A bijection between unicellular and bicellular maps

In this paper we present a combinatorial proof of a relation between the generating functions of unicellular and bicellular maps. This relation is a consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it can be proved using representation theory of the symmetric group. Here we give a bijective proof by rewiring unicellular maps of topological genus $(g+1)$ into bicellular maps of genus $g$ and pairs of unicellular maps of lower topological genera. Our result has immediate consequences for the folding of RNA interaction structures, since the time complexity of folding the transformed structure is $O((n+m)^5)$, where $n,m$ are the lengths of the respective backbones, while the folding of the original structure has $O(n^6)$ time complexity, where $n$ is the length of the longer sequence.

Eq. (1.1) is a consequence of the Schwinger-Dyson equation of matrix theory [3,11]. It can also be proved by extending the representation theoretic framework of Zagier [12]. To the best of our knowledge, our bijection represents the first combinatorial proof of eq. (1.1).
The motivation for this paper stems from the algorithmic folding problem of RNA-pseudoknot structures over one and two backbones [9,1]. The folding of RNA molecules means to identify some minimum energy configuration of a given sequence. These configurations are subject to certain constraints on how two nucleotides can bond [9,1,4]. RNA structures over two backbones are called RNA-RNA interaction structures [5,6] and of importance in the context of many biochemical, regulatory activities.
Theorem 1 provides a rewiring algorithm transforming bicellular maps into certain unicellular maps. It thus allows to reduce the folding problem of RNA-RNA interaction structures to that of RNA-pseudoknot structures over one backbone, see Fig. 1. This rewiring is of practical interest, since the time complexity of folding the rewired interaction structure is given by O((n+m) 5 ), where n, m are the lengths of the respective backbones. The direct folding of the interaction structure however has a time complexity of O(n 6 ) where n is the length of the longer sequence. Since there exist an abundance of "small RNA" interactions between a large and a very small RNA structure, the O((n + m) 5 ) time complexity is oftentimes much smaller than O(n 6 ) [1].
The paper is organized as follows: first we recall some basic facts about diagrams, fatgraphs, unicellular and bicellular maps. We shall work with planted unicellular and double planted bicellular maps. These plants are additional vertices of degree one and emerge naturally in the context of RNA as there is a 5 ′ to 3 ′ orientation of the molecular backbone. Modulo Poincaré-duality the plant "marks" the beginning and ending of the backbone. Interestingly, the plants themselves play a key role in the combinatorial construction.
Second we dissect the bijection into three separate maps by introducing a certain partition of the set of unicellular maps of fixed topological genus. Then we prove our two main lemmas. These show that, with respect to the above mentioned partition, a unicellular map either corresponds uniquely to a pair of unicellular maps of lower genus (Lemma 1) or to a unique bicellular map of lower genus (Lemma 2).  We third prove the main result, Theorem 1, by combining the two lemmas and give eq. (1.1) as an enumerative corollary.

Some basic facts
2.1. Unicellular maps and bicellular maps. Definition 1. A unicellular map, u with n edges is a triple u = ([2n], α, σ), where α is an involution of [2n] without fixed points and σ is a permutation of [2n] such that γ = α • σ has only one cycle. The elements of [2n] are called half-edges of u. The cycles of α and σ are called the edges and the vertices of u, respectively. The permutation γ is called the face or boundary component of u.
Given a unicellular map u = ([2n], α, σ), its associated graph G is the graph whose edges are given by the cycles of α, vertices by the cycles of σ. We can consider a G-edge as a ribbon whose two sides are labeled by the half-edges as follows: if a half-edge h belongs to a cycle e of α and a certain v of σ, then h is the right-hand side of the ribbon corresponding to e, when entering v.
We draw the graph G in such a way that around each vertex v, the counterclockwise ordering of the half-edges belonging to the cycle v is given by the cycle v. This ordering of half-edges enriches the combinatorial graph G to a ribbon graph or fat graph G. Clearly, a fat graph G with one boundary component is tantamount to the unicellular map u, see Fig. 2(a). γ = α • σ is interpreted as the cycle of half-edges visited when making the tour of the graph, keeping the graph on its left.

Definition 2.
A planted unicellular map having n edges is a unicellular map u = ([2n + 2], α, σ), such that (1, 2n + 2) is a cycle of α. We shall label the face of u as and denote (2n R ) as p, the plant of u.
Given a planted unicellular map u the face γ induces a linear order < u on H via: Suppose u has J vertices, v 1 , v 2 , . . . , v J . Then there is a natural equivalence relation of half-edges, given by h ∼ α(h) and in particular, 1 R ∼ 2n R .
For each vertex v j , j ∈ J, let min u (v j ) denote the first half-edge where γ arrives at v j . We write v j , reading the v j -half-edges counter clockwise and starting at h 1 j = min(v j ): v j = (h 1 j , . . . , h mj j ). In particular, the vertex containing the half-edge 1 R is denoted by v 1 . The order < u induces thus a linear order < v on the vertices by setting Definition 3. A planted bicellular map b having n edges is a triple b = (L, β, τ ), where L is a set of cardinality (2n + 4) such that β is a fixed-point free involution containing the cycles (1 R , m R ) and ((m + 1) R , 2n R ). β • τ consists of the two cycles The elements of [2n] are called half-edges of b and there exists some half-edge . We furthermore assume the following linear order of the half-edges of the two faces ω 1 and ω 2 : As in the case of unicellular maps, a bicellular map, b = (L, β, τ ), has an associated connected graph G ′ , whose edges are given by the cycles of   RNA structures and interaction structures contain more information that just the set of contacts between nucleotides. Aside form the 5 ′ to 3 ′ orientation of the backbone itself there is in addition a fixed ordering of the backbone relative to the base pairs. This orientation implies that the contact graph together with the backbone gives rise to a natural fatgraph structure as shown in Fig. 5, G [7,8]. We obtain again the counterclockwise traveling of the half-edges around each vertex as for unicellular maps. This fattening works analogously for RNA diagrams over two backbones [2,1], see Fig. 6.
Euler's characteristic equation shows that, without affecting the topological type of the fatgraph G, one can collapse each backbone into a single vertex with the induced fattening. In other words, there is an equivalent fatgraph representation of RNA-diagrams having a vertex for each respective backbone. Moreover, we may enrich this representation by adding an arc that labels the 5 ′ to 3 ′ end of the backbone. We refer to this arc as rainbow-arc or just rainbow.  Clearly the n arcs of the diagram determine after fattening 2n halfedges and the fatgraph consists of a pair (α ′ , σ ′ ) together with the additional rainbow arc. Then, the mapping is a bijection mapping vertices into boundary components. Topologically this is the Poincaré dual, mapping a fatgraph over one-backbone with rainbow into an planted unicellular map, see Fig. 7. The scenario is analogus for RNA-diagrams over two backbones, where we insert two rainbows over the respective backbones. Formation of the Poincaré dual, as illustrated in Fig. 8 generates a planted bicellular map.
We next replace v 1 by v 1,u2 = (h 1 1 , h 3 1 , . . . , h m 1 ). Then the set of all u-vertices different from v 1 that are contained in H 2 = H \ H 1 and v 1,u2 , together with the restriction of α to H 2 form a new map, u 2 . The latter is by construction unicellular and its boundary component is given by Considering the Euler characteristics we can conclude (g + 1) = g 1 + g 2 .
By construction, ψ • θ = id˙ 0≤g 1 ≤g+1,0≤j≤n (Ug 1 ,j ×Ug+1−g 1 ,n−j ) and The second result relates bicellular maps and unicellular maps of types I and II; the key idea is analogous to that in Lemma 1.
Let B g,n denote the set of bicellular map of genus g with n edges. We observe that B g,n can be written as B g,n = B I g,n∪ B II g,n , where B I g,n denotes the set of bicellular maps of genus g with n edges in which the two plants p 1 and p 2 are incident to two different vertices and B II g,n denotes its complement.
We shall relabel γ as

Lemma 2. There exists a bijection
η : B I g,n∪ B II g,n −→ U I g+1,n+1∪ U II g+1,n+1 , and η induces by restriction the two bijections and η II : B II g,n −→ U II g+1,n+1 .  Proof. Let b = (L, β, τ ) be a planted bicellular map of genus g having n edges with plants p 1 and p 2 and tour β • τ = ω 1 • ω 2 . Let and v i,b , for 1 ≤ i ≤ J be the set of its vertices.
Consider the two vertices (m R b ) and v p2,b , where (m R b ) is the plant p 1 and v p2,b denote the cycle containing half-edge (m + 1) R b . I.e. we have The key operation consists in "gluing " This generates the unicellular map, η(b), with boundary component and the new vertex Note that in case of v 1,b = v p2,b , the gluing does not merge these b-vertices.
The case of b ∈ B II g,n is analogous. Then there also exists some 1 < x < α(1) such that α(x) > α(1) holds, whence η(b) ∈ U II g+1,n+1 . In Fig. 11 and Fig. 12 we depict what happens if we glue (m R b ) into v p2,b in these respective cases.
We next construct the inverse of η. To this end, let u = (H, α, σ) be a unicellular map of genus g having (n + 1) edges with plant p and face Suppose first u ∈ U I g+1,n+1 , then Since α(1 R b ) and 1 R b are incident to the same vertex and h 2 1 = α(1 R b ), we can conclude that 1 R b is a half-edge of v 1 . In other words, there exist some half-edge h a 1 (namely h a 1 = 1 R b ), for some 1 < a < j, such that α(h a 1 ) = m R b .
We now introduce the mapping ς obtained by "cutting" the edge This process generates the new vertex set and represent the two boundary components of the new map. Furthermore, since for u ∈ U I g+1,n+1∪ U II g+1,n+1 , there exists some k, 1 < k < α(1), such that α(1) < α(k). Since u has the boundary component given in eq. (3.4), we have 1 = 1 R b and α(1) = m b . Accordingly, there exists some 1 R b < k < α(1 R b ), such that α(1 R b ) < α(k). ς(u) has the two new boundary components ω 1 and ω 2 , i.e. there exist some k ∈ ω 1 , such that β(k) ∈ ω 2 and ς(u) is bicellular map. By construction, if u ∈ U I g+1,n+1 , then ς(u) has its two plants incident to two distinct vertices, whence ς(u) ∈ B I g,n . In case of u ∈ U II g+1,n+1 then ς(u) has its two plants incident to one vertex and ς(u) ∈ B II g,n .
Thus η is a bijection that induces by construction the bijections η I and η II and the lemma follows.

The main result
Theorem 1. Let U g,n and B g,n denote the sets of unicellular and bicellular maps containing n edges and genus g. Then there is a bijection (U g1,j × U g+1−g1,n−j )∪ B g,n −→ U g+1,n+1 .