Matroids, delta-matroids and embedded graphs

Matroids, delta-matroids and embedded graphs

Carolyn Chun Mathematics Department, United States Naval Academy, Annapolis, MD, USA Iain Moffatt Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, United Kingdom Steven D. Noble Mathematical Sciences, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom Ralf Rueckriemen111Ralf Rueckriemen was financed by the DFG through grant RU 1731/1-1. Department of Mathematics, University of Trier, Trier, Germany
Abstract

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illustrate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that The Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the literature, including the Las Vergnas, Bollabás-Riordan and Krushkal polynomials, are in fact delta-matroidal.

Mathematics Subject Classification: Primary 05B35; Secondary 05C10, 05C31, 05C83

1 Overview

Matroid theory is often thought of as a generalization of graph theory. Many results in graph theory turn out to be special cases of results in matroid theory. This is beneficial in two ways.

First, graph theory can serve as an excellent guide for studying matroids. As reported by Oxley, in [56], Tutte famously observed that, “If a theorem about graphs can be expressed in terms of edges and circuits alone it probably exemplifies a more general theorem about matroids.” Perhaps one of the most spectacular illustrations of the effect of graph theory on matroid theory can be found in Geelen, Gerards and Whittle’s recent and at the time of writing unpublished result that, for any finite field, the class of matroids that are representable over that field is well-quasi-ordered by the minor relation. This profound result is the matroid analogue of an equally profound result that came out of Robertson and Seymour’s Graph Minors Project, in which, they proved that graphs are well-quasi-ordered by the minor relation [58]. Rather than the result itself, here we want to focus on the fact that, to quote a recent statement of Whittle [37] about his work with Geelen and Gerards, “It would be inconceivable to prove a structure theorem for matroids without the Graph Minors Structure Theorem as a guide”.

Second, insights from matroid theory can lead to new results about graphs. For example, Wu [69] established an upper bound for the number of edges of a loopless 2-connected graph, which was an improvement on existing results suggested by matroid duality. Graph theory and matroid theory are mutually enriching, and this is the subject of [56] by Oxley.

The key purpose of this paper is to propose and study a similar correspondence between embedded graphs and delta-matroids.

Delta-matroids, introduced by Bouchet [5], can be seen as a generalization of matroids. Where a matroid has bases, a delta-matroid has feasible sets. These satisfy a symmetric exchange axiom, but do not all have to be of the same size. We give a formal definition in the next section. The greater generality of delta-matroids allows us to capture not only information about a graph, but also about its embedding in a surface. Bouchet was the first to observe a connection between embedded graphs and delta-matroids in [6]. Our approach is more direct than his and has the advantage that it enables us to exploit the theory of ribbon graphs, much of which has developed since Bouchet did his work.

As mentioned above, we will describe embedded graphs as ribbon graphs. The cycle matroid of a connected graph is constructed by taking the collection of spanning trees of the graph as its bases. In a connected ribbon graph, the spanning-trees are precisely the genus-zero spanning ribbon subgraphs that have exactly one boundary component. In the context of ribbon graphs, the genus-zero restriction is artificial, and it is subgraphs with exactly one boundary component, called quasi-trees that play the role of trees. It turns out that the edge set of a ribbon graph together with its spanning quasi-trees form a delta-matroid.

Moreover, we will see that this delta-matroid arises as the natural extension of a cycle matroid to the setting of embedded graphs, and that the delta-matroid structure follows from basic properties of surfaces. We show that various concepts related to cellularly embedded graphs are special cases of concepts for delta-matroids. Because of this compatibility between the two structures, we extend Bouchet’s initial ideas and propose that there is a correspondence between embedded graphs and delta-matroids that is analogous to the one between graphs and matroids. We justify this proposition by illustrating how results from topological graph theory can be used to guide the development of delta-matroid theory, just as graph theory often guides matroid theory. We also see that several polynomials of embedded graphs, including the Tutte, Las Vergnas, Bollobás-Riordan and Krushkal polynomials, are in fact delta-matroidal objects, just as many graph polynomials are matroidal.

The paper is structured as follows. In Section 2, we give an overview of some relevant properties of matroids and delta-matroids. Next, Section 3 contains some background on cellularly embedded graphs. Most of the time, we will use the language of ribbon graphs instead of cellularly embedded graphs. These are equivalent concepts (see Figure 1), but ribbon graphs have the advantage of being closed under the natural minor operations.

In Section 4, we describe how delta-matroids arise from ribbon graphs, emphasising that they arise as the natural extensions of various classes of matroid associated with graphs. We show that some of these delta-matroids, albeit in a different language, appeared in Bouchet’s foundational work in delta-matroids. In Section 5 we discuss their connections with graphic matroids and describe how basic properties of a ribbon graph are encoded in its delta-matroid. We provide evidence of the basic compatibility between delta-matroids and ribbon graphs. In particular, we prove that one of the most fundamental operations of delta-matroids, the twist, is the delta-matroid analogue of a partial dual of a ribbon graph, which turns out to be a key result in connecting the two areas. We describe how to see edge structure and connectivity in a ribbon graph in terms of its delta-matroid, and show how results on delta-matroid connectivity inform ribbon graph theory. We also demonstrate that excluded minor characterisations that have appeared in both the delta-matroid and ribbon graph literature are translations of one another.

In Section 6, we discuss various polynomials. Some well-known graph polynomials, and in particular the Tutte polynomial, are properly understood as matroid polynomials, rather than graph polynomials. There has been considerable recent interest in extensions of the Tutte polynomial to graphs embedded in surfaces. Three generalizations of the Tutte polynomial to embedded graphs in the literature are the Las Vergnas polynomial, the Bollobás-Riordan polynomial, and the Kruskal polynomial. We show that each of these generalizations is determined by the delta-matroids of ribbon graphs, and that the ribbon graph polynomials are special cases of more general delta-matroid polynomials. That is, while the Tutte polynomial is properly a matroid polynomial, its topological extensions are properly delta-matroid polynomials.

Our results here offer new perspectives on delta-matroids. We illustrate here a fundamental interplay between ribbon graphs and delta-matroids, that is analogous to the interplay between graphs and matroids. By doing so we offer a new approach to delta-matroid theory.

We would like to thank Tony Nixon, Sang-Il Oum, and Lorenzo Traldi for many helpful suggestions and comments.

2 Matroids and delta-matroids

Our terminology follows [5] and [57], except where explicitly stated.

2.1 Set systems

A set system is a pair where is a set, which we call the ground set, and is a collection of subsets of . The members of are called feasible sets. A set system is proper if is not empty; it is trivial if is empty. For a set system we will often use to denote its ground set and its collection of feasible sets. In this paper we will always assume that is a finite set and will do so without further comment.

The symmetric difference of sets and , denoted by , is .

Axiom 2.1 (Symmetric Exchange Axiom).

Given a set system , for all and in , if there is an element , then there is an element such that is in .

Note that we allow in the Symmetric Exchange Axiom.

A delta-matroid is a proper set system that satisfies the Symmetric Exchange Axiom (Axiom 2.1). If the feasible sets of a delta-matroid are equicardinal, then the delta-matroid is a matroid and we refer to its feasible sets as its bases. If a set system forms a matroid , then we usually denote by , and often use to denote its collection of bases . It is not hard to see that the definition of a matroid given here is equivalent to the ‘usual’ definition of a matroid through bases given in, for example, [57, 68].

Throughout this paper, we will often omit the set brackets in the case of a single element set. For example, we write instead of , or instead of .

2.2 Graphic matroids

For a graph with connected components, let be the edge sets of the maximal spanning forests of . is obviously non-empty, and its elements are equicardinal since each spanning forest of will have edges. It is not too hard to see that the Symmetric Exchange Axiom holds, and so the set system is a matroid, which is called the cycle matroid of . Any matroid that is the cycle matroid of a graph is a graphic matroid.

Example 2.2.

If is the graph shown in Figure 1(a), then where and .

2.3 Matroid rank

Let be a matroid with ground set . A subset of is an independent set of if and only if it is a subset of a basis of . A rank function is defined for all subsets of the ground set of a matroid. Its value on a subset of is the cardinality of the largest independent set contained in . The rank of a set is written , or just if the matroid is clear from the context. Thus, . We say that the rank of , written , is equal to , which is equal to , for any .

Example 2.3.

For a graph , the rank function of its cycle matroid is given by , where is the number of connected components of the spanning subgraph of , and .

2.4 Width and evenness

For a delta-matroid , let and be the set of feasible sets with maximum and minimum cardinality, respectively. We will usually omit when the context is clear. Let and let . Then is the upper matroid and is the lower matroid for . These matroids were defined by Bouchet in [6]. It is straightforward to show that the upper matroid and the lower matroid are indeed matroids. The width of , denoted by , is defined by

Thus the width of is the difference between the sizes of its largest and smallest feasible sets.

If the feasible sets of a delta-matroid all have the same parity, then we say that the delta-matroid is even. Otherwise, we say that the delta-matroid is odd. In particular, every matroid is an even delta-matroid. It is perhaps worth emphasising that an even delta-matroid need not have feasible sets of even cardinality.

It is convenient to record the following useful result here.

Lemma 2.4.

Let be a delta-matroid, let be a subset of and let . Then for any we have .

Proof.

We proceed by contradiction. If , then there is nothing to prove, so we can assume that . Suppose that and . Choose with and as large as possible. Now there exists and so . Hence there exists belonging to such that . Because , we have . And because , we must have . But then , and , contradicting the choice of . ∎

2.5 Twists, duals, loops, coloops, and minors

Twists, introduced by Bouchet in [5], are one of the fundamental operations of delta-matroid theory.

Definition 2.5.

Let be a set system. For , the twist of with respect to , denoted by , is given by . The dual of , written , is equal to .

It follows easily from the identity that the twist of a delta-matroid is also a delta-matroid. We restate this fact in the following lemma.

Lemma 2.6 (Bouchet [5]).

Let be a delta-matroid and let be a subset of . Then is a delta-matroid.

Although it is always a delta-matroid, a twist of a matroid need not be a matroid. (For example, if then has feasible sets and so is not a matroid.) However, its dual will be. The rank function of is given by

(2.1)

For a delta-matroid , and , if is in every feasible set of , then we say that is a coloop of . If is in no feasible set of , then we say that is a loop of . Note that a coloop or loop of is a loop or coloop, respectively, of .

If is not a coloop, then, following Bouchet and Duchamp [11], we define delete , written , to be

If is not a loop, then we define contract , written , to be

If is a loop or coloop, then .

Both and are delta-matroids (see [11]). Let be a delta-matroid obtained from by a sequence of deletions and contractions. Then is independent of the order of the deletions and contractions used in its construction (see [11]) and is called a minor of . If is formed from by deleting the elements of and contracting the elements of then we write . The restriction of to a subset of , written , is equal to .

Note that . The next result shows that deletion, contraction and twists are also related. It is a reformulation of Property 2.1 of [11].

Lemma 2.7.

For a delta-matroid and distinct elements and of , we have

  1. and ;

  2. and .

Using Lemma 2.7 and induction we obtain the following.

Proposition 2.8.

Let be a delta-matroid and let , and be subsets of with . Then

In particular, and, when is the disjoint union of and , we have

2.6 Delta-matroid rank

Bouchet defined an analogue of the rank function for delta-matroids in [4]. For a delta-matroid , it is denoted by or simply when is clear from the context. Its value on a subset of is given by

An easy consequence of basic properties of the symmetric difference operation is the following.

Lemma 2.9.

Let be a delta-matroid and let be a subset of . Then .

The next two results show how the rank function changes when an element is deleted or contracted.

Lemma 2.10.

Let be a delta-matroid and let be an element in , and a subset of . Then either is a coloop or there exists such that and .

Proof.

Suppose is not a coloop. Then there is a feasible set avoiding . Take such that . If avoids then the lemma holds, so we assume this is not the case. Then , so the Symmetric Exchange Axiom (Axiom 2.1) implies that there exists such that . If , then which is not possible, because . So and , so we deduce that . As was chosen from to minimize , we deduce that . Since , the lemma holds. ∎

Lemma 2.11.

Let be a delta-matroid and let be an element in , and a subset of . Then

(2.2)
and
(2.3)
Proof.

We first establish Equation (2.2). Suppose that is not a coloop. Lemma 2.10 implies that there exists such that and , which in turn implies that .

Suppose that is a coloop of . Take such that . Then is in and clearly has smallest symmetric difference with of all feasible sets in . Thus .

Now Equation (2.3) is obtained by using duality. Lemma 2.9 implies that . Using Equation (2.2), we obtain

The result follows by applying duality again and noting that is a coloop of if and only if it is a loop of . ∎

3 Ribbon graphs

We are concerned here with connections between cellularly embedded graphs and delta-matroids. As it is much more convenient for our purposes, we realize cellularly embedded graphs as ribbon graphs. This section provides a brief overview of ribbon graphs, as well as standard ribbon graph notation and constructions. A more thorough treatment of the topics covered in this section can be found in, for example, [33].

3.1 Cellularly embedded graphs and ribbon graphs

3.1.1 Ribbon graphs

A cellularly embedded graph is a graph drawn on a closed compact surface in such a way that edges only intersect at their ends, and such that each connected component of is homeomorphic to a disc. Note that each connected component of must be embedded in a different component of the surface.

Two cellularly embedded graphs and are equivalent if there is a homeomorphism, , which is orientation preserving if is orientable, and has the property that is a graph isomorphism. We consider cellularly embedded graphs up to equivalence.

Ribbon graphs provide an alternative, and more natural for the present setting, description of cellularly embedded graphs.

Definition 3.1.

A ribbon graph is a surface with boundary, represented as the union of two sets of discs: a set of vertices and a set of edges with the following properties.

  1. The vertices and edges intersect in disjoint line segments.

  2. Each such line segment lies on the boundary of precisely one vertex and precisely one edge.

  3. Every edge contains exactly two such line segments.

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(a) A cellularly embedded graph .

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(b) as a ribbon graph.

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(c) The partial dual .
Figure 1: Embedded graphs and ribbon graphs.

It is well-known that ribbon graphs are just descriptions of cellularly embedded graphs (see for example [40]). If is a cellularly embedded graph, then a ribbon graph representation results from taking a small neighbourhood of the cellularly embedded graph , and deleting the faces. On the other hand, if is a ribbon graph, then, topologically, it is a surface with boundary. Capping off the holes, that is, ‘filling in’ each hole by identifying its boundary component with the boundary of a disc, results in a ribbon graph embedded in a closed surface from which a graph embedded in the surface is readily obtained. Figure 1 shows an embedded graph described as both a cellularly embedded graph and a ribbon graph. We say that two ribbon graphs are equivalent if they define equivalent cellularly embedded graphs, and we consider ribbon graphs up to equivalence. This means that ribbon graphs are considered up to homeomorphisms that preserve the graph structure of the ribbon graph and the cyclic order of half-edges at each of its vertices.

3.1.2 Ribbon subgraphs and edge deletion

Let be a ribbon graph. Then a ribbon graph is a ribbon subgraph of if it can be obtained by removing vertices and edges of . If then is a spanning ribbon subgraph of . Note that every subset of uniquely determines a spanning ribbon subgraph of .

If is an edge of , then delete , written , is defined to be the ribbon subgraph of . Similarly, for , is defined to be . Table 1 shows the local effect of deleting an edge of a ribbon graph.

An important observation about ribbon subgraphs is that if a ribbon graph is realised as a graph cellularly embedded in a surface , and , or a ribbon subgraph of , is realised as a graph cellularly embedded in a surface , then and need not be homeomorphic.

3.1.3 Standard parameters

If is a ribbon graph, then and denote and , respectively. Furthermore, denotes the number of connected components in , and is the number of boundary components of the surface defining the ribbon graph. For example, the ribbon graph of Figure 1(b) has . Note that, if is realised as a cellularly embedded graph, then is the number of its faces. The rank of , denoted by , is defined to be , and the nullity of , denoted by , is defined to be .

A ribbon graph is orientable if it is orientable when regarded as a surface. We define a ribbon graph parameter by setting if is non-orientable, and otherwise.

The genus of a ribbon graph is its genus when regarded as a surface. If is realized as a graph cellularly embedded in , then its genus is exactly the genus of , and is orientable if and only if is. The Euler genus, , of is the genus of if is non-orientable, and is twice its genus if is orientable. Euler’s formula gives . We say that a ribbon graph is plane if . Note that we allow plane graphs to have more than one connected component here. Plane ribbon graphs correspond to graphs that can be cellularly embedded in some disjoint union of spheres.

For each subset of , we let , , , , , and each refer to the spanning ribbon subgraph of , where is given by context. When the choice of is not clear from the context, we write , , etc.. Observe that the function on defined here coincides with the rank function of the cycle matroid of .

3.1.4 Loops and bridges

An edge of a ribbon graph is a bridge if . The edge is a loop if it is incident with exactly one vertex. We will abuse notation and also use the term loop to describe the ribbon subgraph of consisting of and its incident vertex. In ribbon graphs, loops can have various properties. A loop or cycle is said to be non-orientable if it is homeomorphic to a Möbius band. Otherwise it is orientable. Two cycles and in are said to be interlaced if there is a vertex such that , and and are met in the cyclic order when travelling around the boundary of the vertex . A loop is non-trivial if it is interlaced with some cycle in , otherwise it is trivial.

3.1.5 Ribbon graph minors

For a ribbon graph with an edge recall that is obtained by removing from . Similarly, if is a vertex of , then the vertex deletion is defined to be the ribbon graph obtained from by removing the vertex together with all its incident edges.

The definition of edge contraction is a little more involved than that of edge deletion.

Definition 3.2.

Let be a ribbon graph. Let and and be its incident vertices, which are not necessarily distinct. Then denotes the ribbon graph obtained as follows. Consider the boundary component(s) of as curves on . For each resulting curve, attach a disc, which will form a vertex of , by identifying its boundary component with the curve. Delete , and from the resulting complex. We say that is obtained from by contracting .

A ribbon graph is a minor of a ribbon graph if is obtained from by a sequence of edge deletions, vertex deletions, and edge contractions.

The local effect of contracting an edge of a ribbon graph is shown in Table 1. Observe that contracting an edge may change the number of vertices or orientability of a ribbon graph. Since deletion and contraction are local operations, if some edges in a ribbon graph are deleted and some others are contracted, then the same ribbon graph will be produced regardless of the order of operations.

The definition of edge contraction might be a little surprising at first. However, the reader should see that it is natural upon observing that Definition 3.2 is just an expression of the obvious idea of contraction as the ‘identification of and its incident vertices into a single vertex’ in a way that allows it to be applied to loops. (See also the discussion in [33] on this topic.) Unlike for graphs, when working with ribbon graph minors it is necessary to be able to contract loops as otherwise the set of ribbon graphs will contain infinite anti-chains when quasi-ordered using the minor relation (see [54]).

non-loop non-orientable loop orientable loop

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Table 1: Operations on an edge (highlighted in bold) of a ribbon graph. The ribbon graphs are identical outside of the region shown.

3.1.6 Separability

For a ribbon graph and non-trivial ribbon subgraphs and , we write when is the disjoint union of and , that is, when and . A vertex of is a separating vertex if there are non-trivial subgraphs and of such that and . In this case we write .

We say that is the join of and , written , if and no cycle in is interlaced with a cycle in . In other words, can be obtained as follows: choose an arc on a vertex of and an arc on a vertex of such that neither arc intersects an edge, then identify the two arcs merging the two vertices on which they lie into a single vertex of . The join is also known as the “one-point join”, the “map amalgamation”, and the “connected sum” in the literature. Observe that there is ambiguity in the notation as it does not specify where the two arcs are, nor which of the two possible identifications is used. A similar comment holds for . We refer the reader to [52, 53] for a fuller discussion of separability for ribbon graphs.

We can reformulate the definition of a trivial loop in terms of joins. A loop of is trivial if and only if we can write or where and for each and .

3.2 Geometric duals and partial duals

The construction of the geometric dual, , of a cellularly embedded graph is well known: is obtained by placing one vertex in each face of , and is obtained by embedding an edge of between two vertices whenever the faces of in which they lie are adjacent. Geometric duality has a particularly neat description when translated to the language of ribbon graphs. Let be a ribbon graph. Recalling that, topologically, a ribbon graph is a surface with boundary, we cap off the holes using a set of discs, denoted by , to obtain a surface without boundary. The geometric dual of is the ribbon graph . Observe that, for ribbon graphs, the edges of and are identical. The only change is which arcs on their boundaries do and do not intersect vertices. This allows us to consider a subset of edges of as also being a subset of edges of and vice versa. We adopt this convention. Although it is common to distinguish the two sets by writing and , doing so proves to be notationally difficult in the current setting.

Chmutov, in [22], introduced a far reaching generalization of geometric duality, called partial duality. Roughly speaking, a partial dual of a ribbon graph is obtained by forming the geometric dual with respect to only a subset of its edges. Partial duality arises as a natural operation in knot theory, topological graph theory, graph polynomials, and quantum field theory. We will see later that it is also an analogue of a fundamental operation on delta-matroids. Here we define partial duals directly on ribbon graphs. We refer the reader to [22, 32, 51] or the exposition [33] for alternative constructions and other perspectives of partial duals.

Let be a ribbon graph and . The partial dual of is obtained by forming the geometric dual of as described above but ignoring the edges not in as follows. Regard the boundary components of the spanning ribbon subgraph of as curves on the surface of . Glue a disc to along each connected component of this curve and remove the interior of all vertices of . The resulting ribbon graph is the partial dual .

We identify the edges of with those of using the natural correspondence. Table 1 shows the local effect of partial duality on an edge (highlighted in bold) of a ribbon graph . The ribbon graphs are identical outside of the regions shown. In fact Table 1 serves as a perfectly adequate definition of partial duality for this paper.

Observe from Table 1 that is a bridge of if and only if is a trivial orientable loop in ; is a non-loop non-bridge edge of if and only if is a non-trivial orientable loop in ; and is a (non-)trivial non-orientable loop in if and only if is a (non-)trivial non-orientable loop in . We also record the following basic properties of partial duality for use later.

Proposition 3.3 (Chmutov [22]).

Let be a ribbon graph and . Then

  1. and ;

  2. ;

  3. .

Note that it follows from the proposition that partial duals may be formed one edge at a time. Also note that the form of the final part of the proposition is very similar to that of the second part of Lemma 2.7. We will return to this later.

3.3 Quasi-trees

Quasi-trees are one of our fundamental objects of study. They are the analogue of trees for ribbon graphs, and our terminology reflects this. A quasi-tree is a connected ribbon graph with exactly one boundary component. If is a connected ribbon graph, a spanning quasi-tree of is a spanning ribbon subgraph with exactly one boundary component. For disconnected graphs, we abuse notation by saying that is a spanning quasi-tree of if and the connected components of are spanning quasi-trees of the connected components of .

We record the following basic facts about quasi-trees for reference later. For (3), recall that, for ribbon graphs, .

Lemma 3.4.

Let be a ribbon graph, and be a spanning quasi-tree of . Then the following hold.

  1. .

  2. if and only if is a maximal spanning forest of .

  3. is a spanning quasi-tree of of Euler genus if and only if is a spanning quasi-tree of of Euler genus .

  4. If then if and only if is a maximal spanning forest of .

Proof.

Items (1) and (2) follow easily from Euler’s formula. Item (4) is an immediate consequence of (2) and (3). It remains to prove (3). For this first assume that is connected. Consider the intermediate step of the formation of from , as described in Section 3.2, in which the holes of have been capped off with elements of giving a surface . For each , observe that and have the same boundary components. Thus is a spanning quasi-tree of if and only if is a spanning quasi-tree of . Suppose that and are both spanning quasi-trees. Then each of and has one boundary component and is connected. Moreover . Euler’s formula gives and . Thus . Extending the result to disconnected graphs is straightforward because each of the parameters , , and is additive over connected components, and the geometric dual of a disconnected ribbon graph is the disjoint union of the geometric duals of its connected components. ∎

4 Delta-matroids from ribbon graphs

4.1 Defining the delta-matroids

Consider a connected ribbon graph . We start by considering some standard ways that gives rise to a matroid. The most fundamental matroid associated with is its cycle matroid , where consists of the edge sets of the spanning trees of . The matroid contains no information about the topological structure of , only its graphical structure. This is because trees always have genus zero and therefore cannot depend upon the embedding of . Our aim here is to find the matroidal analogue of an embedded graph, and to do this we clearly need to adapt the definitions of . By thinking of the the construction of in terms of ribbon graphs it becomes obvious how this should be done: spanning trees are genus-zero spanning ribbon subgraphs with exactly one boundary component, so to retain topological information, we drop the genus zero condition, consider quasi-trees instead of trees, and obtain the set system , where consists of the edge sets of the spanning quasi-trees of .

There is a natural variation of the construction of a cycle matroid obtained by choosing , taking as the ground set and to be either the edge sets formed by deleting edges from each spanning tree, or the edge sets formed by adding edges to each spanning tree. In the former case, consists of the spanning forests of that have exactly connected components and is shown to be a matroid by noting that it is the th truncation of , see [57]. In the latter case, is the dual of the th truncation of . Consider this construction in terms of quasi-trees: adding or removing an edge from a spanning quasi-tree results in a spanning ribbon subgraph with exactly two boundary components. However once we add or remove further edges, the number of boundary components is not determined by the number of edges added or removed and can be anywhere between and . In the quasi-tree setting it no longer makes sense to make the distinction between adding and removing edges, as we did in the case of matroids and spanning trees. These ribbon graph extensions of matroids naturally lead us to the make the following definition.

Definition 4.1.

Let be a ribbon graph with connected components, and let . Then we define

  1. , and

  2. .

For a connected ribbon graph, is the collection of all edge sets that determine a spanning ribbon subgraph of with exactly boundary components, and is the collection of all edge sets that determine a spanning ribbon graph of with at most boundary components. Note that . This set will be particularly important to us here, and later we will denote it by just . Note that may be empty.

Example 4.2.

For the ribbon graph of Figure 1(b),
,
,
,
, and , for . Then can be found easily from these.

Definition 4.3.

For a ribbon graph and a non-negative integer , let denote the set system , and denote the set system . We call and , the -spread and -toggle, respectively, of .

Theorem 4.4.

Let be a ribbon graph, and . Then

  1. is a delta-matroid, and

  2. is a delta-matroid, if is non-empty and orientable.

The proof of Theorem 4.4 follows from the next lemma. For the next two proofs we use to denote the spanning ribbon subgraph of . Note that does not denote the induced ribbon subgraph .

Lemma 4.5.

Let , , and . If , then there exists such that .

Proof.

The ribbon graph has boundary components and has at most boundary components. Each edge of a ribbon graph is incident with one or two boundary components. So and differ by at most one. Thus has boundary components. We think of as a subset of . We can then consider how the edges in meet the boundary components of .

If there is an edge that intersects two distinct boundary components of , then adding this edge to will give a ribbon subgraph with one fewer boundary component, and so . If there is an edge that meets two distinct boundary components of , then removing this edge from results in a ribbon subgraph with one fewer boundary component, and so .

All that remains is the case in which each edge in intersects exactly one boundary component of . We shall show that this case cannot happen. To see why observe that can be obtained from by first deleting the edges in and then adding the edges in , one by one. Colour the boundary components of so that each one receives a different colour. Whenever an edge is added or deleted, the only boundary components that change are those intersecting an edge that is deleted or those intersecting the two line segments forming the ends of an edge that is added. At each step the number of boundary components may stay the same, or increase or decrease by one. After a step where the number of boundary components increases by one, the two new boundary components are given the same colour as the one they replace. We claim that when the number of boundary components decreases by one, the two boundary components being replaced have the same colour. The single boundary component replacing them may then be given this common colour. Consider a single step in the process of deleting and adding edges, and let be a point that belongs to the boundary both before and after this step. The boundary component containing may change after the step but its colour does not. Thus the claim follows and moreover all the original colours used to colour the boundary components of are used to colour the boundary components of . Therefore has at least as many boundary components as , giving a contradiction. ∎

Proof of Theorem 4.4.

In each case it is enough to show that the given families of feasible sets satisfy the Symmetric Exchange Axiom.

For Item (1), let and . If , then taking gives , as desired. In the exceptional case, . Then Lemma 4.5 guarantees that there is an element such that .

For Item (2), we first observe that it follows easily from Euler’s formula that the parity of is the same as the parity of . In particular, the sizes of all spanning quasi-trees of have the same parity. Furthermore, as every edge of a ribbon graph is incident with at most two boundary components, . Thus, if and , then , so is a proper set system. Let , be members of and . If , then by Lemma 4.5, there exists such that . It remains to consider what happens if . As and have the same parity, there exists . Now, by our earlier observation, . Hence is a delta-matroid. ∎

In general, the set system is not a delta-matroid. For example, if is the plane graph obtained by taking a triangle and adding an edge in parallel with one of the edges, then is not a delta-matroid. Also, if is non-orientable may not be a delta-matroid. Consider, for example, the ribbon graph of Euler genus 2 obtained by adding an interlaced non-orientable loop to a plane 2-cycle.

It is enlightening to observe that the fact that forms a delta-matroid but, in general, does not reflects the fact mentioned at the start of this section that adding one edge to a quasi-tree results in a ribbon subgraph with one additional boundary component, but adding one edges can result in a ribbon subgraph with anything between 1 and boundary components.

4.2 Ribbon-graphic delta-matroids

A main purposes of this article is to illustrate that the delta-matroid plays a role in delta-matroid theory analogous to the role graphic matroids play in matroid theory. In this subsection we set up some additional terminology for these delta-matroids and show that they have appeared in the literature in other guises.

Definition 4.6.

Let be a ribbon graph. We use to denote the set , so that

and to denote the delta-matroid . We say that is a ribbon-graphic delta-matroid.

Example 4.7.

For the ribbon graph of Figure 1(b),
.

To relate the delta-matroid to the literature, particularly to Bouchet’s foundational work on delta-matroids, we take what may appear to be a detour into transition systems. Let be a connected 4-regular graph. Each vertex of is incident with exactly four half-edges. A transition at a vertex is a partition of the half-edges at into two pairs, and a transition system, of is a choice of transition at each of its vertices.

For the purposes of this section, we allow graphs to include free loops, that is edges which are not incident with any vertex. We think of a free loop as a circular edge or as a cycle on zero vertices. Given a transition system of then we can obtain a set of free loops as follows. If and are two non-loop edges whose half edges are paired at the vertex , then we replace these two edges with a single edge . In the case of a loop, we temporarily imagine an extra vertex of degree two on the loop, carry out the operation, and then remove the temporary vertex. Doing this replacement for each pair of half edges paired together in the transition system results in a set of free loops, that we denote by and call a graph state.

Since is 4-regular, at each vertex there are three transitions. Choose exactly two transitions and at each vertex, and consider the set consisting of all transition systems of in which the transition at each vertex is one of the distinguished transitions, or . An element of is called a transversal. Fix some transversal , and let

Kotzig’s Theorem [43] implies that is a proper set system. Bouchet showed in [5] that is a delta-matroid. A delta-matroid that can be obtained in this way is called an Eulerian delta-matroid. (Note that although the definition of Eulerian delta-matroid is not made explicit in [5], that is, Bouchet never uses the term “Eulerian delta-matroid,” it is implied that this is the intended definition by his later work, such as [8].)

Bouchet showed that is a delta-matroid, albeit using a different language. Following [6], let be a connected graph cellularly embedded in a surface , and let be its geometric dual. Consider the natural immersion of in . For each let denote the corresponding set in . A set is said to be a base if is connected, where denotes the closure operator. Let denote the collection of all bases of . Bouchet showed that satisfies the Symmetric Exchange Axiom, and so the pair is a delta-matroid.

By changing from the language of cellularly embedded graph to ribbon graphs we can see that and are identical objects. To see this consider and as ribbon graphs and respectively. Then as described in Section 3.2. It is not hard to see that the number of components of is exactly the number of boundary components of . It follows that defines a base of if and only is a spanning quasi-tree of . Thus and coincide.

Bouchet did not use the language of quasi-trees to show that is a delta-matroid, but rather transition systems and Eulerian delta-matroids, identifying it with a construction from [5]. For this, again let be a connected graph cellularly embedded in a surface. Its medial graph, , is the embedded graph constructed by placing a vertex on each edge of , and then drawing the edges of the medial graph by following the face boundaries of (so each vertex of is of degree ). The medial graph of an isolated vertex is a free loop. The vertices of are 4-valent and correspond to the edges of . Every medial graph has a canonical face 2-colouring given by colouring faces corresponding to a vertex of black, and the remaining faces white. We can use the canonical face 2-colouring to distinguish among the three types vertex transitions. We call a vertex transition white if it pairs half-edges that share a white face, black if it pairs half-edges that share a black face, and crossing otherwise. If consists of all the transition systems that have only white or black transitions at each vertex, and consists only of the white transitions, then it is not hard to see that .

This discussion shows that every ribbon-graphic delta-matroid is Eulerian. In fact, ribbon-graphic delta-matroids are exactly Eulerian delta-matroids.

Theorem 4.8 (Bouchet [6]).

A delta-matroid is eulerian if and only if , for some ribbon graph .

Sketch of proof..

If is Eulerian then, by definition, we can obtain it as some . We need to construct a ribbon graph such that . But such a graph can be obtained as a cycle family graph of , from [32]. (The cycle family graphs of are precisely the embedded graphs that have a medial graph isomorphic to .) The six choices at each vertex in the construction of a cycle family graph correspond to the six choices of the white and black transitions of (c.f. the proof of Theorem 4.12 of [32]). ∎

We have just seen the delta-matroids of ribbon graphs considered here appeared in a rather different framework as Eulerian delta-matroids in Bouchet’s initial work on delta-matroids. Here, we are proposing that for many purposes, Eulerian delta-matroids, and delta-matroid theory in general, is best thought of as extensions of ribbon graph theory. (Saying this, of course there are certainly situations where it is most helpful to think of Eulerian delta-matroids as generalisations of transition systems.) As we will demonstrate here, this is because there is a natural and fundamental compatibility between ribbon graph theory and delta-matroid theory, with many constructions, results, and proofs in the two areas being translations of one another.

From the perspective of Eulerian delta-matroids, is significant since the transition systems of arise canonically. Another setting in which canonical transition systems arise is in digraphs. Suppose that is a 4-regular digraph. At each of its vertices there are two natural transitions that are consistent with the direction of the half-edges of the digraph. We take to be the set of all transition systems that arise from these choices. Then for each , is a delta-matroid. We call a delta-matroid arising in this way a directed eulerian delta-matroid.

Theorem 4.9 (Bouchet [6]).

A delta-matroid is directed eulerian if and only if , for some orientable ribbon graph .

Sketch of proof..

First suppose that , for some orientable ribbon graph . Arbitrarily orient (the surface) and draw its canonically face 2-coloured medial graph on it. Direct each edge of so that it is consistent with the orientation of the black face it bounds.

Conversely, suppose that is directed Eulerian, arising from a digraph . By the proof Theorem 4.8, we know for some ribbon graph , where the underlying graphs of and are isomorphic. The direction of induces a direction of . Furthermore, by forming the twisted duals (see [32]) or , if necessary, we may assume that the transitions that are consistent with the directions of coincide with the black and white transitions of . These directions induce an orientation on each black face of , and hence of each vertex and half-edge of . Since the black and white transitions of are consistent with transitions coming from the directions, these orientations of vertices must be consistent and so is orientable. ∎

Combining Theorems 4.8 and 4.9 immediately gives the following.

Corollary 4.10 (Bouchet [6]).

A delta-matroid is directed eulerian if and only if it is Eulerian and even.

In recent papers, Traldi introduced the transition matroid of an abstract four-regular graph [63] and the isotropic matroid of a symmetric binary matrix [62]. These two matroids have almost identical definitions: both are binary matroids described by a representation, with the only difference being a permutation of some of the columns labels. Moreover, both are relevant to ribbon graphs. We have described the fundamental relationship between a ribbon graph and its medial graph, which is an embedded four-regular graph; in Section 5.7 we describe how a ribbon graph with one vertex may be represented by a symmetric binary matrix. In [16] Traldi describes the construction of the transition matroid of a ribbon graph. We now describe the almost identical construction of the isotropic matroid of a ribbon graph, and discuss the extent to which it determines the ribbon graph.

Let be a connected ribbon graph and be its canonically face 2-coloured medial graph. Let be a transition system in with . In other words, defines an Eulerian circuit in with no crossing transitions. Apply an orientation to the edges of , so that is now a directed Eulerian cycle.

We say that two vertices and of are interlaced with respect to if they are met in the cyclic order when travelling round . Let denote the binary by matrix whose rows and columns are indexed by the elements of . The -entry of is zero if and only if in , opposite edges at the vertex corresponding to have inconsistent orientations. For , the –entry is one if and only the vertices corresponding to and in are interlaced with respect to .

We now let be the matrix

The isotropic matroid of is the binary matroid with representation . Each edge of indexes three columns of , one in each of the three blocks, with the order of the indices consistent with the indices of . Following Traldi, we use , and to denote the columns of corresponding to in , and respectively. For , let . A basis of is called transverse if for each , it contains precisely one of , and .

The isotropic matroid itself does not determine , because knowledge of is required. Let denote the edges of where at the corresponding vertex of , takes the white transition. Then from the discussion above is a feasible set of . We claim that is a feasible set of if and only if the principal submatrix of corresponding to the edges of is non-singular. This is easily verified when , by considering the effect of switching the transitions of from black to white or vice versa at the vertices of corresponding to e