Topological inductive constructions for tight surface graphs

arXiv:1909.06545v3 [math.CO] 6 Mar 2021 TOPOLOGICAL INDUCTIVE CONSTRUCTIONS FOR TIGHT SURFACE GRAPHS JAMES CRUICKSHANK, DEREK KITSON, STEPHEN C. POWER, AND QAYS SHAKIR Abstract. We investigate properties of sparse and tight surface graphs. In particular we derive topological inductive constructions for (2, 2)-tight surface graphs in the case of the sphere, the plane, the twice punctured sphere and the torus. In the case of the torus we identify all 116 irreducible base graphs and provide a geometric application involving contact graphs of configurations of circular arcs. 1. Introduction Inductive characterisations of various families of graphs play an important role in many parts of graph theory. A graph G = (V, E) is said to be (2, 2)-sparse if for any nonempty V ′ ⊂ V , we have |E(V ′ )| ≤ 2|V ′ | − 2. If, in addition, |E| = 2|V | − 2 we say that G is (2, 2)-tight. Such graphs arise naturally in various parts of geometric graph theory, including framework rigidity, circle packings, and also in graph drawing. We will derive inductive characterisations of (2, 2)-tight graphs that are embedded without edge crossings in certain orientable surfaces of genus at most 1. Our characterisations will be based on edge contractions and are in the spirit of well known results of Barnette, Nakamoto and others ([2, 15, 16]) on irreducible triangulations and quadrangulations of various surfaces. It may be worth noting, for example, that a graph is a quadrangulation of the plane if and only if it is (2, 4)-tight. There are similar characterisations of quadrangulations for various other surfaces. Thus it is clear that our results are related to, but distinct from, existing results on quadrangulations. Since our graphs are embedded, we consider inductive characterisations based on topological edge contractions - that is to say that the contraction preserves the embedding of the graph (precise definitions given below). This is a key point, since the well-known inductive characterisation of simple (2, 2)-tight graphs by Nixon, Owen and Power is purely graph theoretic ([17]). To further illustrate the significance of this we give an application of our main result to a recognition problem in graph drawing. The topological nature of the inductive characterisation is crucial in this context. Finally we note that there are similar topological inductive characterisations of Laman graphs in the literature already (see Fekete et al. and Haas et al.), which have interesting geometric applications to pseudotriangulations and auxetic structures. 2010 Mathematics Subject Classification. 05C10, 05C62, 52C30. Key words and phrases. graph, surface, torus graph, rotation system, sparse graph, tight graph, vertex splitting, inductive construction, contact graph, contacts of circular arcs. The second and third authors were supported by the Engineering and Physical Sciences Research Council [grant number EP/P01108X/1]. The fourth author gratefully acknowledges the financial support from the Iraqi Ministry of Higher Education and Scientific Research and Middle Technical University, Baghdad. 1 1.1. Summary of main results. Section 2 and the first part of Section 3 are background material for the rest of the paper. The main contributions of the paper are as follows • Theorem 3.4 presents an elementary but very useful principle concerning sparsity counts and graphs embedded in surfaces. While related results and special cases already exist in the literature, our statement and proof emphasises that this is a general principle that applies to wide range of sparsity counts and to surfaces of all genus. • In Section 4 we analyse the quadrilateral contraction move. This operation is well known in the context of quadrangulations. Here we examine its properties with respect to (2, 2)-sparsity and prove some structural results about non contractible quadrilaterals in this context. • Theorem 6.6 shows that if G is an irreducible (2, 2)-tight surface graph, then any (2, 2)-tight subgraph is also irreducible. This holds for surfaces of arbitrary genus. • We give topological inductive characterisations of (2, 2)-tight graphs embedded in the sphere, plane, annulus (Theorem 5.4) and the torus (Theorem 7.6). The first two of these are relatively standard, whereas the latter two characterisations are new. In the case of Theorem 7.6 we have also identified the 116 irreducible (2, 2)-tight torus graphs. We do not give an explicit description of these graphs in the paper for reasons of space, but the reader is referred to [5] for details. • Finally we present an application of our results to a recognition problem in geometric graph theory. Specifically we show that every (2, 2)-tight torus graph can be realised as the contact graph of a collection of nonoverlapping circular arcs in the flat torus. We note that by passing to the universal cover this result may also be interpreted as a recognition result for contact graphs of doubly periodic collections of circular arcs in the plane. We also conjecture a generalisation of our main results (Conjecture 5.1) to the set of irreducible (2, 2)-tight surface graphs for surfaces of arbitrary genus. This would be analogous to results of Barnette, Nakamoto and others on triangulations and quadrangulations of surfaces. As noted above and at appropriate points in the paper, many of our results are valid for a range of sparsity counts and for surfaces of arbitrary genus and these will be useful in future investigations of this conjecture. 2. Graphs, surfaces and embeddings In this section we fix our conventions and terminology regarding topological graphs. Throughout, we use the word graph for a finite undirected multigraph. So loop and parallel edges are allowed a priori, although loop edges will not arise in most of the cases of interest. If Γ is a graph then |Γ| is its geometric realisation. Suppose that Σ is a compact real 2-dimensional manifold without boundary. A Σ-graph, or surface graph, is a pair (Γ, ϕ) where Γ is a graph and ϕ : |Γ| → Σ is a continuous embedding of the geometric realisation of Γ in Σ. Given Σi -graphs (Γi , ϕi ) for i = 1, 2 we say that they are isomorphic if there is a homeomorphism h : Σ1 → Σ2 and a graph isomorphism g : Γ1 → Γ2 such that h ◦ ϕ1 = ϕ2 ◦ |g|, where |g| is the induced homeomorphism |Γ1 | → |Γ2 |. By the Heffter-Edmonds-Ringel rotation principle, the surface Σ and the Σ-graph (Γ, ϕ) are determined up to isomorphism by data consisting of a rotation system on Γ, a partition of the set facial walks associated to the rotation system and for each part of the partition a nonnegative integer that represents the genus of the corresponding facial region. Note that we do not assume that our surface graphs are cellular. See [14] for details of this. We will 2 be interested in determining certain classes of surface graphs up to isomorphism. The abovementioned principle allows us to argue topologically using properties of surfaces and curves in surfaces to deduce combinatorial information and we choose to write our arguments using topological terminology based on this. Now we clarify the meaning of some standard terms which may have ambiguous interpretations in this topological context. Let G = (Γ, ϕ) be a Σ-graph and let e ∈ E(Γ). By Γ/e we mean the graph obtained by identifying the end vertices of e and deleting (only) the edge e. Thus edge contractions can create parallel edges and/or loops. By G/e we mean the surface graph obtained by collapsing the arc corresponding to e to a single point. Clearly the underlying graph of G/e is Γ/e. A face, F , of G is a connected component of Σ − ϕ(|Γ|). As is well known there is a well defined collection of closed boundary walks associated to F . We say that F is non degenerate if no vertex occurs more than once in this collection of walks. A cellular face is one that is homeomorphic to R2 . For a cellular face F , the degree of F , denoted |F |, is the edge length of its unique boundary walk (which of course may differ from the number of vertices or edges in degenerate cases). We write fi for the number of cellular faces of degree i. Note that if Σ is connected then f0 = 1 if Σ is a sphere and Γ comprises a single vertex, and f0 = 0 otherwise. Finally we note that we extend much of the standard language of graph theory concerning subgraphs, intersections and unions to surface graphs, understanding that these terms apply to the underlying graphs. Thus if G = (Γ, ϕ) is a Σ-graph, a Σ-subgraph of G is a pair (Γ′ , ϕ|Γ′ ) where Γ′ is a subgraph of Γ and ϕ|Γ′ is the restriction of ϕ to |Γ′ |. If H1 , H2 are Σ-subgraphs of G then H1 ∪ H2 , respectively H1 ∩ H2 , is the Σ-graph whose underlying graph is the union, respectively intersection, of the underlying graphs of H1 and H2 . 3. Sparsity For a graph Γ = (V, E, s, t) as above, define γ(Γ) = 2|V | − |E|. For l ≤ 2 we say that Γ is (2, l)-sparse (or just sparse if l is clear from the context) if, γ(Γ′ ) ≥ l for every nonempty subgraph Γ′ of Γ. We say that Γ is (2, l)-tight if it is (2, l)-sparse and γ(Γ) = l. We will be particularly interested in (2, 2)-sparse graphs. Note that (2, 2)-tight graphs cannot have loop edges but can have parallel edges (but not triples of parallel edges). We record some standard elementary facts for later use. The proofs are straightforward and we omit them. Suppose that Γ1 , Γ2 are subgraphs of Γ. Then (1) γ(Γ1 ∪ Γ2 ) = γ(Γ1 ) + γ(Γ2 ) − γ(Γ1 ∩ Γ2 ) Lemma 3.1. Suppose that Γ is (2, 2)-sparse and that γ(Γ′ ) ≤ 3 for some subgraph Γ′ of Γ. Then Γ′ is connected.  Lemma 3.2. Suppose that Γ1 , Γ2 are (2, 2)-tight subgraphs of a (2, 2)-sparse graph Γ. If Γ1 ∩Γ2 is not empty then both Γ1 ∪ Γ2 and Γ1 ∩ Γ2 are (2, 2)-tight.  We also record a straightforward consequence of Euler’s polyhedral formula. Theorem 3.3. If Σ is a connected boundaryless compact orientable surface of genus g and G is a cellular Σ-graph then X (2) (4 − i)fi = 8 − 8g − 2γ(G) i≥0 Proof. Use the polyhedral formula and the fact that 3 P ifi = 2|E|.  Next we will derive an elementary but subsequently very useful principle relating the function γ, which is defined purely in terms of the underlying graph, to the embedding of the graph in the surface. We note that related results and special cases of this have appeared elsewhere (notably [3, 4, 7]). Here we attempt to express this principle in a more general terms: for surfaces of arbitrary genus and for a variety of sparsity counts. In order to state the result we first need some notation. Let H be a subgraph of the Σgraph G and suppose that F is a face of H. Let intG (F ) be the subgraph of G consisting of all vertices and edges of G that lie inside F (the topological closure of F in Σ). Let extG (F ) be the subgraph of G consisting of all vertices and edge of G that lie in Σ − F . Define ∂F = intG (F ) ∩ extG (F ) and note that ∂F is the smallest subgraph of G that supports the boundary walks of F . It follows that G = intG (F ) ∪ extG (F ) since any edge joining intG (F ) to extG (F ) must pass through ∂F . Theorem 3.4. Suppose that l ≤ 2 and that G is a (2, l)-tight Σ-graph. If H is a subgraph of G and F is a face of H, then γ(H ∪ intG (F )) ≤ γ(H). Proof. By (1) we have γ(H ∪ intG (F )) = γ(H) + γ(intG (F )) − γ(H ∩ intG (F )). Now, H ∩ intG (F ) = extG (F ) ∩ intG (F ) and using (1) again, we see that γ(H ∩ intG (F )) = = = = ≥ γ(extG (F ) ∩ intG (F )) γ(intG (F )) + γ(extG (F )) − γ(extG (F ) ∪ intG (F )) γ(intG (F )) + γ(extG (F )) − γ(G) γ(intG (F )) + γ(extG (F )) − l γ(intG (F )). The last inequality above follows from applying the sparsity of G to the nonempty subgraph extG (F ).  Corollary 3.5. Suppose that l ≤ 2 and that G is a (2, l)-tight Σ-graph. If H is a subgraph of G and F is a face of H, then γ(extG (F )) ≤ γ(H). Proof. Let J1 , · · · , Jk be all the faces of H that are different from F . Then extG (F ) = S H ∪ ki=1 intG (Ji ). Now the conclusion follows from repeated applications of Theorem 3.4.  We remark that all of the results of this section admit straightforward adaptations to the function γk which maps H 7→ k|V (H)| − |E(H)|, for any positive integer k. We conclude this section by making some straightforward observations about the subgraphs intG (F ) and extF (G) that will be useful in the sequel. Any face of intG (F ) that is contained in F is also a face of G. On the other hand, there are one or more faces of intG (F ) which are contained in Σ − F . We call such a face an external face of intG (F ). Such an external face need not be a face of G. Note that if F has a unique boundary walk that is a simple cycle, then intG (F ) has just one external face. In general it may have more than one external face. Observe that extG (F ) has one exceptional face, namely F , such that all other faces of extG (F ) are also faces of G. 4 4. Inductive operations on surface graphs In this section we will focus on topological inductive operations on graphs that are natural in the context of (2, l)-tight graphs. We consider three types contractions associated to cellular faces of degree 2, 3 and 4, respectively called digons, triangles and quadrilaterals hereafter. In each case the contraction decreases the number of vertices by one and the number of edges by two. We investigate necessary and sufficient conditions for these moves to preserve the property of being (2, 2)-tight. We pay particular attention to degenerate cases as these play an important role later. 4.1. Digon and triangle contractions. Let G be a Σ-graph and suppose that D is a digon of G with boundary walk v1 , e1 , v2 , e2 , v1 such that v1 6= v2 and e1 6= e2 . Let GD = (G/e1 )−e2 . Observe that (G/e1 ) − e2 is canonically isomorphic to (G/e2 ) − e1 , so GD depends only on the digon and not the particular choice of labelling of the edges. We remark that, for a connected surface Σ, while a digon in a (2, 2)-sparse Σ-graph necessarily has distinct vertices, it may have degenerate boundary, but only in the case that the graph is a single (non loop) edge and Σ is a sphere. The proof of the following is straightforward and we omit it. Lemma 4.1. G is (2, l)-sparse if and only if GD is (2, l)-sparse Now suppose that T is a triangle in G with boundary walk v1 , e1 , v2 , e2 , v3 , e3 , v1 such that v1 6= v2 and e1 6= e2 . Let GT,e1 = (G/e1 ) − e2 . Again we omit the proof of the following lemma as it is a straightforward consequence of the definitions. Lemma 4.2. Suppose that G is (2, l)-sparse and that GT,e1 is not (2, l)-sparse. Then there is a subgraph H of G that contains e1 but not v3 such that γ(H) = l We refer to the graph H whose existence is asserted in Lemma 4.2 as a blocker for the contraction GT,e1 . We note that a triangle in a (2, 2)-sparse surface graph necessarily has a non degenerate boundary walk, since any degeneracy would entail a (forbidden) loop edge. Thus, in this case there are three possible contractions (one for each of the edges) associated to any such face. Lemma 4.3. Suppose that G is a (2, 2)-sparse Σ-graph and that T is a triangle with edges e1 , e2 , e3 . Then at least two of the Σ-graphs GT,e1 , GT,e2 , GT,e3 are (2, 2)-sparse. Proof. Suppose that there are blockers H1 , respectively H2 , for GT,e1 respectively GT,e2 . Then v1 , v3 ∈ H1 ∪ H2 . However v3 6∈ H1 and v1 6∈ H2 so e3 6∈ H1 ∪ H2 . However v2 ∈ H1 ∩ H2 so by Lemma 3.2, H1 ∪ H2 is (2, 2)-tight. This contradicts the sparsity of G.  4.2. Quadrilateral contractions. In the case of quadrilaterals we consider a somewhat different contraction move. In this case the analysis is a little more complicated and we include the details. Suppose that Q is a quadrilateral of G with possibly degenerate boundary walk v1 , e1 , v2 , e2 , v3 , e3 , v4 , e4 , v1 . Suppose that v1 6= v3 and e1 6= e3 . Let d be a new edge that joins v1 and v3 and is embedded as a diagonal of the quadrilateral Q. Define GQ,v1 ,v3 to be (G ∪ {d})/d − {e1 , e3 }. Clearly the underlying graph of GQ,v1 ,v3 is obtained from Γ by identifying the vertices v1 and v3 and then deleting e1 and e3 . Thus γ(G) = γ(GQ,v1 ,v3 ). However this quadrilateral contraction move does not necessarily preserve (2, l)-sparsity. 5 Figure 1. A (2, 2)-tight projective plane graph. Here we are using the representation of the projective plane as a disc with antipodal boundary points identified. This surface graph has a single quadrilateral face, with a degenerate boundary walk. Lemma 4.4. Suppose that G is (2, l)-sparse but GQ,v1 ,v3 is not (2, l)-sparse. Then at least one of the following statements is true. (1) There is some subgraph H of G such that v1 , v3 ∈ H, exactly one of v2 , v4 is in H and γ(H) = l. (H is called a type 1 blocker.) (2) There is some subgraph K of G such that v1 , v3 ∈ K, v2 , v4 6∈ K and γ(K) = l + 1. (K is called a type 2 blocker.) Proof. Let K be a maximal subgraph of GQ,v1 ,v3 satisfying γ(K) ≤ l − 1. Let z be the vertex of GQ,v1 ,v3 corresponding to v1 and v3 . Clearly z ∈ K, otherwise K would also be a subgraph of G. Let H be the maximal subgraph of G satisfying (H ∪ {d})/d − {e1 , e3 } = K. It is clear that H is an induced subgraph, since K is an induced subgraph. If {v2 , v4 } ⊂ H, then γ(H) = γ(K) ≤ l − 1 which contradicts the sparsity of G. So at most one of v2 , v4 belongs to H. Also, it is clear that l ≤ γ(H) ≤ γ(K) + 2 ≤ l − 1. So γ(H) = l or l + 1. If γ(H) = l and one of v2 , v4 ∈ H then (1) is true. If γ(H) = l and neither of v2 , v4 is in H, then let H ′ = H ∪ {v2 } ∪ {e1 , e2 }. Now observe that e1 6= e2 since v1 6= v3 . Thus γ(H ′ ) = γ(H) = l and, again, (1) is true. Finally if γ(H) = l + 1. Then γ(H) = γ(K) + 2 and since H is an induced graph, it follows that neither of v2 , v4 belongs to H. Thus (2) is true in this case.  In the special case that l = 2, various degeneracies are forbidden. Now suppose that G is a (2, 2)-sparse Σ-graph and that Q is a quadrilateral face of G with boundary walk v1 , e1 , v2 , e2 , v3 , e3 , v4 , e4 , v1 . Lemma 4.5. For i = 1, 2, 3, vi 6= vi+1 , and v1 6= v4 . Proof. Loop edges are forbidden in a (2, 2)-sparse graph.  Lemma 4.6. Suppose that Σ is orientable and that G is (2, 2)-tight. Then ei 6= ej for 1 ≤ i < j ≤ 4. Proof. We observe that since Σ is orientable, a repeated edge in ∂Q implies the existence either of a vertex of degree one or of a loop edge. Both of these are forbidden in a (2, 2)-tight graph.  Lemma 4.7. Suppose that Σ is orientable, G is (2, 2)-tight and that v1 = v3 . Then v2 6= v4 . Furthermore GQ,v2 ,v4 is also (2, 2)-tight. 6 Proof. Suppose that v2 = v4 . By Lemma 4.5 and the sparsity of G, ∂Q has exactly two vertices and two edges. This contradicts Lemma 4.6. Thus v2 6= v4 . Now suppose that GQ,v2 ,v4 is not (2, 2)-tight. By Lemma 4.4 there is a blocker for this contraction. Since v1 = v3 by assumption, the blocker must be a type 2 blocker. Thus we have a subgraph K such that γ(K) = 3, v2 , v4 ∈ K and v1 6∈ K. However, by Lemma 4.6 there are at least four edges joining v1 to K, contradicting the sparsity of G.  See Figure 1 for an example of (2, 2)-tight projective plane graph whose only face is a quadrilateral with repeated edges in the boundary walk. This example shows that orientability is a necessary hypothesis in the statements of Lemmas 4.6 and 4.7. Lemma 4.8. Suppose that Σ is orientable, G is (2, 2)-tight and Q is a quadrilateral face of G such that neither GQ,v1 ,v3 nor GQ,v2 ,v4 is (2, 2)-sparse. Then Q has a non degenerate boundary. Furthermore, if H1 and H2 are blockers for GQ,v1 ,v3 respectively GQ,v2 ,v4 , then both H1 and H2 are type 2 blockers and H1 ∩ H2 = ∅. Proof. The non degeneracy of the boundary walk of Q follows immediately from Lemmas 4.5, 4.6 and 4.7. Now suppose that one of the blockers, say H1 , is of type 1 and suppose that v2 6∈ H1 . Then v4 ∈ H1 ∩ H2 . So γ(H1 ∪ H2 ) = γ(H1 ) + γ(H2 ) − γ(H1 ∩ H2 ) ≤ 2 + γ(H2 ) − 2 = γ(H2 ). Now if H2 is also type 1 then γ(H1 ∪ H2 ) = 2. However v1 , v2 , v3 ∈ H1 ∪ H2 but H1 ∪ H2 does not contain one of e1 , e2 which contradicts the sparsity of G. Similarly if H2 is type 2, then γ(H1 ∪ H2 ) ≤ 3, but H1 ∪ H2 does not contain either of e1 , e2 , again contradicting the sparsity of G. So both H1 and H2 are type 2 blockers. Moreover v1 , v2 , v3 , v4 ∈ H1 ∪ H2 but e1 , e2 , e3 , e4 6∈ H1 ∪ H2 . Now 2 ≤ γ(H1 ∪ H2 ∪ {e1 , e2 , e3 , e4 }) = γ(H1 ) + γ(H2 ) − γ(H1 ∩ H2 ) − 4 = 2 − γ(H1 ∩ H2 ) So γ(H1 ∩ H2 ) ≤ 0 which implies that H1 ∩ H2 = ∅.  4.3. Simple loops in surfaces. Now we briefly digress to review some necessary terminology and facts from low dimensional topology. Proofs of all of the assertions below can be found in (or at least easily deduced from) many sources (for example [6]). A loop in a surface Σ is a continuous function α : S 1 → Σ. We say that α is simple if it is injective. We say that α is non separating if Σ − α(S 1 ) has the same number of connected components as Σ. Given simple loops α, β in Σ, recall that the geometric intersection number is defined by i(α, β) = min |α′ (S 1 ) ∩ β ′ (S 1 )| where α′ , respectively β ′ , varies over all simple loops that are homotopic to α, respectively β. If i(α, β) 6= 0 then both α and β are essential: that is to say they are not null homotopic. If i(α, β) = 1 then both α and β are non separating in Σ. In the special case that Σ is the torus, if i(α, β) = 0 and i(β, δ) = 0 then i(α, δ) = 0. Given a simple loop α in a surface Σ we say that Σ − α(S 1 ) is the surface obtained by cutting along α. Given a surface Σ with boundary we can cap a boundary component by gluing a copy of a closed disc to the surface along the given boundary component. 7 If Σ is an orientable surface of genus g and α is a non separating simple loop in Σ then we form Σα by removing a tubular neighbourhood of α and then capping the two resulting new boundary components. Clearly Σα is an orientable surface of genus g − 1. Suppose that G is a Σ-graph and let F be a face of G. Further suppose that α is a non separating loop in Σ such that α(S 1 ) ⊂ F . By cutting and capping Σ along α we can form a Σα -graph, denoted Gα , which has the same underlying graph as G. Observe that all faces of Gα except the one(s) corresponding to F are also faces of G. Finally some terminology. If G = (Γ, ϕ) is a Σ-graph and α is a loop in Σ, we say that α is contained in G if α(S 1 ) ⊂ ϕ(|Γ|). Now we return to the situation of Lemma 4.8. Suppose that Q is a quadrilateral in G as in the statement of that lemma. We say that the quadrilateral Q is blocked. By Lemma 3.1 the blocker H1 is connected so it is possible to find a simple walk from v1 to v3 in H1 . By concatenating the geometric realisation of this walk with the diagonal of Q joining v3 and v1 we obtain a simple loop in Σ, which we denote by α1 . Note that we can choose different parameterisations of this loop, but this ambiguity will make no difference in our context. Similarly we construct another simple loop, denoted α2 , by concatenating a walk in H2 with the diagonal of Q that joins v4 and v2 . Now since H1 ∩ H2 is empty by Lemma 4.8, we can choose these loops so that they intersect transversely at exactly one point (where the diagonals meet). Thus these loops have geometric intersection number equal to one. In particular, we note that both α1 and α2 must be non separating loops in Σ. These loops will play an important role in the following sections. 5. Irreducible surface graphs Let G be a (2, 2)-tight Σ-graph. In light of Lemmas 4.1 and 4.3 we say that G is irreducible if it has no digons, no triangles and if, for every quadrilateral face of G, both of the possible contractions result in graphs that are not (2, 2)-sparse. For each of the contractions described in Section 4 there are the corresponding vertex splitting moves. More precisely, if G′ = GD , respectively G′ = GT,e , respectively G′ = GQ,u,v for some digon D, respectively triangle T and edge e ∈ ∂T , respectively quadrilateral Q and vertices u, v ∈ ∂Q, then we say that G is obtained from G′ by a digon, respectively triangle, respectively quadrilateral split. Thus every (2, 2)-tight Σ-graph can be constructed from some irreducible by applying a sequence of digon/triangle/quadrilateral splits. Our goal is to identify, for various surfaces, the set of irreducibles. Conjecture 5.1. If Σ is a surface with finite genus and finitely many boundary components and punctures, then there are finitely many distinct isomorphism classes of irreducible (2, 2)tight Σ-graphs. We will address some special cases of Conjecture 5.1 in this and later sections. Let S be the 2-sphere. Theorem 5.2. If G is a (2, 2)-tight S-graph with at least two vertices then G has at least two faces of degree at most 3. In particular, any (2, 2)-tight S-graph can be constructed from a single vertex by a sequence of digon and/or triangle splits. Proof. By Lemma 3.1, G is connected and therefore cellular. Since G has at least two vertices, f0 = 0. Also f1 = 0 by sparsity, so by Theorem 3.3, we see that 2f2 + f3 ≥ 4.  The case of plane graphs is similarly straightforward. 8 Figure 2. The two non cellular irreducible torus graphs. Here and in subsequent diagrams we use the standard representation of the torus as a square with opposite edges identified appropriately. Note that by cutting the torus along a non separating loop these graphs can also be viewed as graphs in the twice punctured sphere. Corollary 5.3. If G is a (2, 2)-tight R2 -graph with at least two vertices then G has at least one cellular face of degree at most 3. In particular, any (2, 2)-tight R2 -graph can be constructed from a single vertex by a sequence of digon and/or triangle splits. Proof. Cap (i.e fill in the puncture of) the non cellular face of G and then apply Theorem 5.2.  Theorem 5.2 and Corollary 5.3 are implicit already in other places in the literature, we include them here for completeness. We note that in both cases the quadrilateral splitting move is not required in the inductive characterisation. Now let A be the twice punctured sphere R2 − {(0, 0)}. Observe that for any positive integer n, it is straightforward to construct an A-graph that has no digons or triangles, but has n quadrilateral faces. So, in contrast to the cases of the sphere or plane, we do require the quadrilateral contraction move in order to have finitely many irreducible (2, 2)-tight A-graphs. There are two obvious examples of irreducible (2, 2)-tight A-graphs, with one vertex and two vertices respectively: see Figure 2. Theorem 5.4. If G is an irreducible (2, 2)-tight A-graph, then G is isomorphic to one of the A-graphs shown in Figure 2. Proof. There are two cases to consider. First suppose that G does not separate the two punctures of A. In this it follows easily from Theorem 5.2 that if G has at least vertices then it has at least one triangular face and so is not irreducible. Now suppose that G does separate the punctures of A. Clearly G has exactly two non cellular faces. By capping these two faces, we create a (2, 2)-tight S-graph G̃. This graph satisfies 2f2 + f3 = 4 + f5 + 2f6 + · · · and since all but two of the faces of G̃ are also faces of the irreducible G, it follows that the two exceptional faces of G̃ are digons and all other faces are quadrilateral faces of G. Thus it suffices to show that there cannot be any quadrilateral faces in G. For a contradiction, suppose that Q is a quadrilateral. Since G is irreducible, both possible contractions of Q are blocked and we infer the existence of simple loops α1 and α2 as described at the end of Section 4. Recall that these loops intersect transversely at exactly one point and thus α1 is non separating in A. However the Jordan Curve Theorem tells us that any simple loop in A must be separating.  9 6. Subgraphs of irreducibles Throughout this section, let Σ be an orientable boundaryless surface and let G = (Γ, ϕ) be an irreducible (2, 2)-tight Σ-graph. The goal of this section is to show that any (2, 2)-tight subgraph of G is also irreducible. Let H = (Λ, ϕ||Λ| ) be a subgraph of G. We say that H is inessential if there is some embedded open disc U ⊂ Σ such that ϕ(|Λ|) ⊂ U . If there is no such disc then H essential. Observe that if F is a cellular face of G that has a non degenerate boundary walk, then ∂F is inessential: let U be an open disc neighbourhood of the embedded closed disc F . We also note that if H is inessential and connected then it has at most one non cellular face F . Moreover if we cut and cap along a maximal non separating set of loops in F we obtain an S-graph which, in this section, we will denote by Ĥ. Let K1 be the graph with one vertex and no edges. Let K2 be the complete graph on two vertices. For n ≥ 2 let Cn be the n-cycle graph (in particular C2 has exactly two parallel edges). Lemma 6.1. Suppose that H is a subgraph of G whose underlying graph is isomorphic to either C2 or C3 . Then H is essential. Proof. Suppose that the underlying graph of H is isomorphic to C2 . The other case is similar. Suppose that H is inessential. Let U be an open disc that contains ϕ(Λ). Clearly there is a digon face D of H that is contained in U . Now let K be the S-graph obtained by cutting and capping the external face of intG (D). By Theorem 3.4, γ(K) = 2 and by Theorem 3.3, K has at least two faces of degree at most 3. One of these faces is also a face of G contradicting the irreducibility of G.  Lemma 6.2. Suppose that H is an inessential subgraph of G and that γ(H) = 2. Then the underlying graph of H is K1 . Proof. Suppose that H has at least two vertices. Then by Theorem 3.3, Ĥ has at least two faces of degree at most 3. If one of these is a triangle or a digon with non degenerate boundary then the underlying graph of H contains a copy of C2 or C3 which contradicts Lemma 6.1. Therefore Ĥ must have two digon faces both of which have degenerate boundaries. However, as pointed out in Section 4.1, no S-graph can have more than one degenerate digon.  Lemma 6.3. Suppose that H is an inessential subgraph of G and that γ(H) = 3. Then the underlying graph of H is K2 . Proof. By Theorem 3.3, Ĥ satisfies 2f2 + f3 = 2 + f5 + 2f6 + · · ·. As in the proof of Lemma 6.2 we see that Ĥ cannot have a triangle or a digon with non degenerate boundary. So the only possibility is that Ĥ has a digon face with degenerate boundary. As pointed out in Section 4.1„ there is only one S-graph with a degenerate digon face and its underlying graph is indeed K2 .  The case of a subgraph isomorphic to C4 is a little more involved. Lemma 6.4. Suppose that H is an inessential subgraph of G whose underlying graph is isomorphic to C4 . Then H is the boundary of some quadrilateral face of G. Proof. Suppose that U is an embedded disc containing ϕ(|Λ|) and let R be the face of H that is contained in U . First observe that γ(H) = 4, so by Theorem 3.4, γ(intG (R)) ≤ 4. Now, by Lemma 6.1, intG (R) has no digons or triangles and it follows easily from Theorem 3.3 that 10 γ(intG (R)) = 4 and that all the cellular faces of intG (R) are quadrilaterals: that is to say that intG (R) is in fact a quadrangulation of R. Now, let Q (with boundary vertices v1 , v2 , v3 , v4 ) be a quadrilateral face of intG (R) that is contained in R. Since G is irreducible, we have blockers H1 and H2 for the two possible contractions of Q, as described in Lemma 4.8. Also we have simple loops α1 and α2 as described in Section 4. These loops intersect transversely at one point in Q. If w1 , w2 , w3 , w3 are the vertices of ∂R in cyclic order, it follows that one of the loops, say α1 , contains w1 and w2 and that α2 contains w2 and w4 . Thus α2 divides R into disjoint open subsets R1 and R3 (see Figure 3) where w1 , v1 ∈ R1 and w3 , v3 ∈ R3 . Now we can decompose the blocker H1 as Ke ∪ K1 ∪ K3 , where Ke = extG (R) ∩ H1 , K1 is the part of H1 contained in R1 and K3 is the part of H1 contained in R3 . It is clear that Ke ∩ K1 = {w1 } and Ke ∩ K3 = {w3 }. Therefore, by (1), 3 = γ(H1 ) = γ(Ke ) + γ(K1 ) + γ(K3 ) − 4. Using the sparsity of G it follows that at least one of γ(K1 ) or γ(K3 ) is equal to 2. Now K1 and K3 are both inessential subgraphs of G since R1 and R3 are both embedded closed discs in Σ. It follows from Lemma 6.2 that at least one of K1 or K3 is a single vertex. So either v1 = w1 or v3 = w3 . We have shown that at least one of v1 or v3 actually lies in the boundary of R. Similarly at least one of v2 or v4 lies in the boundary of R. Thus we have shown that if Q is any quadrilateral face of G contained in R then ∂Q and ∂R share at least one edge. Now it is an elementary exercise to show that in any quadrangulation of R that has this property, either there are no quadrilaterals properly contained in R, or some quadrilateral has a boundary vertex with degree 2. Clearly, by Lemma 4.8, no quadrilateral face of the irreducible graph G can have a boundary vertex of degree 2. It follows that there are no quadrilateral faces of G that are properly contained in R and so R is itself a face of G.  We say that a subgraph H = (Λ, ϕ||Λ| ) of G is annular if it is essential and ϕ(|Λ|) is contained in some embedded open annulus of Σ. Let B be the (unique) (2, 2)-tight graph with 3 vertices, one of which has degree 4. Lemma 6.5. Suppose that H is a subgraph of G whose underlying graph is isomorphic to B. Then H is not annular. Proof. Suppose, seeking a contradiction, that H is annular. Let U be an open annulus containing ϕ(|Λ|) and let R be the face of H that is contained in U . Observe that γ(H) = 2, so by Theorem 3.4, γ(intG (R)) = 2. Let K be the S-graph obtained by cutting and capping the external faces of intG (R) (there could be more than one in this case). Now K is a (2, 2)-tight S-graph with two digon faces. Since all other faces of K are also faces of the irreducible G, it follows easily from Theorem 3.3 that all other faces of K are quadrilaterals. Thus, all faces of G that are contained in R are in fact quadrilaterals. Now we can argue, using a straightforward modification of the argument from the proof of Lemma 6.4, that any quadrilateral face of G that is contained in R must in fact share a boundary edge with R. Again, following the proof of Lemma 6.4 it follows that R itself must be a face of G. However this contradicts Lemma 4.8 where we showed that any quadrilateral face of an irreducible has a non degenerate boundary.  Now the main result of this section: a tight subgraph of an irreducible is also irreducible. 11 w2 v2 Ke v1 w1 v3 α1 K1 K3 w3 Ke α2 v4 w4 Figure 3. From the proof of Lemma 6.4: the shaded region represents the blocker for the contraction GQ,v1 ,v3 . Theorem 6.6. Suppose that G = (Γ, ϕ) is an irreducible (2, 2)-tight Σ-graph and Λ is a (2, 2)-tight subgraph of Γ. Then H = (Λ, ϕ||Λ| ) is an irreducible Σ-graph. Proof. We see that H cannot have any triangle or digon, since the boundary of such a face would contradict Lemma 6.1. Now suppose that Q is a quadrilateral face of H. It is not clear, a priori, that the boundary of Q is non degenerate, so we must prove that before proceeding. Applying Lemma 4.6 to H, we see that there are no repeated edges in the boundary of Q. Thus the only possibility for a degenerate boundary is that one vertex is repeated and that ∂Q has underlying graph isomorphic to B. If ∂Q is inessential then, since B contains a copy of C2 , this contradicts Lemma 6.1. On the other hand, if ∂Q is inessential then it must be annular and this contradicts Lemma 6.5. Thus we see that in fact Q must have a non degenerate boundary. By Lemma 6.4 this means that Q is also a face of G and so there are blockers H1 , H2 as described by Lemma 4.8. Now consider the Σ-graph K = H1 ∪ H2 ∪ ∂Q. This is (2, 2)-tight, so, by Lemma 3.2, K ∩ H is also (2, 2)-tight. Now, K ∩ H = (H1 ∩ H) ∪ (H2 ∩ H) ∪ ∂Q. Using (1), H1 ∩ H2 = ∅, H1 ∩ H ∩ ∂Q = {v1 , v3 } and H2 ∩ H ∩ ∂Q = {v2 , v4 }, we have 2 = γ(K ∩ H) = γ(∂Q) + γ(H1 ∩ H) + γ(H2 ∩ H) − γ(H1 ∩ H ∩ ∂Q) − γ(H2 ∩ H ∩ ∂Q) = 4 + γ(H1 ∩ H) + γ(H2 ∩ H) − 4 − 4. 12 Thus γ(H1 ∩ H) + γ(H2 ∩ H) = 6. If γ(H1 ∩ H) = 2 then (H1 ∩ H) ∪ {v2 } ∪ {e1 , e2 } would be a type 1 blocker for the contraction GQ,v1 ,v3 , contradicting Lemma 4.8. So γ(H1 ∩ H) ≥ 3 and similarly γ(H2 ∩ H) ≥ 3. It follows that γ(H1 ∩ H) = γ(H2 ∩ H) = 3 and that H1 ∩ H and H2 ∩ H are blockers for the contractions HQ,v1 ,v3 and HQ,v1 ,v3 respectively. Thus both possible contractions of Q are blocked in H as required.  For example, suppose that Γ is the simple (2, 2)-tight graph obtained by adding a vertex of degree two to K4 . Is it possible to embed Γ into the torus to create an irreducible torus graph? If so, how many nonisomorphic embeddings exist? There are several possible embeddings of Γto consider, however we can significantly narrow the search space by observing that since K4 is tight, by Theorem 6.6, any irreducible embedding of Γ must extend an irreducible embedding of K4 . It is not difficult to show that, up to isomorphism there is a unique irreducible embedding of K4 in the torus (see figure 10). Thus any irreducible embedding of Γ must restrict to this embedding of K4 . Using this observation, it is not difficult to show that, up to isomorphism there are exactly two distinct irreducible torus embeddings of Γ. 7. Irreducible torus graphs S1 S1 Let T = × be the torus. Throughout this section let G = (Γ, ϕ) be an irreducible (2, 2)-tight T-graph. Our goal in this section is to show that there are only finitely many isomorphism classes of such graphs by establishing an upper bound for the number of vertices of G. In the case that G is not cellular we will see that we can essentially reduce the problem to the sphere or the annulus. If G is cellular then using Theorem 3.3 and f2 = f3 P = 0 we see that G satisfies f5 + 2f6 + 3f7 + 4f8 = 4 and fi = 0 for i ≥ 9. Since |V | = 2 + i≥2 fi , the problem reduces to establishing a bound for the number of quadrilateral faces that an irreducible T-graph can have. First we deal with the non cellular case. Lemma 7.1. Suppose that G is not cellular. Then Γ is either isomorphic to K1 or to C2 . Furthermore, in the latter case, G is annular. Proof. Since Γ is connected it is clear G has a single non cellular face. By cutting along a non separating loop in this face we obtain an A-graph Ĝ. Observe that any face of Ĝ that is not also a face of G is non cellular. It follows that Ĝ is an irreducible A-graph. Now the conclusion follows from Theorem 5.4.  For the remainder of the section, assume that G is cellular. Let Q be a quadrilateral face of G with boundary walk v1 , e1 , v2 , e2 , v3 , e3 , v4 , e4 , v1 . As described in Section 4 we have blockers H1 and H2 and simple loops α1 and α2 that intersect transversely at one point. Lemma 7.2. At least one of H1 or H2 is an inessential subgraph of G. Proof. Suppose that both are essential. Then there are non separating simple loops β1 contained in H1 and β2 contained in H2 . Now H1 ∩ H2 = ∅, so i(β1 , β2 ) = 0. However, it is also clear that i(α1 , β2 ) = i(α2 , β1 ) = 0. As pointed out in Section 4.3 this implies that i(α1 , α2 ) = 0, contradicting the fact that these curves intersect transversely at one point.  For the remainder of the section, suppose that H1 is an inessential blocker. By Lemma 6.3, the graph of H1 is K2 . Furthermore we will assume that H2 is a maximal blocker with respect 13 H1 v1 H2 v4 α1 v2 α2 v3 Figure 4. A quadrilateral face with an essential blocker. The shaded region represents the essential blocker H2 . to inclusion and let J be the face of H2 , that contains v1 , v3 . See Figure 4 for an illustration of these assumptions in the case where H2 is an essential blocker. Lemma 7.3. Any face of H2 that is not J is also a face of G. Proof. Suppose that F 6= J is a face of H2 . Then γ(H2 ∪ intG (F )) ≤ γ(H2 ) = 3, by Theorem 3.4. Also v1 , v3 6∈ intG (F ), since F 6= J. If follows that H2 ∪ intG (F ) is a blocker for GQ,v2 ,v4 and so by the maximality of H2 , intG (F ) ⊂ H2 as required.  Next we want to examine the structure of H2 . It turns out that there are exactly ten distinct possibilities. If H2 is inessential then, by Lemma 6.3 it has graph K2 (Figure 5). On the other hand, if H2 is essential we have the following. Lemma 7.4. Suppose that H2 is essential. Then it is isomorphic to one of the nine torus graphs shown in Figures 6 and 7. Proof. Since γ(H2 ) = 3, it is connected by Lemma 3.1. Let K be the S-graph obtained by cutting and capping H2 along a non separating loop in J. Clearly K has two exceptional faces J + and J − such that all other faces of K are faces of G (using Lemma 7.3). Now, since J + and J − are the only faces of K that could have degree less than 4, Theorem 3.3 implies that K satisfies (3) 2f2 + f3 = 2 + f5 + 2f6 and fi = 0 for i ≥ 7. There are two cases to consider. (a) There is no quadrilateral face of G in H2 . There are various subcases: (1) |J + | = |J − | = 2. Then, from Equation 3 we get f5 + 2f6 = 2. So either f5 = 0 and f6 = 1 and we have the example shown in Figure 6 (a), or, f5 = 2 and f6 = 0 and we have one of the examples shown in Figure 6 (b) or (c). (2) |J + | = 2 and |J − | = 3. Then we have f5 = 1. There is one possibility: Figure 6 (d). (3) |J + | = |J − | = 3. In this case, Equation 3 implies that J + and J − are the only faces of K. So we have the example shown in Figure 6 (e). 14 (4) |J + | = 2 and |J − | = 4. In this case, Equation 3 implies that J + and J − are the only faces of K and we have the example shown in Figure 6 (f). (b) There is some quadrilateral face of G in H2 . This case requires a little more effort as we must first establish that there is no more than one such face. Let G′ = ∂Q ∪ H1 ∪ H2 . Clearly G′ is (2, 2)-tight and so by Theorem 6.6 it is also irreducible. Suppose that R is a quadrilateral face of G, with boundary vertices w1 , w2 , w3 , w4 , that is contained in H2 (and so is also a face of G′ ). By Lemma 7.2 we know that there is a blocker for one of the contractions of R in G′ whose graph is K2 . Without loss of generality assume that a blocker L1 for the contraction G′R,w1 ,w3 has graph K2 . Now we claim that L1 ⊂ H2 . If not then it is clear that L1 must intersect H1 . Since the vertices of L1 are both in H2 this contradicts H1 ∩ H2 = ∅, thus establishing our claim. Now consider a maximal blocker, L2 , for the contraction G′R,w2 ,w4 . We have 3 = γ(L2 ) = γ(L2 ∩ (∂Q ∪ H1 )) + γ(L2 ∩ H2 ) − γ(L2 ∩ (∂Q ∪ H1 ) ∩ H2 ) = γ(L2 ∩ (∂Q ∪ H1 )) + γ(L2 ∩ H2 ) − γ(L2 ∩ {v2 , v4 }) Now it is clear that {v2 , v4 } ⊂ L2 since L2 is connected, so we have (4) γ(L2 ∩ H2 ) = 7 − γ(L2 ∩ (∂Q ∪ H1 )) Furthermore, it is also clear that L1 separates v2 from v4 in H2 , so L2 ∩ H2 has at least two components. Also L2 ∩ (∂Q ∪ H1 ) is a subgraph of ∂Q ∪ H1 that contains the vertices v2 , v4 . It follows easily that γ(L2 ∩ (∂Q ∪ H1 )) ≥ 3 with equality only if L2 ∩ (∂Q ∪ H1 ) = ∂Q ∪ H1 . Therefore the only way that (4) can be satisfied is that ∂Q ∪ H1 ⊂ L2 and L2 ∩ H2 has exactly two components X2 ∋ v2 and X4 ∋ v4 such that γ(X2 ) = γ(X4 ) = 2. In particular it follows from Theorem 6.6 and Lemma 7.1 that the underlying graph of X2 , and also of X4 , is isomorphic to K1 or C2 . Now since L1 also separates w2 and w4 in H1 we can, without loss of generality, assume that v2 , w2 ∈ X2 and v4 , w4 ∈ X4 . Let Z2 , respectively Z4 , be the maximal (2, 2)-tight subgraph of H2 that contains v2 , respectively v4 . By Lemma 3.2 we see that X2 ⊂ Z2 and X4 ⊂ Z4 . Furthermore we see that since Z2 and Z4 are both disjoint from α1 , they are either annular or inessential. By Lemma 7.1, Z2 has graph K1 (inessential case) or C2 (annular case). Similar comments apply to Z4 . Now the argument in the paragraph above shows that every quadrilateral face of H2 has a boundary vertex in Z2 and a diagonally opposite vertex in Z4 . It follows easily that there is at most one such quadrilateral face in H2 . Now we can argue as in case (a) but with the proviso that there is exactly one quadrilateral face, R, of H2 that is also a face of G. We observe that there is a cycle of length 3 in H2 (formed by two edges of ∂R and the inessential blocker for R) and so also in K. It is not hard to see that it follows that K must have at least two faces of odd degree: at least one on either ‘side’ of the cycle of length 3. We find the following subcases. (1) |J + | = |J − | = 2. From Equation (3) we have f5 + 2f6 = 2. Since K has some face of odd degree we can rule out the possibility f5 = 0, f6 = 1. Therefore f5 = 2 and f6 = 0. There is only one possibility for H2 : Figure 7 (a). (2) |J + | = 2 and |J − | = 3. Then, as in case (a) we have f5 = 1 and there is one possibility: Figure 7 (b). 15 Figure 5. The unique inessential blocker (a) (b) (c) (d) (e) (f) Figure 6. Essential blockers with no quadrilateral face (a) (b) (c) Figure 7. Essential blockers with a quadrilateral face (3) |J + | = |J − | = 3. In this case, Equation 3 implies that R, J + and J − are the only faces of K: Figure 7 (c). (4) |J + | = 4 and |J − | = 2. Since K must have at least two faces of odd degree, Theorem 3.3 would imply that there is a triangle or digon in K that is also a face of G, contradicting its irreducibility. Thus this subcase cannot arise.  It remains to rule out the possibility of a quadrilateral face that is neither Q nor a face of H2 . In fact we can prove something a little more general than that. 16 Lemma 7.5. Let K be a (2, 2)-tight subgraph of G and suppose that F is a cellular face of K. There is no quadrilateral face of G properly contained within F . Proof. Suppose that R, with vertices w1 , w2 , w3 , w4 , is a quadrilateral face of G properly contained within F and let B1 and B2 be blockers for contractions of R in G. If B1 ⊂ F , then since F is cellular, B1 would separate w2 from w4 which contradicts B1 ∩ B2 = ∅. Therefore B1 is not contained in F or equivalently, since B1 is connected, B1 ∩ K 6= ∅. Similarly B2 ∩ K 6= ∅. Now, let M = ∂R ∪ B1 ∪ B2 and observe that M is (2, 2)-tight and therefore, by Lemma 3.2, M ∩ K is also (2, 2)-tight. Now it is clear that M ∩ K = (B1 ∩ K) ∪ (B2 ∩ K) ∪ E(∂R ∩ K). Therefore (5) 2 = γ(M ∩ K) = γ(B1 ∩ K) + γ(B2 ∩ K) − |E(∂R ∩ K)| Now, we observe that |E(∂R ∩ K)| ∈ {0, 1, 2, 4} since K is an induced subgraph of G. If |E(∂R ∩ K)| = 4 then clearly R must be a face of K which contradicts our assumption that R is properly contained within F . On the other hand if |E(∂R ∩ K)| ≤ 1, then (5) yields γ(B1 ∩ K) + γ(B2 ∩ K) ≤ 3 which contradicts the fact that both B1 ∩ K and B2 ∩ K are nonempty. Finally if |E(∂R ∩ K)| = 2 then it is clear that K contains exactly three of the vertices w1 , w2 , w3 , w4 . However in this case (5) implies that γ(B1 ∩ K) = γ(B2 ∩ K) = 2. It follows that K contains at most one of the vertices w1 , w3 , otherwise (B1 ∩ K) ∪ {w2 } would span a type 1 blocker for GR,w1 ,w3 , contradicting Lemma 4.8. Similarly K contains at most one of the vertices w2 , w4 . Thus K contains at most two of the vertices w1 , w2 , w3 , w4 yielding the required contradiction.  Finally we have our main theorem about torus graphs. Theorem 7.6. Suppose that G is an irreducible (2, 2)-tight T-graph. Then G has at most two quadrilateral faces. Proof. Suppose, as above, that Q is a quadrilateral face of G, with maximal blockers H1 and H2 . Also assume that H1 is inessential. We have seen that there is at most one other quadrilateral face of G contained among faces of H2 . Now let K = ∂Q ∪ H1 ∪ H2 . Clearly K is a (2, 2)-tight subgraph of G. Now we consider the faces of K that are not also faces of G. If H2 is also inessential then there is at most one such face and this face is cellular of degree 8. If H2 is essential then there at most two such faces and each such face is cellular and has degree at least 5. So by Lemma 7.5, there is no quadrilateral face of G that is not also a face of K.  Corollary 7.7. There are finitely many distinct isomorphism classes of irreducible (2, 2)-tight torus graphs. In particular any such irreducible torus graph has at most eight vertices. Proof. We may as well assume that G is cellular, since in the nonPcellular case we know that G has at most two vertices. Since γ(G) = 2 we have |V | = 1 + 41 ifi , so we must maximise P ifi . Since G is irreducible, fi = 0 for i = 0, 1, 2, 3 and f4 ≤ 2. From TheoremP3.3 we have f5 + 2f6 + 3f7 + 4f8 = 4 and fi = 0 for i ≥ 9. Clearly the maximum value for ifi is attained by having f4 = 2, f5 = 4 and fi = 0 for i 6= 4, 5. In that case |V | = 8. Now there are finitely many isomorphism classes of (2, 2)-tight graphs with at most eight vertices. Moreover, for each such graph, there are finitely many isomorphism classes of torus graphs with that underlying graph.  17 Figure 8. Irreducible torus graphs with eight vertices Figure 9. Irreducible torus graphs with at most three vertices 7.1. Identifying irreducibles. Given Corollary 7.7, a naive algorithm to find all the irreducibles mentioned therein would be (1) Find all (2, 2)-tight graphs with at most 8 vertices. (2) For each such graph, find all isomorphism classes of torus embeddings. (3) Eliminate all embeddings that are not irreducible. It is impractical to carry out this procedure without the assistance of a computer as step (1) will already yield many thousands of distinct graphs, each of which could have many different torus embeddings. However, since we have a lot of structural information about irreducibles, we can narrow the search space significantly. For example, it is clear from the proof of Corollary 7.7 that any irreducible with 8 vertices must have 2 quadrilateral faces, 4 faces of degree 5 and no other faces. Moreover, we know that each quadrilateral face has one essential blocker and one other blocker which must be one of the 10 graphs described in Section 7. It is not too difficult to deduce that any 8 vertex irreducible must be isomorphic to one of the examples shown in Figure 8. Similarly for torus graphs with at most 4 vertices there are relatively few possibilities for the underlying graph: 13 in total. Now, using Lemmas 6.1, 6.4 and 6.5 we can easily deduce that an irreducible with at most 4 vertices is isomorphic to one of the examples shown in Figures 9 or 10. For the cases of 5, 6 and 7 vertices the naive this approach yields a relatively manageable problem in computational graph theory. We have used the computer algebra system SageMath [18] to automate much of the search process in these cases. The interested reader can find full details of the search algorithm and its implementation at [5]. As a result of this computation we have the following. Theorem 7.8. There are 116 distinct isomorphism classes of (2, 2)-tight irreducible torus graphs.  18 Figure 10. Irreducible torus graphs with four vertices 8. Application: contacts of circular arcs In this section we describe an application to the study of contact graphs. The foundational result in this area is the well known Koebe-Andreev-Thurston Circle Packing Theorem ([12]). More recently, contact graphs for many different classes of geometric objects have been studied, with various restrictions placed on the allowed contacts. See for example [1, 8, 9] We consider contact graphs arising from certain families of curves in surfaces of constant curvature. We begin by giving a model for a general class of contact problems and then specialise to a case of particular interest. Let α : [0, 1] → Σ be a curve. We say that α is non selfoverlapping if it is injective on the open interval (0, 1). Now suppose that α, β : [0, 1] → Σ are distinct curves in Σ. We say that α and β are non overlapping if α((0, 1)) ∩ β((0, 1)) = ∅. Let C be a collection of curves in Σ having the following properties • Every α ∈ C is non selfoverlapping. • For every distinct α, β ∈ C, α and β are non overlapping. We want to construct a combinatorial object that describes the contact properties of such a collection. In order to do this we impose some further non degeneracy conditions on C as follows. • α(0) 6= α(1) for every α ∈ C • For every distinct α, β ∈ C, {α(0), α(1)} ∩ {β(0), β(1)} is empty. In other words, we allow the end of one curve to touch another curve (or to touch itself), but the point that it touches cannot be an endpoint of that curve. We say that C is a non degenerate collection of non overlapping curves. Note that if a collection fails the non degeneracy conditions, it can typically be made degenerate by an arbitrarily small perturbation. A contact of C is a quadruple (α, β, x, y) where α, β ∈ C, x ∈ {0, 1} and α(x) = β(y). Now we can define a graph ΓC as follows. The vertex set is C and the edge set is T , the set of contacts of C. We define the incidence functions s, t : T → C by s(α, β, x, y) = α and t(α, β, x, y) = β. The quadruple (C, T , s, t) is a directed multigraph and we let ΓC be the graph obtained by forgetting the edge orientations. We can construct an embedding |ΓC | → Σ as follows. For β ∈ C, suppose that t−1 (β) = {(α1 , β, x1 , y1 ), · · · , (αk , β, xk , yk )}. Let Jβ be a nonempty closed subinterval of [0, 1] with the following properties. (1) {y1 , · · · , yk } ⊂ Jα . (2) 0 ∈ Jβ if and only there is no contact (β, γ, 0, y) in T . (3) 1 ∈ Jβ if and only there is no contact (β, γ, 1, y) in T . 19 α β α γ γ β Figure 11. The construction of the contact graph associated to a collection of curves. On the left we have a collection of curves. The bold section of α represents α(Jα ). On the right is the corresponding graph with edge orientations as indicated. In other words Jβ is a subinterval that covers all the ‘points of contact’ in β together with any endpoints of β that do not touch a curve. Now we observe that β : Jβ → Σ is a homeomorphism onto its image. So it follows from the Jordan-Schoenflies Theorem that Σ/β(Jβ ) is homeomorphic to Σ. Furthermore, since β(Jβ ) ∩ δ(Jδ ) = ∅ for β 6= δ it follows that Σ is homeomorphic to Σ/ ∼ where ∼ is the equivalence relation that collapses each β(Jβ ) to a point, for all β ∈ C. Using this homeomorphism we construct an embedding by mapping each vertex of ΓC (i.e. element of C) to the corresponding point of Σ/ ∼. Since an edge of ΓC is a contact (α, β, x, y), we can construct the corresponding edge embedding by using the restriction of α to the component of [0, 1] − Jα that contains x: see Figure 11. We are interested in the recognition problem for contact graphs: can we find necessary and/or sufficient conditions for a surface graph to be the contact graph of a collection of curves? Typically we are looking for conditions for which there are efficient algorithmic checks. As noted in the introduction, there are efficient algorithms for checking whether or not a given graph is (2, l)-sparse. See [13] and [10] for details. Hliněný ([11]) has shown that a plane graph admits a representation by contacts of curves if and only if it is (2, 0)-sparse. We note that he states the result only for simple planar graphs. It is easy to see however the the result applies to the multigraph model of contact graphs described above and to graphs embedded in arbitrary surfaces. We include the statement to provide some context for our later result. Theorem 8.1. Let G be a Σ-graph. Then G ∼ = GC for some C as above if and only if G is (2, 0)-sparse.  It is worth noting here that graphs studied in [11] and elsewhere are typically the intersection graphs of a collection of curves. While this is usually appropriate in the plane, for non simply connected surfaces we propose that it is more natural to consider the contact graph as above. We observe that the intersection graph is obtained from the surface graph GL by taking the underlying graph, deleting all loops and replacing any sets of parallel edges by a single edge. Now we suppose that Σ is also equipped with a metric of constant curvature. In this context we can distinguish many interesting subclasses of non selfoverlapping curves. For example, a circular arc is a curve of constant curvature and a line segment is a locally geodesic curve. For collections of such curves the representability question can depend on the embedding of the graph and not just the graph itself (in contrast to Theorem 8.1). For example, the graph C2 cannot be represented by a collection of line segments in the flat plane. However, if C2 is 20 Figure 12. A torus graph and a corresponding CCA representation in the flat torus. The orientation of the graph edges is the orientation induced by the CCA representation. embedded as a non separating cycle in the torus, then it is easy to construct a representation of the resulting surface graph as a collection of line segments in the flat torus. Given a Σ-graph G and a non degenerate non overlapping collection of circular arcs C such that G ∼ = GC we say that C is a CCA representation of C (abbreviating Contacts of Circular Arcs). See Figure 12 for an example in the torus. Alam et al. ([1]) have shown that any (2, 2)-sparse plane graph has a CCA representation in the flat plane. We prove an analogous result for the flat torus. First we need a lemma to show that every sparse surface graph can be obtained by deleting only edges from a tight surface graph. Lemma 8.2. Suppose that Σ is a connected surface, l ≤ 2 and G is a (2, l)-sparse Σ-graph. There is some (2, 2)-tight Σ-graph H such that V (H) = V (G) and G is a subgraph of H. Proof. Clearly it suffices to show that if γ(G) ≥ l + 1 then we can add an edge e within some face of G so that G ∪ {e} is (2, l)-sparse. Now if G has no tight subgraph then we can add any edge without violating the sparsity count. So we assume that G has some nonempty tight subgraph. Let L be a maximal tight subgraph of G. If L spans all vertices of G then L = G and G is already tight, so we assume that L is not spanning in G. Since Σ is connected there is some face F of G whose boundary contains vertices u ∈ L and v 6∈ L. Let e be a new edge that joins u and v through a path in F . We claim that G ∪ {e} is (2, l)-sparse. If not then there must be some tight subgraph K of G such that u, v ∈ K. But K ∩ L is nonempty, so by Lemma 3.2, K ∪ L is (2, l)-tight. This contradicts the maximality of L.  Theorem 8.3. Every (2, 2)-sparse torus graph admits a CCA representation in the flat torus. Proof. First observe that edge deletion is CCA representable: just shorten one of the arcs slightly. So by Lemma 8.2 it suffices to prove the theorem for (2, 2)-tight torus graphs. To that end we must show that (a) each irreducible (2, 2)-tight torus graph has a CCA representation. (b) if G → G′ is a digon, triangle or quadrilateral contraction move and G′ has a CCA representation, then G also has a CCA representation. In other words the relevant vertex splitting moves are CCA representable. 21 v2 v2 v3 v1 z v4 v4 Figure 13. A CCA representation of a quadrilateral splitting move. The contact graph of the configuration on the left is a quadrilateral contraction of the contact graph of the configuration on the right. For (a) it is possible to give an explicit CCA representation for each of the 116 irreducibles listed in Appendix A. We will not describe those here but below we shall explain a simple method to make these constructions easily. Full details are given in [19]. For (b), see Figure 13 for an illustration of the CCA representation of the quadrilateral split. It is easily seen that any quadrilateral split is similarly CCA representable. The digon and triangle vertex splits are also representable and indeed have already been dealt with in the plane context in [1]. We observe that the constructions described there work equally well for torus graphs.  In order to construct the CCA representations of the irreducibles mentioned in the proof we can make use of topological Henneberg moves. We remind the reader that a Henneberg vertex addition move is the operation of adding a new vertex to a graph and two edges from that vertex to the existing graph. Note that we allow the two new edges to be parallel. Moreover in the context of surface graphs we insist that the new vertex is placed in some face of the existing graph and the two edges are incident with vertices in the boundary of that face. We refer to such an operation as a topological Henneberg move. Clearly a Henneberg move is the inverse operation to divalent vertex deletion. It is well known (and elementary) that divalent vertex deletions preserve (2, l)-sparseness for all l. On the other hand Henneberg moves preserve (2, l)-sparseness for l ≤ 2, and for l = 3 if we also insist that the new edges are not parallel. It turns out that there are just 12 irreducibles that have no vertices of degree 2. In Figure 15 we give diagrams of each of these torus graphs and in Figure 16 we give sample CCA representations of each of these in the flat torus. We observe that each of the 116 irreducible graphs can be constructed by a sequence topological Henneberg moves from one of the torus graphs in Figure 15: indeed one easily sees that at most five Henneberg moves are required. It remains to show that the required topological Henneberg moves are CCA representable. A CCA representation of a topological Henneberg move is illustrated in Figure 14. In general of course, topological Henneberg moves can fail to be CCA representable given a fixed representation of the initial graph. It is easy to construct an CCA representation of a graph that has a highly nonconvex region and so may not admit the required circular arc to realise 22 Figure 14. A CCA representation of a topological Henneberg move. The bold arc on the right represents the new vertex, that touches two of the initial arcs. G11 G41 G42 G51 G52 G61 G62 G63 G64 G65 G71 G81 Figure 15. The irreducibles that have no vertex of degree 2. a topological Henneberg move. However, it is readily verified that given the CCA representations in Figure 16, it is possible to represent all the necessary Henneberg moves that are required to construct CCA representations of the full set of 116 irreducible graphs. See [19] for complete details of this. Finally we observe that allowing divalent vertex additions we have the following inductive construction for (2, 2)-tight torus graphs. 23 C11 C14 C24 C15 C25 C16 C26 C36 C46 C56 C17 C18 Figure 16. CCA representations of the 12 irreducible torus graphs with no vertices of degree two: Cji is a CCA representation of Gij from Figure 15. Theorem 8.4. If G is a (2, 2)-tight torus graph then G can be constructed from one of the torus graphs in Figure 15 by a sequence of moves each of which is either a digon split, triangle split, quadrilateral split or a divalent vertex addition.  24 References [1] Md. Jawaherul Alam, David Eppstein, Michael Kaufmann, Stephen G. Kobourov, Sergey Pupyrev, André Schulz, and Torsten Ueckerdt, Contact graphs of circular arcs, Algorithms and data structures, Lecture Notes in Comput. Sci., vol. 9214, Springer, Cham, 2015, pp. 1–13. MR3677547 [2] D. W. Barnette and Allan L. Edelson, All 2-manifolds have finitely many minimal triangulations, Israel J. 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MR3022162 [18] The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.1), 2017, available at https://www.sagemath.org [19] Qays Shakir, Sparse topological graphs, Ph.D. Thesis, National University of Ireland Galway, 2020. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Ireland. Email address: [email protected] Department of Mathematics and Computer Studies, Mary Immaculate College, Thurles, Ireland. Email address: [email protected] Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, U.K. Email address: [email protected] Middle Technical University, Baghdad, Iraq Email address: [email protected] 25