associated triangulated disc for the detachment map for G. In the latter case, it follows that G contains a fully triangulated torus graph, which violates (3,6)-sparsity, and so this case does not occur. In the former case, G contains the graph of a triangulated sphere all of whose faces are faces of G. Such graphs have edges which are of F F type and do not lie on a non-facial 3-cycle, and so we may rechoose e to be such an edge.
Since the annular graph contains an edge of F F type which is not on a non-facial 3-cycle we may either contract with this edge to a smaller graph in T or, by the critical cycle lemma, obtain another proper critical cycle, c 2 say, and an associated division G → {H 1 , H • 2 } with H 1 ∈ T , where these graphs have fewer vertices than G.
Note that we can assume that c 2 lies inside c 1 , or, more precisely, that the detached triangulated disc for c 2 is contained in the detached triangulated disc for c 1 . One way to see this is to note that the union J of G 1 and H 1 is a proper subgraph of G which lies in T . Thus, we may replace c 2 by the proper critical cycle for the detachment map for J. By the reasoning above there is a contractible edge in the associated annular graph which does not lie on a non-facial 3-cycle.
Repeating this process, identifying nested critical cycles c 1 , c 2 . . . , we must eventually obtain a contractible edge for which the contracted graph lies in T . Indeed, if this did not occur, then the process provides an infinite strictly increasing sequence of proper subgraphs of G and this contradicts the finiteness of the graph.
We have shown that the uncontractible graphs in T are 3-rigid by the analysis in Sections 4 and 5. This, together with Lemma 6.1 provides a second proof of the equivalence of (i) and (ii) in the main theorem, which avoids the use of fission moves.
In fact we can obtain a stronger result by enlarging the analysis of the small graphs to identify the uncontractible graphs of T . To obtain this fact note first that H 16 and H 17 are uncontractible. Also it is straightforward to check, with the assistance of Lemmas 5.5 and 5.6, that the particular graphs H 2 , . . . , H 15 are contractible in T , and indeed, are completely contractible in the class T to one of these two graphs. However, it remains to check that every graph with no interior vertices and one of these boundary types is similarly contractible. We show this in the second part of the Appendix by arguments which become progressively simpler as the number of vertices decrease. Remark 6.3. It is natural to seek a combinatorial characterisation of 3-rigid torus with hole graphs along similar lines. Natural examples, which are not minimally 3-rigid, can be found amongst those graphs, G + say, which may be obtained by merging the boundaries of a (3,6)-tight torus with hole graph G and an annular graph H with a 9-cycle boundary and an r-cycle boundary, where 4 r 8. Here, of course, the identified directed 9-cycles for G and H should be of the same detachment type α i , for some 1 i 17. For such graphs G + , there is a simple necessary and sufficient condition for 3-rigidity, namely that the annular graph should have girth r, meaning that there is no hole separating cycle c for G + which lies in H and has length less than r. We expect that an examination of the relevant small graphs will provide a basis for a proof that the graphs G + are precisely the 3-rigid torus with hole graphs.   Since the graphs in T are simple it follows that the sum of any two of a, b, c, d is at least 3. In fact this is the only constraint, and up to cyclic order and reversals there are five types for such quadruples (a, b, c, d), namely (1, 2, 2, 4), (1, 2, 3, 3), (1, 2, 4, 2), (1, 3, 2, 3), (2, 3, 2, 2), and there are five associated detachment maps, of types v1w2v2w4, etc. The graphs in Figures A.3, A.4 give representative vertex minimal graphs, H 5 , . . . , H 9 for these attachment types.
We now look at the non-alternating case so that the cyclic word for the detachment map α is vavbwcwd. Since G is simple it follows that a and c are at least 3 and so there is a unique form up to relabelling and ordering, namely v3v2w3w1. (See Figure A .5.) In this case, we see that G contains a triangulated sphere which is formed by the part of the triangulated torus for G which lies 'between' the two non-facial 3-cycles associated with the subwords v3v and w3w. Moreover, G contains the augmentation of this subgraph by the edge vw and so G cannot be (3,6)-tight. Figure A.6 also indicates a perspective view of such a graph. Since G is assumed to be (3,6)-tight, this type of detachment map cannot occur.    There is one further case of a torus with hole graph with |V (∂G)| = 7, |E(∂G)| = 9. The detachment map is of type v3v3v3 and G can be thought of as a torus graph with a 6-cycle hole to which a triangle has been attached by a single vertex. In particular, such graphs are not (3,6)-tight.
As an aside we remark that it follows from the main theorem that every graph in T is 3-connected. However, this fact may also be proved directly by an embedded graph argument analogous to the one used in the exclusion of the small graph in Figure A Hole types v3e2v1e1 etc., for H12, H13, H14.  with this vertex and edge count for ∂G. Indeed, in analogy with the graph types of Figure A.6, a torus with hole graph whose cyclic word contains disjoint subwords of the form w1v and v2w is not (3,6)-tight.
We next consider further types with fewer than nine edges in the boundary graph. The first case to consider is |V (∂G)| = 6, |E(∂G)| = 8.
In this case, there are three types of detachment map, for simple torus with hole graphs, and these correspond to the cyclic words v3e2v1e1, v3e1v2e1, v2e2v2e1.
The structure of these words is depicted in Figure A Here the detachment map covers one edge of the boundary graph twice and two further vertices, v, w are covered twice. Such a graph in T is the graph H 15 in Figure A.10. This has cyclic type v1e1w2v1e1w1 and we note that the v and w vertices are alternating. The only other possible cyclic type is the non-alternating case v1e1v2w1e1w1. Arguing as in the previous non-alternating case (depicted in Figure A.5) it follows that the graph G cannot be (3,6)-tight. We next consider the case |V (∂G)| = 5, |E(∂G)| = 7.  Note that up to relabelling and order there is one form of cyclic word, namely e1f 2e1f 1, and so one form of detachment map for graphs in T . Figure A.11 indicates a vertex minimal representative, H 16 , for this type. Finally, we consider the case of boundary graphs with four vertices. There is one possible form of detachment map, with cyclic word ef 1ge1fg1. A vertex minimal representative is given by the graph H 17 in T in Figure A.12.
Note that K 3 is not a torus with hole graph G arising from any triple (T, D, i) where D is a 9-cycle. To see this, note that the map i : ∂D → ∂G must cover at least one edge of ∂G more than twice. On the other hand, each edge of G is incident to at most two faces, and i is injective on faces, so this is not possible.

A.1. The uncontractible graphs in T
We next identify the graphs in T that are uncontractible.
Note first that H 16 and H 17 are uncontractible. Also it is straightforward to check, with the assistance of Lemmas 5.5 and 5.6, that the particular graphs H 2 , . . . , H 15 are contractible in T , and indeed, are completely contractible to one of these two graphs. However, it remains to check that every graph with no interior vertices is similarly contractible. To see this, we argue in a case-by-case manner according to boundary type. The arguments become progressively simpler as the number of vertices decreases.
The 9v case, corresponding to the detachment map α 1 is covered by Lemma 5.7. (Alternatively one can effect a proof in the style of the following argument for the v3v6 case.) A.2. The α 2 , α 3 cases; types v3v6, v4v5, with |V (G)| = 8 Assume that G ∈ T is of type v3v6 and that G is uncontractible. We shall obtain a contradiction.
Since the boundary is of type v3v6 the vertex v is of degree 4 or more. Also there are no vertices of degree 3 incident to an F F edge, in view of Lemma 5.5. We now consider the possibilities for an embedded graph representation of G which has been 'standardised' so that (i) the v3v subcycle of the boundary cycle, with vertex sequence v, x, y, v, appears as the right-hand boundary of a representing rectangle for S 1 × S 1 ; (ii) the first two edges, vw, wz of the v6v subcycle appear on the lower boundary of the representing rectangle for S 1 × S 1 .
The embedded representation of G may be partially indicated, as in Figure A.13. The upper boundary of the diagram is represented by the embeddings of the edges vw, wz and a path τ , which is not necessarily a path of embedded edges. The remaining embedded edges of G are representable by paths in the region R which may pass across τ a number of times. (That the degree of vertices is at least 4 is suggested by emerging edge paths.) Suppose first that the crossover edge uw exists. It does not lie on a critical cycle, by Lemma 5.6 and so uw lies on a non-facial 3-cycle (since otherwise G is contractible). From the standardised diagram, it follows that this non-facial 3-cycle has vertices u, w, z. We may assume that the embedded path in the diagram representing the edge wz is a path from w to z in R. This now implies that there is a hole separating cycle of length less than 9, a contradiction.
Since the edge uw does not exist the face incident to vu has edges va and ua with a = w, and va must be represented by an embedded curve in R starting from the upper left representative of v. If a is one of r, s or z, then we obtain a contradiction as before. On the other hand, since G is simple, without loop edges or multiple edges between the same vertices, the vertex a cannot be equal to v, x or y, and so we have the desired contradiction.
There is similar argument for the case v4v5.
A.3. The α 4 case; type e3e4 with |V (G)| = 7 Let G ∈ T be a graph of this type and assume that G is not contractible. Once again we may partially represent G as an embedded graph on the torus as indicated in Figure A.14. By Lemma 5.5, the degree of each of the vertices z, r, s, x, y is at least 4.
In particular, there are pairs of edges from each vertex r, s, y, x which are of F F type and which (as before) must lie on non-facial 3-cycles. Considering an F F edge out of y the only possibilities are (i) yw, or (ii) yz, which requires ys (for the non-facial 3-cycle), or (iii) ys which requires either sv or yz. Since there are at least two edges out of y it follows that (ii) or case (iii) must hold. However, case (ii), with yz and ys, cannot hold for the following reason. The  triangulation would require the edge ry and this edge would not lie on a non-facial 3-cycle (in any triangulation). Similarly, the remaining case (iii), with edges sy and sv, would require the edge sx and in any completing triangulation this would not lie on a non-facial 3-cycle. Thus, we obtain a contradiction in all cases.
A.4. The cases α 5 , . . . , α 9 Suppose, by way of contradiction that there exists an uncontractible graph G ∈ T where the detachment map has one of these forms, so that there are no repeated edges and exactly two repeated vertices. Since every edge of ∂G lies on a face it follows from Lemma 5.5, that we may assume that each vertex is of degree at least 4. Since there are seven vertices it follows that there are at least seven crossover edges. This is a contradiction since there are nine non-crossover edges and by (3,6)-tightness there are 15 edges in total.
A.5. The cases α 10 , . . . , α 15 Consider a graph G ∈ T with no interior vertices and detachment map α 10 . In an embedded graph representation on the torus, we can assume that (in the edge identified rectangular representation) the repeated vertex x appears in the corners and the repeated vertices v and w appear on opposite sides. Thus, we have the representation in Figure A.15. In this case, note that we can assume that the boundary of the rectangle is provided by the edges of the 9-cycle for the hole together with some edge repetitions.
As before if we assume that G is uncontractible, then the degrees of the vertices r, s and z are at least 4. By (3,6)-tightness there are three crossover edges. It follows that the degrees are exactly 4. We thus see that there is only one triangulation and so G is equal to H 10 which is contractible, a contradiction.