Topological constructions for tight surface graphs

TOPOLOGICAL CONSTRUCTIONS FOR TIGHT SURFACE GRAPHS arXiv:1909.06545v1 [math.CO] 14 Sep 2019 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 to configurations of circular arcs in the spirit of the Koebe-Andreev-Thurston circle packing theorem. 1. Introduction Given a graph Γ and a non-negative integer k, define γk (Γ) = kn − m where Γ has n vertices and m edges. Intuitively we think of γk as a functional that measures the kdimensional ‘degrees of freedom’ of the graph. If l ≤ k, we say that Γ is (k, l)-sparse if γk (Γ′ ) ≥ l for every nonempty subgraph Γ′ of Γ. If, in addition, γk (Γ) = l, then we say that Γ is (k, l)-tight. The definitions can be modified a little to allow l > k although we will not need that generality. These families of graphs have been the subject of much research, starting with the well known tree packing theorem of Nash-Williams and Tutte: (k, k)-tight graphs are precisely those that can be decomposed into k edge disjoint spanning trees ([17] and [25]). More recently Lee and Streinu have investigated the matroids whose independent sets are the (k, l)-sparse graphs ([14]). They also describe a family of algorithms, generalising earlier work of Hendrickson and Jacobs ([10]), called pebble game algorithms. These determine, among other things, whether or not a given graph is sparse or tight. Much of the interest in these graphs has been inspired by the mathematical theory of structural rigidity, latterly known simply as geometric rigidity theory. For example Pollaczek-Geiringer and Laman ([21], [22] and [13]) showed that (2, 3)-tight graphs are precisely those for which a minimally rigid plane bar and joint framework model exists. Such graphs are now typically referred to as Laman graphs. Many generalisations and variations of this fundamental theorem have appeared since. We also mention recent work by Nixon, Owen and the third author on the rigidity of bar joint frameworks satisfying various geometric constraints. Their work naturally involves inductive characterisations of various classes of (2, l)-sparse graphs ([18], [19] and [20]). 2010 Mathematics Subject Classification. 05C10, 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 fourth author gratefully acknowledges the financial support from the Iraqi Ministry of Higher Education and Scientific Research and Middle Technical University, Baghdad. 1 Another important reference point for the present work is the investigation of plane Laman graphs by Fekete, Jordán and Whiteley ([8]). We have adapted and extended some of their ideas and techniques to genus one and to different sparsity counts. Sparse and tight graphs play significant roles in other aspects of geometric graph theory. We note in particular recent work on a connection with contact graphs of curves. A starting point here is the observation that (2, 0)-tight plane graphs arise naturally as the contact graphs of configurations of curves in the plane. Other sparsity counts can arise in connection with configurations of curves with specific geometric properties–for example the contact graphs of configurations of line segments are exactly the plane Laman graphs. Recently Alam et al. have investigated various contact representations of sparse planar graphs ([1]). Among other results, they show that (2, 2)-tight plane graphs can always be represented by configurations of circular arcs. Our geometric application in Section 9 is a genus one analogue of their result. Our interest in higher genus constructions is partly motivated by the connection to gain graphs and related sparsity counts although these do not play an explicit role in the present work. In particular torus graphs, which are the subject of later sections of the present paper, correspond via the universal covering construction to doubly periodic plane graphs. Thus our geometric application can be viewed as a partial characterisation of a certain type of crystallographic structure in the flat plane. A recurring theme in all of this literature is the importance of inductive arguments. Of course, such arguments are ubiquitous throughout discrete mathematics. However in the context of sparse graphs and geometric graph theory, particular types of inductive ‘moves’ play a prominent role. These are the Henneberg moves and the vertex splitting moves. In the case of a Henneberg move on a graph, we delete some specified number of edges and add a vertex of a specified degree. Vertex splitting on the other hand generally involves replacing a vertex by one or more vertices and some edges between them and distributing the incident edges of the original vertex among the new vertices in some way. In certain geometric contexts vertex splitting operations are better behaved than Henneberg moves: in Section 9 we will see such an application. Furthermore in the setting of topological graph theory, vertex splitting, or its inverse edge contraction, are very natural and extensively studied operations. In the study of triangulations and quadrangulations of surfaces vertex splitting has an important role. Barnette and Edelson ([2]) have shown that any compact surface has finitely many triangulations that are minimal with respect to edge contractions (which are inverse to vertex splitting). In a similar vein Nakamoto and Ota ([16]) have given a bound for the size of an irreducible quadrangulation of a compact boundaryless surface. Indeed these results are partly the inspiration for Conjecture 5.1 below. 1.1. Outline of the paper and summary of main results. Section 2 summarises the necessary background material from graph theory and topology in particular. Section 3 contains a review of the basic facts about (2, l)-sparse graphs. We also prove a key result, Theorem 3.5, which provides an important connection between the topological 2 and combinatorial properties of a tight surface graph. Special cases of this have appeared elsewhere (notably in [8], [3] and [4]). In Section 4 we give a careful analysis of certain operations which later form the basis of our main inductive construction results. Having defined our class of inductive moves we turn, in Section 5, to the investigation of those surface graphs that are irreducible with respect to the specified set of operations. We conjecture that for any surface of finite type there are finitely many such irreducibles and we are easily able to establish this in the case of the sphere, plane and twice punctured sphere. In Section 6 we show that a tight subgraph of an irreducible surface graph is also irreducible (Theorem 6.6). This is an important result for our later work and we note that we can prove this for arbitrary orientable surfaces (not just for small genus) which we hope may prove useful in future work on higher genus. In Section 7 we consider the class of irreducibles on the torus which, in contrast to the three other surfaces considered earlier, requires considerable effort. We are able to establish an upper bound for the size of an irreducible, thus reducing the problem to a finite search. In Section 8 we outline the data structures and algorithms that we used in a computer assisted search to find all examples. Finally in Section 9 we give a geometric application of our inductive construction for (2, 2)-tight torus graphs. 2. Graphs, surfaces, embeddings and rotation systems In this section we review the basic concepts that we need from topological graph theory. 2.1. Graphs and surface graphs. A graph is a quadruple Γ = (V, E, s, t) where V, E are finite sets (vertices and edges respectively) and s, t : E → V are functions that encode the incidence relation between the edges and vertices. Thus, a priori, graphs can have multiple parallel edges and/or loop edges. In other contexts such an object might be referred to as a one dimensional simplicial set. If v ∈ {s(e), t(e)} then we say that the vertex v is incident to the edge e or vice versa. We say that edges e, f are adjacent if they are incident to a common vertex and we say that vertices u, v are adjacent if they are incident to a common edge. Given E ′ ⊂ E, V (E) is the set of vertices spanned by E ′ . Given V ′ ⊂ V , E(V ) is the set of edges spanned by V ′ . A walk of length k in Γ is a sequence w = v1 , e1 , v2 , e2 , · · · , ek , vk+1 where for i = 1, · · · , k, ei is incident to both vi and vi+1 . Note that in the case that ei is a loop edge, some extra care is required to specify which direction of ei is intended. However since loop edges cannot arise in the context of our later results we ignore this ambiguity in the notation. We say that w is a cycle if vk+1 = v1 and we say that such a cycle is simple if that is the only repeated vertex. The geometric realisation of Γ is the compact topological space |Γ| = V ∪ (E × [0, 1])/ ∼ where (e, 0) ∼ s(e) and (e, 1) ∼ t(e). Suppose that Σ is a surface (a real two dimensional smooth manifold). A Σ-graph is a pair G = (Γ, ϕ) where Γ is a graph and ϕ : |Γ| → Σ is an embedding (that is to say a continuous injective map). Suppose that for i = 1, 2, Gi = (Γi , ϕi ) is a Σi -graph. We say that G1 and G2 are isomorphic if there is a homeomorphism 3 h : Σ1 → Σ2 and a graph isomorphism l : Γ1 → Γ2 such that h ◦ ϕ1 = ϕ2 ◦ |l|. Up to isomorphism, surface graphs can be described by a combinatorial data structure called a rotation system and it is common in the topological graph theory literature to work exclusively with rotation systems: see Section 8 for the basic definition and [15] for a more thorough treatment of rotation systems and combinatorial maps. Generally we will use topological descriptions of the objects and trust that the reader can, if desired, make the appropriate translation to the language of rotation systems and combinatorial maps. In Section 8 we do make use of rotation systems in order to describe a suitable data structure for making computations with surface graphs. Note that we will apply some of the standard terminology of graph theory to surface graphs with the understanding that we are referring to the underlying graph where appropriate. For example if G = (Γ, ϕ) we say that G is connected if Γ is connected in the standard sense of undirected graphs. Similarly we will understand vertices and edges of G to mean vertices and edges of Γ, or possibly their images under ϕ. We will also apply standard set theoretic operations such as subset, union or intersection, understanding that we refer to the underlying graph. For example if Gi = (Γi , ϕi ), i = 1, 2 are both subgraphs of a Σ-graph G = (Γ, ϕ), then G1 ∪ G2 is the Σ-graph (Γ1 ∪ Γ2 , ϕ||Γ1∪Γ2 | ), where Γ1 ∪ Γ2 = (V1 ∪ V2 , E1 ∪ E2 , s, t). 2.2. Deletions and contractions. Let e be an edge of Γ that is not a loop edge. As is standard in the topological graph theory literature, let G/e be the Σ-graph obtained by collapsing ϕ(e) to a single point. So the underlying graph of G/e is Γ/e (the standard graph theoretic edge contraction). Clearly ϕ induces an embedding |Γ/e| → Σ/ϕ(e) and we compose this with the canonical homeomorphism (using the assumption that e is not a loop edge) Σ/ϕ(e) → Σ to obtain an embedding |Γ/e| → Σ. Also, if e is any edge (loop or not) of G we define G − e to be the Σ-graph (Γ − e, ϕ||Γ−e|). In later sections we adopt the following notational convenience: if H is a subgraph of G and e is an edge of G then H/e = H − e = H in the case where e is not an edge of H. 2.3. Faces and subgraphs. A face of G is a component of Σ − ϕ(|Γ|). The boundary of F , denoted ∂F is the Σ-subgraph of G consisting of all those edges and vertices in F , the topological closure of F in Σ. There is a well defined collection of closed walks, called the boundary walks of F that cover the underlying graph of ∂F (see [9] for details). We say that F has a nondegenerate boundary if no vertex is repeated among all the boundary walks of F . We say that a face is cellular if it is homeomorphic to R2 and we say that G is cellular if every face of G is cellular. For a cellular face F , the degree, denoted |F | is the length of the unique boundary walk of F . We write fi for the number of cellular faces of degree i. Note that if Σ is connected then f0 ≥ 1 if and only if Σ is a sphere and Γ is a single vertex. Suppose that G = (Γ, ϕ) is a Σ-graph as above. A subgraph Ω of Γ induces a Σ-graph H = (Ω, ϕ||Ω| ) and we refer to H as a subgraph of G. Let F be a face of H. Let Λ be the subgraph of Γ consisting of all those vertices and edges of Γ whose image under ϕ is 4 contained in F . Define intG (F ) = (Λ, ϕ||Λ|). Observe that 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. Similarly, let Φ be the subgraph of Γ consisting of those vertices and edges of Γ whose image under ϕ is contained in Σ − F . Define extG (F ) = (Φ, ϕ|Φ| ) and observe that extG (F ) has one exceptional face, namely F , such that all other faces of extG (F ) are also faces of G. Finally we observe that ∂F = intG (F ) ∩ extG (F ) = intG (F ) ∩ H. 2.4. Simple loops in surfaces. A loop in a surface Σ is a continuous map α : S 1 → Σ. We say that α is simple if it is injective. We say that α is nonseparating if Σ − α(S 1 ) has the same number of connected components as Σ. 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. If Σ is a surface without boundary and of genus g and α is a nonseparating simple loop in Σ then we form Σα by cutting along α and then capping the two resulting boundary components. Clearly Σα is a surface without boundary of genus g − 1. 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. We review some basic facts about this invariant that we will need later. Proofs of all of the assertions below can found in (or at least easily deduced from) many sources (for example [7]). 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 nonseparating in Σ. Finally in the special case that Σ is the torus, if i(α, β) = 0 and i(β, δ) = 0 then i(α, δ) = 0. Suppose that G = (Γ, ϕ) is a Σ-graph and let F be a face of G. Further suppose that α is a nonseparating 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. In this way we can, given any Σ-graph G whose underlying graph is connected, construct a cellular graph G̃, by cutting and capping along a collection of nonseparating curves contained in the noncellular 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|). 5 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 elementary lemmas for later use. The proofs are straightforward and we omit them. Lemma 3.1. Suppose that Γ1 , Γ2 are subgraphs of Γ. Then (1) γ(Γ1 ∪ Γ2 ) = γ(Γ1 ) + γ(Γ2 ) − γ(Γ1 ∩ Γ2 )  Lemma 3.2. Suppose that Γ is (2, 2)-sparse and that γ(Γ′ ) ≤ 3 for some subgraph Γ′ of Γ. Then Γ′ is connected.  Lemma 3.3. 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.  Now we consider sparsity in the context of surface graphs. Theorem 3.4. 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 Euler polyhedral formula and the fact that P ifi = 2|E|.  Now our first significant new result. We note that related results and special cases of this have appeared elsewhere (notably [8] and [3]). Theorem 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(H ∪ intG (F )) ≤ γ(H). Proof. By Lemma 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 Lemma 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 )). 6 The last inequality above follows from applying the sparsity of G to the nonempty subgraph extG (F ).  Corollary 3.6. 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 3.5.  We remark that all of the results of this section admit straightforward adaptations to the function γk for any positive integer k. We have specialised to the case γ = γ2 since this will be our main interest later and we wish to avoid excessive notational clutter. 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. Let G be a Σ-graph. A digon, respectively triangle, respectively quadrilateral is a cellular face of degree two, respectively three, respectively four. 4.1. Digon contractions. 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 edge and Σ is a sphere. Lemma 4.1. G is (2, l)-sparse if and only if GD is (2, l)-sparse Proof. Let z be the vertex of GD that corresponds to the contracted edge e1 . It is clear that any subgraph of GD that does not contain z is isomorphic to a subgraph of G. Also if H is a subgraph of GD that does contain z then there is a subgraph K of G such that e1 , e2 ∈ K and (K/e1 ) − e2 = H. Thus γ(K) = γ(H). So we have shown that if G is sparse then so is GD . For the converse, suppose that H is a subgraph of G such that γ(H) < l. If {v1 , v2 } 6⊂ H, then H is isomorphic to a subgraph of GD . If {v1 , v2 } ⊂ H, then γ((H/e1 )−e2 ) ≤ γ(H) < l. So in either case GD is not sparse.  4.2. Triangle contractions. 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 . We note that a triangle in a (2, 2)-sparse surface graph necessarily has a nondegenerate boundary walk, since any degeneracy would entail a (forbidden) loop edge. 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 7 Proof. Let K be a subgraph of GT,e1 satisfying γ(K) ≤ l −1 and let z be the vertex of GT,e1 corresponding to the contracted edge e1 . Clearly z ∈ K, otherwise K is also a subgraph of G. Also, we can clearly assume that K is a spanning subgraph of G. Now let H be the maximal subgraph of G such that H/e1 −e2 = K. Now γ(K) ≥ γ(H)−1 with equality if and only if e2 6∈ H. But γ(K) ≤ l − 1 and γ(K) ≥ l, so we do indeed have equality. Since K and therefore H are spanning subgraphs, it follows that v3 6∈ H.  We refer to the graph H whose existence is asserted in Lemma 4.2 as a blocker for the contraction GT,e1 . As noted above, in a (2, 2)-sparse surface graph a triangle necessarily has a nondegenerate boundary walk. Thus 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 3.3, H1 ∪ H2 is (2, 2)-tight. This contradicts the sparsity of G.  4.3. Quadrilateral contractions. Finally, suppose that Q is a quadrilateral of G with 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. 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 a spanning subgraph, since K is a spanning 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 8 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. and since H is a spanning 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 . We adopt the notational convenience that v5 = v1 and e5 = e1 . Lemma 4.5. For i = 1, 2, 3, 4, vi 6= vi+1 . 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 entails the existence either of a vertex of degree one or 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. 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. Then 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 not a redundant hypothesis in the statements of Lemmas 4.6 or 4.7. Lemma 4.8. Suppose that Σ is orientable, G is (2, 2)-tight and Q is a quadrilateral face of G such neither GQ,v1 ,v3 nor GQ,v2 ,v4 is (2, 2)-sparse. Then Q has a nondegenerate boundary. 9 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 nondegeneracy 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 = ∅.  In the situation described in the statement of Lemma 4.8 we say that the quadrilateral Q is blocked. Observe that 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 . 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, 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 nonseparating 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. 10 Figure 2. The two noncellular 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 nonseparating loop these graphs can also be viewed as graphs in the twice punctured sphere. 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 3.2, G is connected and therefore cellular. Since G has at least two vertices, f0 = 0. Also f1 = 0 by sparsity, so by Theorem 3.4, we see that 2f2 + f3 ≥ 4.  The case of plane graphs is similarly straightforward. 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) the noncellular face of G and then apply Theorem 5.2.  Let A be the twice punctured sphere R2 − {(0, 0)}. 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. Then there is a unique noncellular face of G. By capping this face (i.e. filling in the two punctures) we create a cellular (2, 2)-tight S-graph G̃. As in the proof of Theorem 5.2, we see that either this graph has a single vertex or it has at least two faces 11 that are digons or triangles. In the latter case, one of these faces must also be a face of G and so, in this case, if G has at least two vertices then it is not irreducible. Now suppose that G does separate the punctures of A. Clearly G has exactly two noncellular 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 nonseparating in A. However the Jordan Curve Theorem tells us that that any simple loop in A must be separating.  We note 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. 6. Subgraphs of irreducibles Throughout this section, let Σ be an orientable surface and let G = (Γ, ϕ) be an irreducible (2, 2)-tight Σ-graph. 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 U then H essential. We observe that if F is a cellular face of G that has a nondegenerate boundary walk, then ∂F is inessential: let U be an open disc neighbourhood of the the embedded closed disc F . We also note that if H is inessential and is connected then it has at most one noncellular face F . Moreover if we cut and cap along a maximal nonseparating 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 3.5, γ(K) = 2 and by Theorem 3.4, 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 . 12 Proof. Suppose that H has at least two vertices. Then by Theorem 3.4, Ĥ has at least two faces of degree at most 3. If one of these is a triangle or a digon with nondegenerate 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 3.4, 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 nondegenerate 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 3.5, γ(intG (R)) ≤ 4. Now, by Lemma 6.1, intG (R) has no digons or triangles and it follows easily from Theorem 3.4 that 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 Lemma 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 13 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 . 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 graph ({u, v, w}, {e, f, g, h}, s, t), where s(e) = s(f ) = s(g) = s(h) = u, t(e) = t(f ) = v and t(g) = t(h) = w. 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 3.5, 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 3.4 that all other faces of K are quadrilaterals. Thus, all faces of G that are contained in R are in fact quadrilaterals. 14 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 nondegenerate boundary.  Now the main result of this section: a tight subgraph of an irreducible is also irreducible. 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 nondegenerate, 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 nondegenerate 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 3.3, K ∩ H is also (2, 2)-tight. Now, K ∩ H = (H1 ∩ H) ∪ (H2 ∩ H) ∪ ∂Q. Using Lemma 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. 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? There are several possible embeddings to 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 . Now, it is not hard to see that there is only one irreducible embedding of K4 in the torus, up to isomorphism. Moreover, one readily checks that there is no way to add the remaining vertex of Γ to this embedding 15 without creating a face of degree at most 4. Thus there is no irreducible embedding of Γ in the torus. 7. Irreducible torus graphs Let T = S 1 × S 1 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 3.4 and f2 = f3P = 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 bounding the number of quadrilateral faces that an irreducible T-graph can have. First we deal with the noncellular 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 noncellular face. By cutting along a nonseparating 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 noncellular. 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 nonseparating 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 2.4 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 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 3.5. 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. 16 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 . 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 or 7. Proof. Since γ(H2 ) = 3, it is connected by Lemma 3.2. Let K be the S-graph obtained by cutting and capping H2 along a nonseparating loop in J. Clearly K has two exceptional faces J + and J − that are not faces of G (using Lemma 7.3). Now, by Theorem 3.4, K satisfies (3) 2f2 + f3 = 2 + f5 + 2f6 and fi = 0 for i ≥ 7. Moreover, J + , J − are the only faces that can have degree 2 or 3. 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). (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). 17 (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. 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 3.3 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, Q, of H2 that is also a face of G. We observe that there is a closed walk 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. 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). 18 Figure 5. The unique inessential blocker (a) (b) (c) (d) (e) (f) Figure 6. Essential blockers with no quadrilateral face (3) |J + | = |J − | = 3. In this case, Equation 3 implies that Q, 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 3.4 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.  Finally we have our main theorem about torus graphs. 19 (a) (b) (c) Figure 7. Essential blockers with a quadrilateral face Theorem 7.5. Suppose that G is an irreducible (2, 2)-tight T-graph. Then G has at most two quadrilateral faces. Proof. There is much routine checking to rule out various possibilities so we will outline the main argument and omit some of the detail. Suppose that Q is a quadrilateral of G and that H1 and H2 are maximal blockers for the two possible contractions of G. By Lemma 7.2 we can assume that H1 is inessential and therefore has graph K2 . As above, let J be the face of H2 that contains v1 , v3 . Now we know that all other faces of H2 are also faces of G. In particular G is built up by adding vertices and edges to the T-graph G′ = ∂Q ∪ H1 ∪ H2 , but only vertices and edges inside J − Q. Now at most one of these faces of H2 is a quadrilateral. The only other possibility is that there is a quadrilateral face of G that is contained in J − Q. Say that there is such a face R. Now R must have blockers as described above: one inessential blocker and one other blocker that is one of the ten possibilities shown in Figures 5, 6 and 7. It is a straightforward if tedious task to show that there is no way to extend G′ by adding vertices and edges in J − Q so as to create a quadrilateral face R with the requisite pair of blockers. We give details of one such case. Suppose that G′ is the torus graph as shown in Figure 8: so H2 is the blocker from Figure 6 (a) and suppose, for a contradiction, that R is a quadrilateral face of G such that R ⊂ M where M is the the degree 5 face of G′ as indicated in Figure 8. Now, let L1 be an inessential blocker for the contraction GR,w1 ,w3 and let L2 be the blocker for the contraction GR,w2 ,w4 . First observe that since L1 together with the diagonal of R forms a nonseparating loop in T it is clear that one of w1 or w3 lies in ∂M, and that the other is adjacent to a vertex of ∂M via the edge of L1 (as illustrated in Figure 8). Now using an argument similar to that of Lemma 6.4 we can show that for each of v2 , v4 , either it lies in ∂M or is adjacent to a vertex in ∂M. Now however it is clear that this contradicts Lemma 7.6 below. 20 R Q M Figure 8. A case in the proof of Theorem 7.5. The solid graph is G′ . The dashed edges are a hypothetical quadrilateral together with its inessential blocker that are contained in the degree 5 face M of G′ . It is clear that there are several other cases to consider depending on the structure of G′ and on the placement of the hypothetical quadrilateral face R. However all cases can be dealt with in a similar way.  Lemma 7.6. Suppose that H is a (2, 2)-tight subgraph of G and that F is a cellular face of H with |F | ≥ 5. Suppose that R is a nondegenerate quadrilateral face of G that is contained in F and for every vertex v ∈ ∂R such that v ∈ F there is an edge e of G such that e is incident to v and to ∂F but e 6∈ ∂R. Then, in fact, v ∈ F for all vertices v ∈ ∂R. Proof. Observe that H is an induced subgraph of G, since both G and H are (2, 2)-tight. Now since |R| < |F |, not all vertices of ∂R belong to ∂F . So there must be at least one vertex of ∂R in F . On the other hand, if there is at least one vertex of ∂R in ∂F , and I is the subgraph of G spanned by H ∪ ∂R, it is easy to see that γ(I) ≤ 1, contradicting the sparsity of G. Thus all vertices of ∂R lie in F .  Corollary 7.7. There are finitely many distinct isomorphism classes of (2, 2)-tight torus graphs. In particular any such torus graph has at most eight vertices. Proof. We may as well assume that G is cellular, since in the noncellular P case we know 1 that G hasPat most two vertices. Since γ(G) = 2 we have |V | = 1 + 4 ifi , so we must maximise ifi . Since G is irreducible, fi = 0 for i = 0, 1, 2, 3 and f4 ≤ 2. From Theorem 3.4 P 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: see Figure 21 for examples for which these bounds are achieved. In that case |V | = 8. Now there are finitely 21 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.  8. Searching for irreducibles Following the proof of 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 21. 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 16 or 17. For the cases of 5, 6 and 7 vertices the naive approach becomes excessively laborious. We have used the computer algebra system SageMath [23] to automate much of the search process in these cases. We briefly outline the relevant data structures and algorithms here. 8.1. Data structures. In order to carry out our computer assisted search we needed to implement two key data structures, one to model graphs (the native SageMath Graph class is not particularly well adapted to our purposes), and one to model surface graphs. 8.1.1. Graphs. A dart (or half-edge) of a graph Γ = (V, E, s, t) is a pair (e, r) where e ∈ E and r ∈ {s, t}. Let D be the set of darts of Γ and observe that there is a partition V of D defined by Pv = {(e, r) ∈ D : r(e) = v}. There is another partition E of D defined by Qe = {(e, s), (e, t)}. Using this construction one readily sees that there is a correspondence between graphs and triples (X, P, Q) where X is a set, P is a partition of X and Q is a partition of X each of whose parts has two elements. We use this observation to implement a class in SageMath that accurately models our notion of graph. We have subclassed the native SageMath Graph class in order to take advantage of the built in graph theoretic functionality in SageMath. We have also implemented methods modelling various standard graph theoretic operations including vertex splitting and edge contractions. We adapted the built-in SageMath graph isomorphism checker to work with our subclass. 22 We also need to check (2, 2)-sparsity for our graphs and for this purpose we created a very basic implementation of the pebble game algorithm of Lee and Streinu ([14]). 8.1.2. Surface graphs. Let Sk be the group of permutations of the set {1, · · · , k}. An oriented rotation system is a pair (σ, τ ) where σ is some element of S2n and τ is a fixed point free involution in S2n . By a theorem of Edmonds ([6]) there is a correspondence between isomorphism classes of oriented rotation systems and isomorphism classes of cellular surface graphs whose underlying surface is orientable. For a contemporary exposition of this theory see [15]. Oriented rotation systems provide a convenient data structure for carrying out computations with surface graphs. In particular it is straightforward to compute boundary walks of faces, the genus and components of the underlying surface etc. Furthermore it is straightforward to implement the topological edge contraction and deletion operations discussed in Section 4, as well as other standard operations such as adding a new vertex within a specified face and adding a new edge that subdivides a face in a specified way. We have implemented this data structure in SageMath along with methods corresponding to the invariants and operations mentioned above. 8.2. The search algorithm. In order to search for irreducibles, we make use of Theorem 6.6 in a relatively straightforward way. Observe that if a (2, 2)-tight graph has a vertex of degree 2 then deleting this vertex yields a smaller (2, 2)-tight graph. If we know all possible irreducible torus embeddings of this smaller graph then we need only work out all possible ways to add back in the deleted vertex ‘topologically‘. That is to say we must add the vertex within a face together with edges to the required neighbours that must lie in the boundary of the face. This is substantially more efficient than searching among all possible embeddings of the original graph. On the other hand if the graph has minimum degree 3 then we carry out a brute force search among all possible rotation systems whose underlying graph is the given one. A slightly more formal description of this idea is given in Algorithm 1. 8.3. Computational results. Using a SageMath implementation of Algorithm 1 we have found the complete list of irreducible torus graphs: there are 116 in total. See [5] for the SageMath code together with data files describing the rotation systems corresponding to each of cellular irreducible torus graphs and corresponding diagrams. 9. 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]) which says that every plane simple graph can be realised as the contact graph of some arrangement of circles with nonoverlapping interiors in the Euclidean plane. Following this theorem, contact graphs arising in many other geometric contexts have been investigated. 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. 23 Algorithm 1 An inductive algorithm for finding all irreducibles with n vertices 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: Input: lists Gn−1 , respectively Gn , of all (2, 2)-tight graphs with n − 1, respectively n vertices, a list In−1 of all the irreducible torus graphs with n−1 vertices and a mapping fn−1 : In−1 → Gn−1 that maps each irreducible to its underlying graph. Output: A list In of all the irreducible torus graphs with n vertices together with a mapping fn : In → Gn . for Γ ∈ Gn do if Γ has a vertex of degree 2 then Let Θ be the graph obtained by deleting the vertex of degree 2 −1 for G ∈ fn−1 (Θ) do Identify any face whose boundary contains both neighbours of the deleted vertex. See if a new vertex can be added within the face and adjacent to the two neighbours without creating any face of degree 2, 3 or 4. Add all resulting rotation systems to the list In , check to see if any new entry is isomorphic to any existing one and remove the new one if it is. Update the mapping fn mapping all the new entries in In to the appropriate Henneberg extension of Θ in Gn . end for else Label the darts of Γ, 1, · · · , 2n and identify the partitions V, respectively E corresponding to the vertices, respectively edges. Let τ be the involution whose cycle partition is E. for each σ ∈ S2n whose cycle partition is V do Check that the rotation system (σ, τ ) corresponds to an irreducible torus graph. Check to see if it is isomorphic to any existing entry in In . If not then add to In and update fn appropriately. end for end if end for Let α : [0, 1] → Σ be a curve. We say that α is nonselfoverlapping 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 nonoverlapping if α((0, 1)) ∩ β((0, 1)) = ∅. Let C be a collection of curves in Σ having the following properties • Every α ∈ C is nonselfoverlapping. • For every distinct α, β ∈ C, α and β are nonoverlapping. 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 nondegeneracy 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. 24 α β α γ γ β Figure 9. 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, 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 nondegenerate collection of nonoverlapping curves. Note that if a collection fails the nondegeneracy 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 required incidence functions by s(α, β, x, y) = α and t(α, β, x, y) = α. Finally we can construct an embedding |ΓC | → Σ. 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 . 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 9. 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 [14] and [10] for details. 25 Figure 10. 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. Hliněný ([11]) has shown that a plane graph admits a representation by contacts of curves if and only if it is (2, 0)-sparse. This result easily generalises to other surfaces. We include the statement to provide some context for our later result. Lemma 9.1. 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 the definition of the contact graph used in [11] and elsewhere is different to the one we have given above. In the literature the contact graph is typically defined as the intersection graph of the collection of curves. This definition works well in the plane. However for non simply connected surfaces we propose that it is more natural to define the contact graph as above. Now we suppose that Σ is also equipped with a metric of constant curvature. In this context we can distinguish many interesting subclasses of nonselfoverlapping 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 Lemma 9.1). For example, if Π is the graph consisting of two vertices joined by two parallel edges, then Π cannot be represented by a collection of line segments in the flat plane. However, if Π is embedded as a nonseparating 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 nondegenerate nonoverlapping 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 10 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 9.2. 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. 26 v2 v2 v3 v1 z v4 v4 Figure 11. 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. 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 3.3, K ∪ L is (2, l)-tight. This contradicts the maximality of L.  Theorem 9.3. 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 9.2 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. 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 [24]. For (b), see Figure 11 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 27 Figure 12. 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. 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 13 we given diagrams of each of these torus graphs and in Figure 14 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 13: 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 12. In general of course, topological Henneberg moves can fail to be CCA representable given a fixed representation of the initial graph (Figure 15). However, it is readily verified that given the CCA representations in Figure 14, 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 [24] for complete details of this. 28 G11 G41 G42 G51 G52 G61 G62 G63 G64 G65 G71 G81 Figure 13. The irreducibles that have no vertex of degree 2. Finally we observe that allowing divalent vertex additions we have the following inductive construction for (2, 2)-tight torus graphs. Theorem 9.4. If G is a (2, 2)-tight torus graph then G can be constructed from one of the torus graphs in Figure 13 by a sequence of moves each of which is is either a digon split, triangle split, quadrilateral split or a divalent vertex addition.  Appendix A. Irreducible torus graphs Up to isomorphism there are 116 distinct irreducible (2, 2)-tight torus graphs. We describe them all in this section, grouped according to the number of vertices. Our descriptions will consist of a diagram of a representative of each class. Each diagram is a standard representation of a torus as a rectangle with the usual side identifications. For graphs with at most three vertices (Figure 16) the situation is straightforward. There are four (2, 2)-tight graphs and each has a unique irreducible embedding in the torus. Among the graphs with four vertices (Figure 17) we see the first instance of a irreducible torus graph with a quadrilateral face. Also we have a pair of nonisomorphic irreducibles that have the same underlying graph. Given that there are 23 irreducibles with five vertices and 47 with six, one might expect even larger numbers for the cases of seven or eight vertices. However irreducibles with seven, respectively eight, vertices must contain at least one, respectively two, quadrilateral faces. This follows easily from Theorem 3.4. The presence of these quadrilateral faces enforces a lot of additional structure, hence the relatively small number of examples in these cases. 29 C11 C14 C24 C15 C25 C16 C26 C36 C46 C56 C17 C18 Figure 14. CCA representations of the 12 irreducible torus graphs with no vertices of degree two: Cji is a CCA representation of Gij from Figure 13. 30 v u Figure 15. It is impossible to insert a circular arc that touches both u and v. This illustrates a topological Henneberg move that cannot be represented by contacts of circular arcs given this representation of the initial graph. Figure 16. Irreducible torus graphs with at most three vertices Figure 17. Irreducible torus graphs with four vertices 31 Figure 18. Irreducible torus graphs with five vertices 32 Figure 19. Irreducible torus graphs with six vertices 33 Figure 20. Irreducible torus graphs with seven vertices Figure 21. Irreducible torus graphs with eight vertices 34 References [1] Md. 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E-mail address: [email protected] Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, U.K. E-mail address: [email protected] Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, U.K. E-mail address: [email protected] School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Ireland. E-mail address: [email protected] 36