Quasi-tree expansion for the
Bollobás and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are discs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing Tutte’s concept of activity to quasi-trees, we prove a quasi-tree expansion of the Bollobás–Riordan–Tutte polynomial.
An oriented ribbon graph is a multi-graph (loops and multiple edges allowed) that is embedded in an oriented surface, such that its complement in the surface is a union of –cells. The embedding determines a cyclic order on the edges at every vertex. Terms for the same or closely related objects include: combinatorial maps, fat graphs, cyclic graphs, graphs with rotation systems, and dessins d’enfant (see [10, 2] and references therein).
The Tutte polynomial is a fundamental and ubiquitous invariant of graphs. Bollobás and Riordan  extended the Tutte polynomial to an invariant of oriented ribbon graphs in a way that takes into account the topology of the ribbon graph. In , they generalized it to a four-variable invariant of non-orientable ribbon graphs. We only consider the Bollobás–Riordan–Tutte polynomial for the orientable case, and henceforth all ribbon graphs will be oriented.
The Tutte polynomial can be defined by a state sum over all subgraphs, by contraction-deletion operations, and by a spanning tree expansion (see  for a detailed introduction)222The rank polynomial, formulated independently by H. Whitney , equals the Tutte polynomial after rescaling.. Tutte’s original definition in  was the spanning tree expansion, discussed below, which relies on the concept of activity of edges with respect to a spanning tree. In [2, 3] the Bollobás–Riordan–Tutte polynomial was shown to satisfy many essential properties of the Tutte polynomial, including a spanning tree expansion using Tutte’s activities.
For planar graphs, a spanning tree is a spanning subgraph whose regular neighborhood has one boundary component. For ribbon graphs, the analogue of a spanning tree is a quasi-tree, which is a spanning subgraph with one face, introduced in . Just as the spanning trees of a graph determine many of its important properties, topological properties of a ribbon graph are determined by the set of its quasi-trees. A natural question is whether the Bollobás–Riordan–Tutte polynomial has a quasi-tree expansion analogous to the spanning tree expansion for the Tutte polynomial.
In Section 2, we extend Tutte’s concept of activity (with respect to a spanning tree) to activity with respect to a quasi-tree by expressing the quasi-tree as an ordered chord diagram. For a genus zero ribbon graph, spanning trees and quasi-trees coincide, and the two notions of activity are the same. However, for ribbon graphs of higher genus, spanning trees are a proper subset of quasi-trees, and the two definitions of activity are quite distinct (see Remark 1 and Section 6).
In Section 3, we give an expansion of the Bollobás–Riordan–Tutte polynomial over quasi-trees. Each term in the expansion is determined by a particular quasi-tree as a product of factors with a topological meaning. In the genus zero case, we recover Tutte’s original spanning tree expansion. In general, our expansion is different from the spanning tree expansion given in . For example, in the case of one-vertex ribbon graphs, the spanning tree expansion is the same as the expansion over all subgraphs, but the quasi-tree expansion has fewer terms (see Remark 2). In addition, we show that a specialization of the Bollobás–Riordan–Tutte polynomial gives the number of quasi-trees of every genus.
2 Activities with respect to a quasi-tree
A ribbon graph can be considered both as a geometric and as a combinatorial object. Starting from the combinatorial definition, let be permutations of , such that is a fixed-point free involution and . We define the orbits of to be the vertex set , the orbits of to be the edge set , and the orbits of to be the face set . Let , and be the numbers of vertices, edges and faces of . The preceding data determine an embedding of on a closed orientable surface, denoted , as a cell complex. The set can be identified with the directed edges (or half-edges) of . Thus, is connected if and only if the group generated by acts transitively on . The genus of is called the genus of , . If has components, , where denotes the nullity of . Henceforth, we assume that is a connected ribbon graph. See Table 1 for an example of distinct ribbon graphs with the same underlying graph.
Any subgraph of the underlying graph of determines a ribbon subgraph of with underlying graph . We can construct its embedding surface as follows. A regular neighborhood of can be constructed on the surface by gluing discs at each vertex and rectangular bands whose midlines are the edges of . Let be the union of simple closed curves that bound such a regular neighborhood of on . By attaching a disc to every boundary component of this regular neighborhood, we construct , whose genus may be smaller than . By definition, the faces are the complementary regions of on . Thus, the components of correspond exactly to the faces . So if denotes the number of its components, . In particular, . Note that ribbon subgraphs may be disconnected. Also note that an isolated vertex cannot be represented by ; in this case, and .
A ribbon subgraph is called a spanning subgraph if . In this case, is a ribbon graph formed from by deleting some set of the edges, and keeping all vertices. The following concept was introduced and related to the determinant of a link in , and also related to Khovanov homology in . Following Definition 3.1 of ,
A quasi-tree is a connected spanning subgraph of with .
Equivalently, a spanning subgraph of is a quasi-tree if its regular neighborhood on has exactly one boundary component, . Also, a spanning connected ribbon graph is a quasi-tree if and only if . If the genus is zero, then the underlying graph of is a spanning tree. In Table 1, only the ribbon graph on the right is itself a quasi-tree.
Geometrically, is a simple closed curve on that divides as the connect sum of two surfaces with complementary genera. Traversing along , we can mark every half-edge of on its first encounter. Therefore, determines an ordered chord diagram , which is a circle marked with in some order, and chords joining all pairs . We say that is parametrized by . For example, see Figure 1.
Let be a connected ribbon graph. For every quasi-tree of , is parametrized by the ordered chord diagram , whose consecutive markings in the positive direction are given by the permutation:
Proof: Since is a quasi-tree, is one simple closed curve. If we choose an orientation on , we can traverse along successive boundaries of bands and vertex discs, such that we always travel around the boundary of each disc in a positive direction (i.e., the disc is on the left). If a half-edge is not in , will pass across it travelling along the boundary of a vertex disc to the next band. If a half-edge is in , traverses along one of the edges of its band. On , we mark a half-edge not in when passes across it along the boundary of the vertex disc, and we mark a half-edge in when we traverse an edge of a band in the direction of the half-edge. If the half-edge is not in , travelling along the boundary of a vertex disc, the next half-edge is given by . If the half-edge is in , traversing the edge of its band to the vertex disc and then along the boundary of that disc, the next half-edge is given by .
As is a quasi-tree, each of its half-edges must be in the orbit of its single face, while the complementary set of half-edges are met along the boundaries of the vertex discs. Since we mark all half-edges traversing , the chord diagram parametrizes .
We now define activity with respect to a quasi-tree:
Fix a total order on the edges of a connected ribbon graph . For every quasi-tree of , this induces an order on the chords of . A chord is live if it does not intersect lower-ordered chords, and otherwise it is dead. For any , an edge is live or dead when the corresponding chord of is live or dead; and is internal or external, according to or , respectively.
If is given by as above, we will order the edges by , though any ordering convention will work as well. For every quasi-tree of , the induced order on chords of is also given by . In Figure 1, we show such that the only edge live with respect to is , which is internally live.
Tutte  originally defined activities as follows. For every spanning tree of , each edge has an activity with respect to . If , is the set of edges that connect . If , is the set of edges in the unique cycle of . Note if and only if . An edge (resp. ) is internally active (resp. externally active) if it is the lowest edge in its cut (resp. cycle), and otherwise it is inactive.
Because the two types of activities are distinct, we will use the notation active/inactive when referring to activities in the sense of Tutte with respect to a spanning tree, and live/dead for activities with respect to a quasi-tree, as in Definition 2.
If , then the underlying graph is planar, and is given by a fixed planar embedding of . In this case, every quasi-tree of is a spanning tree of . It is easy to check that live (resp. dead) edges of with respect to are active (resp. inactive) in with respect to .
A spanning tree of any ribbon graph is also a quasi-tree (of genus zero). In this case, the activities using Tutte’s original definition are different from the activities using our definition. For the example in Figure 1, the only spanning tree is the one with no edges. Using Tutte’s definition, all four edges are externally active, but using our definition, the activities are , where and denote externally live and dead, respectively. See Section 6 for examples of non-trivial spanning trees whose activities are different from those of the corresponding quasi-trees.
As for planar graphs, the activities with respect to a quasi-tree depend on the edge order. In the case of a spanning tree of a planar graph, when the edge order is changed, Tutte proved there is a corresponding spanning tree whose activity in the new edge order matches the activity of in the old order. However, for general quasi-trees, such a correspondence may not exist: In the example in Section 6, switching the edge order by the permutation changes the activity of the unique genus 2 quasi-tree from to .
3 Main results
The Bollobás–Riordan–Tutte polynomial is recursively defined by the disjoint union, , and the following recursion for edges of and subgraphs of , where and denote deletion and contraction, respectively:
where an edge is a bridge if deleting it increases the number of components. Note that is assigned to a bridge, and to a loop. For the Tutte polynomial , these are usually and , respectively. If is the underlying graph of a ribbon graph , then .
The Bollobás–Riordan–Tutte polynomial has a spanning subgraph expansion given by the following sum over all spanning subgraphs of (p.85 of )333In , this expansion is given for . To relate to , we replace by (p.89 of ).,
The Tutte polynomial has a spanning tree expansion given by the following sum over all spanning trees of a connected graph with an order on its edges ,
where is the number of internally active edges and is the number of externally active edges of for a given spanning tree of . Similarly, the Bollobás–Riordan–Tutte polynomial has the following spanning tree expansion (p.93 of ),
where is the set of externally active edges of with respect to a spanning tree of .
We will use (1) to prove a quasi-tree expansion for the Bollobás–Riordan–Tutte polynomial, which is different from the expansion (2). Fix a total order on the edges of a connected ribbon graph . In Definition 2, we defined activities (live or dead) for edges of with respect to . Let be the spanning subgraph whose edges are the dead edges in (internally dead edges). Let be the set of live edges in (internally live edges). Let be the set of live edges in (externally live edges).
For a given quasi-tree let denote the graph whose vertices are the components of and whose edges are the internally live edges of . Let denote the Tutte polynomial of . Our main result is the following:
Let be a connected ribbon graph. The Bollobás–Riordan–Tutte polynomial is given by the following sum over all quasi-trees of ,
Let and be the set of internally live edges of that are, respectively, bridges and edges that join the same component of . Thus, has bridges and loops, which contribute factors and to in Theorem 1.
In the case when has a single vertex, there are only loops, so we have the following simplification:
Let be a connected ribbon graph with one vertex. Taking the sum over all quasi-trees of ,
For one-vertex ribbon graphs, the only spanning tree is the subgraph with no edges. All edges are loops, so all edges are externally active in the sense of Tutte. The spanning tree expansion (2) becomes the expansion (1) over all subgraphs. In contrast, the quasi-tree expansion in Corollary 2 has fewer terms because some subgraphs are not quasi-trees.
3.1 Counting quasi-trees
The Tutte polynomial counts the number of spanning trees of a connected graph by the specialization . Below, we show that specializing the Bollobás–Riordan–Tutte polynomial counts the number of quasi-trees of every genus.
Let . Then is a polynomial in and such that
where is the number of quasi-trees of genus . Consequently, equals the number of quasi-trees of .
Proof: The surviving terms in the expansion (1) of satisfy , so they correspond to connected spanning subgraphs. Hence,
where the sum is taken over connected spanning subgraphs. Since , it follows that . This proves that is a polynomial. The terms of are those whose exponent vanishes, which come from spanning subgraphs with . These are precisely quasi-trees, whose genus is given by the exponent on .
The Tutte polynomial satisfies an important duality property; for a dual graph , . Since a ribbon graph is embedded in a surface, there is a natural dual ribbon graph. Bollobás and Riordan  found a 1–variable specialization of the Bollobás–Riordan–Tutte polynomial that is invariant under this duality.
Building on the work of Ellis-Monaghan and Moffat, Chmutov found the Bollobás–Riordan–Tutte polynomial satisfies a much more general duality with respect to any subset of edges of a ribbon graph (see  and references therein). When all the edges are dualized, this construction yields the usual dual ribbon graph. Let denote the genus of . In our notation, we have
More recently, Krushkal  introduced a four-variable polynomial invariant of orientable ribbon graphs that satisfies a duality relation like the Tutte polynomial, and that specializes to the Bollobás–Riordan–Tutte polynomial.
The quasi-trees of a ribbon graph and its dual are in one-one correspondence. Since is a simple closed curve on that divides as the connect sum of two surfaces with complementary genera, the genus of the dual quasi-tree (Theorem 4.1 of ). It is an interesting question to understand the above duality in terms of the quasi-tree expansion, and whether this expansion gives rise to new duality properties.
4 Binary tree of spanning subgraphs
The spanning subgraphs of a given ribbon graph form a poset (of states) isomorphic to the boolean lattice, of subsets of the set of edges. The partial order is given by provided for all . In this section, we define a binary tree , which is similar to the skein resolution tree for diagrams widely used in knot theory (see, e.g., ). By the construction below, the leaves of correspond exactly to quasi-trees of .
A resolution of is a function , which determines a spanning subgraph . Let be a partial resolution of , with edges called unresolved if they are assigned . Let . A partial resolution determines an interval in the poset, , the interval between with all unresolved edges of set to zero, and with all unresolved edges of set to one. Given a partial resolution , we call both and split if for all subsets of unresolved edges.
If is an unresolved edge in a partial resolution , let be partial resolutions obtained from by resolving to be and , respectively. Then is called nugatory if either one of or is split.
Note that an unresolved edge of is nugatory if and only if one of the intervals or contains no quasi-trees. Figure 2 shows two possibilities for a nugatory edge.
For example, when and is not split, an edge is nugatory in if and only if adding it completes a cycle in , or is disconnected and no unresolved edges can connect it back.
For any connected ribbon graph with ordered edges, there exists a rooted binary tree whose nodes are partial resolutions of , and whose leaves correspond to quasi-trees of . If the leaf corresponds to , then its unresolved edges are nugatory, and they can be uniquely resolved to obtain . In , these are exactly the live edges with respect to .
Proof: We prove this theorem in a sequence of two lemmas below.
Let the root of be the totally unresolved partial resolution, for all . We resolve edges by changing to or in the reverse order (starting with highest ordered edge). If an edge is nugatory, the edge is left unresolved, and we proceed to the next edge. For a given node in , if is not nugatory then the left child is and the right child is . We terminate this process at a leaf when all subsequent edges are nugatory, and return as far back up as necessary to a node with a non-nugatory edge still left to be resolved. Therefore, the leaves of are spanning subgraphs of all of whose unresolved edges are nugatory.
Let , which was defined previously as the boundary of a certain regular neighborhood of , and let denote the number of its components. By definition, , which is the number of faces on , the associated surface for .
Let , where denotes the set of interiors of all unresolved edges on . Note that is connected if and only if we can join the components of by resolving some edges of . Since , is split if and only if is disconnected.
Let be any partial resolution that is not split, with an unresolved edge . For , let , and let . The edge is nugatory if and only if either or is disconnected on . If is disconnected then and . If is disconnected then and .
Proof: For , is split if and only if is disconnected. Since is not split, or is split if and only if deleting or cutting along , respectively, disconnects .
If is disconnected then is the only edge connecting two components of . Hence, these two components are connected in . This gives and . On the other hand, if is disconnected, then intersects a component of twice without linking any other unresolved edge, so this component becomes disconnected in . This gives and .
We can now see that the partial resolution of a leaf can be resolved uniquely to give a quasi-tree. By construction, for a leaf of , is not split, so there exists a resolution such that . In particular, since all unresolved edges are nugatory, by Lemma 1, there is a unique resolution such that is minimized. Including nugatory edges for which is connected, and excluding nugatory edges for which is disconnected, . Hence, is a quasi-tree.
Let be a leaf of , and let be the corresponding quasi-tree. If then is live with respect to , and otherwise is dead with respect to .
Proof: If and are any edges of , we will say that and link each other if, when uniquely resolved to obtain , their endpoints alternate on . Equivalently, their corresponding chords intersect in . This notion does not depend on whether the edges are resolved in . If , then is a spanning tree, and edges link each other if and only if they satisfy a cut-cycle condition with respect to : or .
Let and be unresolved edges of , which are therefore nugatory. Let be the unique resolution such that . Let be the resolution obtained from by changing the states of both and . If and link each other, then . Hence, is a quasi-tree for a second resolution , which is a contradiction. Thus, unresolved edges can only link resolved edges.
Suppose is unresolved and links a resolved edge with . There exists a unique closest parent of in , such that is a non-nugatory unresolved edge in . Since edges are resolved in the reverse order, is nugatory in . As links , and are both connected, which contradicts Lemma 1. Thus, if and are linked then , so is live.
Now, let be a resolved edge of . There exists a unique closest parent of in , such that is a non-nugatory unresolved edge in . By Lemma 1, and are both connected. Hence, there exists , which is unresolved in , such that and are linked. If is resolved after in , then . Since and are linked, is dead. On the other hand, if is left unresolved in , then is live by the argument in the previous paragraph with and reversed. Since and are linked, and is live, it follows that is dead.
This completes the proof of Theorem 3.
5 Proof of Theorem 1
Let be a spanning subgraph. Let , and denote the nullity, genus and number of components of , respectively. Since ,
Let be a quasi-tree of . Let and be the internally and externally live edges with respect to . Let be the spanning subgraph whose edges are the dead edges in .
By Theorem 3, there is a unique partial resolution of that is a leaf of , for which , and all resolutions for are of the form where . All resolutions are elements of the state poset , so the sum in (1) is a state sum for . The sum in Theorem 1 is a state sum for . Below, we prove that these two state sums are equal.
For a quasi-tree of , let , where and .
Since , by Lemma 1, , hence
For a quasi-tree of , let . Let be the spanning subgraph of whose edges are the edges in .
Proof: For spanning subgraph of , . Hence,
Proof (Theorem 1): The sum in Theorem 1 is over quasi-trees, which correspond to leaves of . It suffices to show that for any quasi-tree, its summand in Theorem 1 equals the sum over all for in equation (1).
Below, we will use the spanning subgraph expansion of the Tutte polynomial (see, e.g., p.339 of ),
Let denote the graph whose vertices are the components of and whose edges are the edges in . is a connected subgraph of , so is a connected graph, hence . The subgraphs are in one-one correspondence with spanning subgraphs . Let and . By Lemma 4,
The last step is obtained from the spanning subgraph expansion of the Tutte polynomial with and .
We compute the quasi-tree and spanning tree expansions for a ribbon graph with quasi-trees having a variety of topological types. has three vertices and six edges, given by , , so . We order the edges of by . The ribbon graph and its surface are shown below:
In the table below, we denote quasi-trees using the edge order; e.g., 001010 denotes consisting of only the third and fifth edges, and . For each , we compute the chord diagram, activites ( for internally and externally live; for internally and externally dead), numbers , graph , and its weight in the sum of Theorem 1. For the chord diagrams, we give the cyclic permutation of the half-edges. The types of graphs that occur in this example are as follows:
|1. vertex||2. edge|
|3. two edges with a vertex in common||4. two edges with both vertices in common|
|5. 2-cycle joined to a bridge||6. loop|
|7. loop joined to a bridge.|
Adding the weights in the last column, the Bollobás–Riordan–Tutte polynomial of is
By Proposition 2, , which counts the quasi-trees of every genus.
As an example, let be the eighth quasi-tree, denoted 011101. The associated partial resolution is . has three components, consisting of two isolated vertices and a loop. has three vertices and three edges, two connected in parallel and a second edge to the remaining vertex. The Tutte polynomial . Thus, the contribution from