# Delta-matroids: Origins.

Guest post by Carolyn Chun.

In a previous post, Dr. Pivotto posted about multimatroids.  Her post includes a definition of delta-matroid, and a natural way that delta-matroids arise in the context of multimatroid theory.  I recommend her post to readers interested in multimatroids, which generalize delta-matroids.  I will discuss delta-matroids in this post, their discovery and natural ways that a mathematician may innocently stumble into their wonderful world.

Delta-matroids were first studied by Andre Bouchet [BG].  I use $X\bigtriangleup Y$ to denote symmetric difference of sets $X$ and $Y$, which is equal to $(X\cup Y)-(X\cap Y)$. To get a delta-matroid, take a finite set $E$ and a collection of subsets $\mathcal{F}$, called feasible sets, satisfying the following.

I) $\mathcal{F}\neq \emptyset$.

II) If $F,F’\in \mathcal{F}$ and $e\in F\bigtriangleup F’$, then there exists $f\in F\bigtriangleup F’$ such that $F\bigtriangleup \{e,f\}$ is in $\mathcal{F}$.

Then $D=(E,\mathcal{F})$ is a delta-matroid.

It is worth noting that the feasible sets of a delta-matroid can have different cardinalities.  Taking all of the feasible sets of smallest cardinality gives the bases of a matroid, namely the lower matroid for $D$.  Likewise the feasible sets with maximum cardinality give the bases of the upper matroid of $D$.  No other collections of feasible sets of a given size are guaranteed to comprise the bases of a matroid.

Taking minors in delta-matroids is modeled well by considering the bases of a matroid minor.  Take $e\in E$.  As long as $e$ is not in every feasible set (that is, $e$ is not a coloop), the deletion of $e$ from $D$, written $D\backslash e$, is the delta-matroid $(E-e,\{F \mid F\in\mathcal{F}\text{ and }e\notin F\}).$  As long as $e$ is not in no feasible set (that is, $e$ is not a loop), then contracting $e$ from $D$, written $D/e$,  is the delta-matroid $(E-e,\{F-e \mid F\in \mathcal{F}\text{ and }e\in F\})$.  If $e$ is a loop or coloop, then $D\backslash e=D/e$.

There are several natural ways to get to delta-matroids.  They keep showing up, like the page where you die in a choose-your-own-adventure book.  The stairs grow dimmer and dimmer as you walk down the stone staircase into darkness.  You hear what may be screams in the distance.  You finally reach a closed door and hold your candle up to read the label, scrawled in blood.  The label on the door in this metaphor is “delta-matroids,” and they are not as scary as I portrayed them in that story.

***

***

1) “I studied embedded graphs and now I see delta-matroids everywhere.”

One way to arrive at delta-matroids is by considering cellularly embedded graphs, which I like to think of as ribbon graphs, following [EMM].

To get a cellularly embedded graph, start with a surface (compact, connected 2-manifold), then put vertices (points) and edges (curves between vertices) onto the surface so that no edges cross and each face (unbroken piece of the surface enclosed by edges and vertices) is basically a disk.  That is, no face contains a handle or cross-cap.

The particular embedding of a graph encodes more information than the abstract graph, which just encodes adjacencies.  There’s an order to the edges incident with a vertex as you circumnavigate the vertex in the embedding, but not in the abstract graph.  If you take the matroid of an embedded graph, then you lose the extra information stored in the embedding and you wind up with the matroid of the abstract graph.  For example, a pair of loops is indistinguishable from a pair of loops, even though the first pair of loops are embedded in a sphere so that the graph has three faces, and the second pair of loops is embedded in a torus so that the graph has one face.  By looking at the matroid of an embedded graph, you can’t even tell if the graph is embedded in an orientable surface or a non-orientable surface.  So matroids are the wrong object to model embedded graphs.

Here is a figure by Steven Noble, where $\mathcal{R}$ is the set of ribbon graphs.  The correspondence between graphs and matroids is akin to the correspondence between ribbon graphs and question mark.  Likewise, graphs are to embedded graphs as matroids are to question mark. Andre Bouchet showed that delta-matroids are the question mark.

To get a ribbon graph, begin with a cellularly embedded graph, cut around the vertices and edges, and throw away the faces.  The vertices have become disks, and the edges have become ribbons connecting disks. Each missing face counts as a boundary component in the ribbon graph.  We have not lost any of the information from our embedding, since the faces were just disks and can be glued back along the boundary components to return to the original presentation.  Spanning forests in a ribbon graph are exactly what you expect, and the edge sets of spanning forests of a ribbon graph give the bases of a matroid.  To get a quasi-tree, we are allowed to delete edges (remove ribbons) from our ribbon graph so that we leave behind a ribbon graph with exactly as many boundary components as the original graph had connected components.  Note that each spanning forest is a quasi-tree.  The edge sets of quasi-trees are the feasible sets of a delta-matroid.  The reader may take a break to draw a ribbon graph with quasi-trees of multiple sizes.  For more information along these lines, I refer you to [CMNR1] or [CMNR2].

You may be familiar with the mutually enriching relationship between graphs and matroids.  There appears to be a similar mutually enriching relationship between ribbon graphs and delta-matroids.  Tutte said, “If a theorem about graphs can be expressed in terms of edges and circuits only it probably exemplifies a more general theorem about matroids.”  To alter this quote for our purposes, we say, “If a theorem about ribbon graphs can be expressed in terms of edges and quasi-trees only it probably exemplifies a more general theorem about delta-matroids.”

Protip:  Not every delta-matroid can be represented by a ribbon graph.  Geelen and Oum gave an excluded minor characterization for ribbon-graphic delta-matroids in [GO].

***

2) “Partial duality brings me to delta-matroids.”

A planar graph has a nice, well-defined dual.  Not all graphs have well-defined duals.  A graph that is not planar that is cellularly embedded in a surface has a well-defined dual, but the dual depends on the surface.  The matroid of a graph has a well-defined dual, as do all matroids.  Matroids are nice and general in that sense.  The notion of partial duality was developed by Chmutov [CG] in the context of embedded graphs, which can be viewed as ribbon graphs, as discussed in ending (1).  To get the dual from a ribbon graph, replace the boundary components with vertices, and the vertices with boundary components.  Now the ribbons still link up the vertices, but they are short and thick, rather than being long and ribbony.  In fact, one way to look at taking a dual is to focus on the ribbon edges, and simply switch the parts of each edge that are incident with vertices with the parts of the edge that are incident with boundary components.  Furthermore, there’s nothing particularly special about switching parts of the ribbon edges for the entire ribbon graph, rather than just a subset of the edges.  We use $G^A$ to denote the partial dual of a ribbon-graph, $G$, with respect to the edge set $A$.

Here is a drawing of a partial dual for a ribbon graph that Iain Moffatt drew.  Actually, it is a slide from a talk by Steven Noble using Iain Moffatt’s figures, with a dash of copyright infringement.  Luckily, this is being used for educational purposes and I’m not being paid for this.  Unless Spielberg buys the movie rights to this.  Then I will cut Noble and Moffatt in on the profits.

Partial duality is also natural enough in matroids, but the partial dual of a matroid is rarely a matroid.  Recall that $X\bigtriangleup Y$ denotes symmetric difference of sets $X$ and $Y$, which is equal to $(X\cup Y)-(X\cap Y)$.  A matroid $M=(E,\mathcal{B})$ defined in terms of its bases has a dual that may be written $(E,\{E\bigtriangleup B \mid B\in\mathcal{B}\})$.  The dual of a matroid is a matroid.  Now, for a set $A\subseteq E$, the entity $(E,\{A\bigtriangleup B\mid B\in\mathcal{B}\}):=M*A$ is the partial dual with respect to $A$.  There is a way to make sure that the partial dual with respect to $A$ is a matroid.  The following result is Theorem 3.10 in [CMNR2].

Theorem.  Let $M=(E,\mathcal{B})$ be a matroid and $A$ be a subset of $E$.  Then $M*A$ is a matroid if and only if $A$ is separating or $A\in\{\emptyset,E\}$.

Whenever $A$ is not empty or the ground set of a component of the matroid, then the partial dual with respect to $A$ is (scrawled in blood) a delta-matroid!  Matroids may be too abstract for most human beings, but they are not quite abstract enough to accommodate partial duality, which is a natural notion generalizing from ribbon graphs.  Delta-matroids are the right object, and we tend to view the set of partial duals of a delta-matroid as all belonging to the same equivalence class, just as matroid theorists often view a matroid and its dual as belonging to one equivalence class.

***

3) “I left out a basis axiom when defining matroids.  Ahoy, delta-matroids!”

For a set $E$ and a collection $\mathcal{B}$ of subsets of $E$, the set $\mathcal{B}$ form the bases of matroid $(E,\mathcal{B})$ exactly when the following hold.

I) $\mathcal{B}\neq \emptyset$.

II) If $B,B’\in \mathcal{B}$ and $e\in B\bigtriangleup B’$, then there exists $f\in B\bigtriangleup B’$ such that $B\bigtriangleup \{e,f\}$ is in $\mathcal{B}$.

III) The sets in $\mathcal{B}$ are equicardinal.

Omit (III) and hello, sailor!  You have the definition of a delta-matroid!  Just change the word “bases” to the phrase “feasible sets.”

***

4) “Circle graphs seemed like fun.  Until they hatched into delta-matroids.”

You approach the nest full of circle graphs with the stealth and speed of a mongoose, only to discover they are cracking open, each containing an even delta-matroid, where even will be defined in a moment.  You should have known that a circle graph is a ribbon-graph in disguise, and a ribbon-graph is, in turn, just a dressed-up delta-matroid.  Geelen and Oum used this relationship in [GO] to find an excluded-minor characterization for ribbon-graphic delta-matroids.

A delta-matroid is even exactly when all of its feasible sets have the same parity.  They do not all have to have even parity, odd parity is also fine in an even delta-matroid, as long as the parity is exclusive.  Maybe a better title would be a monogamous delta-matroid, but maybe not.  A circle graph is easy to view as a ribbon-graph.  To get a circle graph, you start with a circle, and draw chords (straight lines) across it, and then you check to see which ones cross each other.  Your circle graph has a vertex for each chord, and an edge between each pair of vertices corresponding to chords that cross.  Go back to the chord diagram and fatten up your chords into ribbons, which cross each other.  Where two chords cross, just let one ribbon go over the other, we don’t restrict ourselves to two-dimensions.  It doesn’t matter which ribbon is higher than the other, but don’t put any twists into the edges.  Now view the circle as a big vertex.  Your chord diagram has become a ribbon-graph.  It is worth noting that the ribbon-graph corresponds to a graph embedded in an orientable surface.

The edges in your circle graph now correspond to pairs of intertwined loops in your ribbon-graph.  By intertwined, I mean that two loops, $a$ and $b$, incident with a single vertex so that, when you circumnavigate the vertex they share, you hit $a$, $b$, $a$, and then $b$; rather than $a$, $a$, $b$, and $b$.  Now the feasible sets of your delta-matroid include the empty set (because your vertex has a single boundary component), no single-element sets, and each pair $\{v,w\}$ where $vw$ is an edge in your circle graph.  Check this by drawing a big vertex and two interlaced ribbon-graph loops and tracing around the boundary components.  You will find there’s only one boundary component.  The rest of the feasible sets of the delta-matroid come from the remaining quasi-trees in the ribbon-graph, but you’ll find that, mysteriously, there are no quasi-trees with an odd number of edges.  Ribbon-graphs from orientable surfaces give even delta-matroids.  Bouchet showed that even ribbon-graph delta-matroids also come naturally from 4-regular directed graphs.  For more information along these lines, see Section 4.2 and Section 5.2 of [CMNR1].

Bouchet and Duchamp showed in [BD] that ribbon-graphs correspond to a subset of binary delta-matroids, which will be considered in (5).  They did this by giving an excluded minor characterization for binary delta-matroids.  In [GO], Geelen and Oum built on the work of Bouchet [BC] in the area of circle graphs and found pivot-minor-minimal non-circle-graphs. As an application of this they obtained the excluded minors for ribbon-graphic delta-matroids.

***

5) “C’mon, skew symmetric matrices.  This can’t end in delta-matroids.  Or can it?”

A lot of matroid theorists enjoy representable matroids, which have matrix representations.  Delta-matroids do not disappoint.  Take an $E$x$E$ skew-symmetric matrix over your favorite field.  For $A\subseteq E$, consider the $A$x$A$ submatrix obtained by restricting to the rows and columns labeled by elements in $A$.  If this submatrix is non-singular, then put $A$ into the collection $\mathcal{F}$.  Guess what $(E,\mathcal{F})$ is.  A delta-matroid!  Every ribbon-graphic delta-matroid has a partial dual that has a binary matrix representation.  If you pick a field with characteristic other than two, then your delta-matroids representable over that field will be even.  This follows from the nature of skew-symmetric matrices.  For more information along these lines, see Section 5.7 in [CMNR1]

***

6) “DNA recombination in ciliates is my cup of tea.  Who knew I was brewing delta-matroids?”

The title to this section may sound like a good pick-up line, but I have had no success with it.  Ciliates (phylum Ciliophora) are single-celled organisms that experience nuclear dimorphism.  Their cells each contain two nuclei, which contain different, but related, genomes. The DNA reconstruction in ciliates has something to do with 4-regular graphs, which can be thought of as medial graphs of ribbon graphs.  I’m out of my depth here, so I will refer you to the amazing work of people who know more about this subject.  Jonoska and Saito put together a book on biomathematics that is on my reading list.  I’ll highlight in particular an article by Brijder and Hoogeboom [BH] in this book for more delta-matroids.  While you’re waiting for your local library to order that book, I suggest checking out [AJS] by Angeleska, Jonoska, and Saito.

***

7) “I abandon this quest and run away.”

Very well, you decide to abandon this quest and run away.  You drop your axe.  You put down your boomerang.  You throw away your ninja stars.  You retire the commander of your armies, and donate your blowtorches to charity.  You turn from the Siren-like call of the delta-matroids, but what is that sound?  Is the song growing stronger even as you run away?  Yes, delta-matroids seem to be in front of you every direction you face.  After a meltdown or two, you pull yourself together and return to (0), resolved to pick a different course of action.

***

[AJS] A. Angeleska, N. Jonoska, and M. Saito. DNA recombination through assembly graphs. Discrete Applied Mathematics. 157:14 (2009) 3020–3037.

[BC] A. Bouchet, Circle graph obstructions, J. Combin. Theory Ser. B. 60 (1994) 107–144.

[BG] A. Bouchet, Greedy algorithm and symmetric matroids, Math. Program. 38 (1987) 147– 159.

[BD] A. Bouchet and A. Duchamp, Representability of delta-matroids over GF(2), Linear Algebra Appl. 146 (1991) 67–78.

[BH] R. Brijder and H. Hoogeboom, The algebra of gene assembly in ciliates.  In: N. Jonoska and M. Saito (eds.) Discrete and Topological Models in Molecular Biology. Natural Computing Series, Springer, Heidelberg (2014) 289—307.

[CG] S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial, J. Combin. Theory Ser. B. 99 (2009) 617–638.

[CMNR1] C. Chun, I. Moffatt, S. Noble, and R. Rueckriemen, Matroids, delta-matroids, and embedded graphs, arXiv:1403.0920.

[CMNR2] C. Chun, I. Moffatt, S. Noble, and R. Rueckriemen, On the interplay between embedded graphs and delta-matroids, arXiv:1602.01306.

[EMM] J. Ellis-Monaghan and I. Moffatt, Graphs on surfaces: Dualities, Polynomials, and Knots, Springer, (2013).

[GO] J. Geelen, S. Oum, Circle graph obstructions under pivoting. J. Graph Theory. 61 (2009) 1–11.

# An intro to multimatroids

In March Steve Noble came to the University of Western Australia for a few weeks and introduced me and Gordon to the <<add your favourite adjective here>> world of multimatroids. Multimatroids generalise matroids and even Delta matroids. Multimatroids were extensively studied by Bouchet in a series of papers (the first one being [B97]) and I will follow his terminology here.

Definitions

To define multimatroids we need a bit of setup. Let $K$ be a partition of a finite set $U$; in our context we call each class $k \in K$ a skew class and we say that two elements form a skew pair if they are contained in the same skew class. A subtransversal of $K$ is a subset of $U$ that contains at most one element from each skew class. A transversal is a maximal subtransversal (i.e. a subset of $U$ that contains exactly one element from each skew class). Denote by $\mathcal{S}(K)$ and $\mathcal{T}(K)$ the set of all subtransversals and transversals of $K$ respectively. Now let $r$ be a function from $\mathcal{S}(K) \rightarrow \mathbb{N}$ such that the following hold:

1. For every transversal $T$, the restriction of $r$ to $T$ induces a matroid on $T$.
2. For every subtransversal $S$ and any skew pair $\{x,y\}$ in a skew class disjoint from $S$, we have $r(S\cup x)-r(S)+r(S\cup y)-r(S)\geq 1$.

In other words, property 2. is saying that if we have a subtransversal $S$ and a skew class  $k$ not covered by $S$, then $r(S \cup x)=r(S)+1$ for all but possibly one $x \in k$.

Some properties and constructions

A multimatroid in which every skew class has size $p$ is called a $p$-matroid. From the definition it’s immediate that a 1-matroid is simply a matroid. There is another way of obtaining a multimatroid (in fact, a 2-matroid) from a matroid, as follows. Say that $N=(E,r)$ is a matroid and $N^*=(E,r^*)$ is its dual. Make two copies $E_1$ and $E_2$ of $E$, where, for every $e\in E$ we denote by $e_i$ the copy of $e$ in $E_i$ and similarly, for every subset $A \subseteq E$ we denote by $A_i$ the subset of $E_i$ corresponding to $A$ (and write $r(A_i)$ for $r(A)$, and similarly for $r^*$). Define a 2-matroid as follows: the ground set is $E_1 \cup E_2$, the skew classes are all pairs of the form $\{e_1,e_2\}$ for $e \in E$ and, for every subtransversal $S$, the rank of $S$ in the 2-matroid is $r(S\cap E_1) + r^*(S \cap E_2)$. Bouchet showed that, not only this is multimatroid, but in fact we can do a similar construction for any set of mutually orthogonal matroids $N_1,\ldots, N_n$. He also show that, conversely, if $T_1$ and $T_2$ are two disjoint transversals of a multimatroid $M$, then the matroids induced on $T_1$ and on $T_2$ are orthogonal (see [B98]).

Later in the post I’ll explain how Delta matroids correspond to 2-matroids, but first let’s talk about general multimatroids a bit more.

I defined multimatroids in terms of rank and as a matroid theorist the first question that pops to mind is whether we can talk about other matroid-like objects, such as independent sets, bases, etc. The answer is yes, but only while being extra careful. For example, we need to keep in mind that these terms only make sense when talking about subtransversals, not any subset of the ground set that you could think of. As one may expect, a subtransversal $S$ is independent if $r(S)=|S|$, otherwise it’s dependent. The bases of a multimatroid are the maximal independent sets and (by 2. above) if every skew class has size at least 2 then every base is a transversal. Finally a circuit is a minimal transversal that is not independent. In [B97] Bouchet gave equivalent definitions of multimatroids in terms of bases, independent sets and circuits.

One can also define minor operations for multimatroids, as in [B98], but here things are a bit different than for matroids: we don’t have a deletion and a contraction operation, but just a minor operation. Let $x$ by an element of a multimatroid $M=(U,K,r)$ and $k_x$ be the skew class containing $x$. Then the multimatroid $M|x$ has ground set $U’=U-k_x$, skew classes $K’=K-k_x$ and, for every subtransversal $S$ of $K’$ we have $r'(S)=r(S\cup x)-r(x)$. This looks a bit like contracting $x$ and deleting all other elements in $k_x$. In fact, if $M$ is the multimatroid obtained from a matroid $N$ and its dual as above, then $M|x_1$ is the multimatroid obtained from $N/x$ and its dual, while $M|x_2$ is the multimatroid obtained from $N\backslash x$ and its dual.

Delta matroids

First let me define what Delta matroids are. A Delta matroid is a pair $(E,\mathcal{F})$ where $E$ is a finite set and $\mathcal{F}$ is a nonempty family of subsets of $E$ (called feasible sets) satisfying the following property: if $F,F’\in \mathcal{F}$ and $x \in F\Delta F’$, then there exists $y \in F \Delta F’$ such that $F\Delta \{x,y\} \in \mathcal{F}$. Here $x$ could be equal to $y$, though if that doesn’t happen what we get is an even Delta matroid. The terminology is a bit unfortunate here, but basically a Delta matroid is even if all feasible sets have the same parity.

Now say that $(E,\mathcal{F})$ is a Delta matroid. As before, make two copies $E_1$ and $E_2$ of $E$ and define a 2-matroid as follows: the ground set is $E_1 \cup E_2$, the skew classes are all pairs of the form $\{e_1,e_2\}$ for $e \in E$ and the set of bases of the 2-matroid is $\{F_1 \cup (E_2-F_2) : F \in \mathcal{F}\}$. Then what we get is a 2-matroid and in fact all 2-matroids may be obtained this way from Delta matroids.

References

[B97] A. Bouchet, Multimatroids I. Coverings by independent sets.  SIAM J. Discrete Math 10 (1997) 626-646.

[B98] A. Bouchet, Multimatroids II. Orthogonality, minors and connectivity.  Electron. J. Combinatorics 5 (1998) R8.

# Matroids representable over the Hydra-5 and 2-regular partial fields.

Guest post by Ben Clark.

# 1. Introduction

This post is an informal discussion of an ongoing project to find the excluded minors for the classes of matroids that arise from two partial fields, the Hydra-5 partial field and the 2-regular partial field. Stefan’s earlier posts on partial fields and stabilizers are highly recommended as background.

# 2. The Classes

The Hydra-5 partial field, $\mathbb{H}_5$, was introduced by Pendavingh and Van Zwam [PZ10]. The class of $\mathbb{H}_5$-representable matroids can essentially be thought of as the class of matroids having six inequivalent representations over $\text{GF}(5)$. The excluded minors for this class of matroids are the first step in a 5-step process to find the excluded minors for the class of $\text{GF}(5)$-representable matroids, as described in [MWZ12].

The 2-regular partial field, $\mathbb{U}_2$, was introduced by Semple [S97]. Representations over this partial field are the natural generalisation of totally unimodular’ and totally near-unimodular’ representations over the regular and near-regular partial fields. It is therefore natural to ask if $\mathbb{U}_2$ captures the class of matroids representable over all fields of size at least four, but sadly that’s not the case. However, we do expect the excluded minors for the class of $\mathbb{U}_2$-representable matroids will shed light on the following problem.

Problem 1. Find the partial field $\mathbb{P}$ such that the set of $\mathbb{P}$-representable matroids is exactly the set of matroids representable over $\text{GF}(4)$ and all larger fields.

The classes of matroids that arise from $\mathbb{H}_5$ and $\mathbb{U}_2$ are closely related (the latter is a subset of the former), and it seems that any differences in the excluded-minor analyses can be confined to finite case checks.

# 3. Inequivalent representations

Broadly speaking, we follow the strategy used in Geelen, Gerards and Kapoor’s excluded-minor characterisation of the $\text{GF}(4)$-representable matroids [GGK00]. The first step is to construct a candidate matrix for an excluded minor $M$. That is, a candidate for a representation of $M$ that we construct by piecing together representations for some of its minors, say $M\backslash a$ and $M\backslash b$. The construction of this candidate representation is complicated by the possibility of inequivalent representations, that is, the minors $M\backslash a$ and $M\backslash b$ could have representations that we cannot piece together.

The way around the problem of inequivalent representations is to keep a certain minor and sufficient connectivity. We have a class of $\mathbb{P}$-representable matroids for some fixed partial field $\mathbb{P}$. A matroid $N$ is called a strong stabilizer for the class of $\mathbb{P}$-representable matroids if every $\mathbb P$-representation of $N$ extends uniquely to a $\mathbb{P}$-representation of any 3-connected $\mathbb{P}$-representable matroid having $N$ as a minor. In [GGK00], the strong stabilizer is $U_{2,4}$. For the partial fields $\mathbb{U}_5$ and $\mathbb{U}_2$, the strong stabilizer is $U_{2,5}$ (and its dual $U_{3,5}$).

We can therefore construct a candidate matrix $A$ for the excluded minor $M$. Since $M$ is an excluded minor, there must be some difference between the subdeterminants of $A$ and the corresponding subsets of $M$. In fact, we can choose $A$ so that this difference is certified by a $2\times 2$ subdeterminant of $A$ (see [MWZ12]). We call the corresponding set of elements an incriminating set for $(M,A)$.

# 4. Fragility

Since an excluded minor $M$ is minimal with respect to being non-representable, we deduce that there are no elements that can be removed (in the right way relative to the candidate matrix $A$) keeping all three of: the strong stabilizer $N$ minor, sufficient connectivity, and the incriminating set. Considering only the elements that are required to keep the $N$-minor, it follows that $M$ has some minor $M’$ with the property that $M’\backslash e$ or $M’/e$ has no $N$-minor for every element $e$ of $M’$. Matroids with this property are said to be $N$-fragile, and we can think of them as the foundation for building excluded minors.

In [GGK00], it is shown that the $\text{GF}(4)$-representable $U_{2,4}$-fragile matroids are whirls. Here, we are interested in the structure of the $\mathbb{H}_5$- and $\mathbb{U}_2$-representable $\{U_{2,5}, U_{3,5}\}$-fragile matroids.

A matroid has path width $3$ if there is an ordering $(e_1, e_2, \ldots, e_n)$ of its ground set such that $\{e_1, \ldots, e_t\}$ is $3$-separating for all $t$. Gluing a wheel to $M$ is the process of taking the generalized parallel connection of $M$ with $M(\mathcal{W}_n)$ along a triangle $T$, and deleting any subset of $T$ containing the rim element. We proved the following.

Theorem 1. [CMWZ16] Let $\mathbb{P} \in \{\mathbb{H}_5, \mathbb{U}_2\}$. Let $M$ be a $3$-connected $\{U_{2,5}, U_{3,5}\}$-fragile $\mathbb{P}$-representable matroid with $|E(M)| \geq 10$. Then either

1. $M$ or $M^*$ can be obtained by gluing up to three wheels to $U_{2,5}$; or
2. $M$ has path width $3$.

In fact, we can describe the structure of the matroids in this class in much more detail, using the concept of generalized $\Delta$-$Y$ exchange.

The idea of the proof of Theorem 1 is to bound the size of a minimal counterexample to at most 12 elements, then obtain a contradiction by finding all of the $\mathbb{H}_5$-representable $\{U_{2,5}, U_{3,5}\}$-fragile matroids up to 12 elements and showing they have the right structure. The exhaustive search was made feasible by the development of Sage Matroid.

# 5. Further work

The foremost issue is another fragility problem. We would like to use Theorem 1 to determine the minimal $\{U_{2,5}, U_{3,5}\}$-fragile $\mathbb{P}$-representable matroids that could be used to build an excluded minor. This is relatively simple for whirls in the GF$(4)$ case, and also for the matroids in the first case of Theorem 1, using the structure to find pivots’ in all but the smallest cases. However, there are families of fragile matroids of path width $3$ where these pivoting arguments are unsuccessful. These families of matroids have a flan’-type path of $3$-separations.

Beyond that, there is an intertwining problem to solve to bridge the gap from the fragile minor to the incriminating set. This is attacked using blocking sequences in [GGK00]. An approach that we hope will simplify the problem is to make stronger connectivity assumptions on $M\backslash a,b$. We proved the following.

Theorem 2. [COSW16] Let $M$ be a sufficiently large excluded minor for the class of matroids representable over $\mathbb{H}_5$ or $\mathbb{U}_2$. Then $M$ has a $\{U_{2,5}, U_{3,5}\}$-minor, and if $M$ has a pair of elements $a,b$ such that $M\backslash a,b$ is $3$-connected with a $\{U_{2,5}, U_{3,5}\}$-minor, then $M\backslash a,b$ is a $\{U_{2,5}, U_{3,5}\}$-fragile matroid.

Roughly, the strategy for proving Theorem 2 is to assume there are elements that are not fragile’, that is, elements we can remove to keep the stabilizer minor and the incriminating set. Removing such elements must destroy the $3$-connectivity, so we are able to deduce that $M\backslash a,b$ has a certain path of $3$-separations. Analysing this structure we find pivots if $M\backslash a,b$ is sufficiently large.

To date, we have no guarantee that a $3$-connected deletion pair exists, but progress towards finding such a pair is happening. Williams [W15] proved the following theorem. We say that a pair $a,b$ of elements of a 3-connected matroid $M$ is detachable if either $M\backslash a,b$ of $M/a,b$ is 3-connected.

Theorem 3. [W15] Let $M$ be a $3$-connected matroid with $|E(M)|\geq 12$. Then one of the following holds.

1. $M$ is a spike; or
2. $M$ contains a detachable pair; or
3. There is a matroid obtained by performing a single $\Delta$-$Y$ operation on$M$ that contains a detachable pair.

The next step in this direction is to obtain a splitter’ analogue of Theorem 3 that preserves a copy of a minor $N$.

# References

[COSW16] B.Clark, J. Oxley, C. Semple, G. Whittle. Excluded minors are almost fragile. Submitted.

[CMWZ16] B.Clark, D. Mayhew, G. Whittle, S.H.M. van Zwam. The structure of $\{U_{2,5}, U_{3,5}\}$-fragile matroids. SIAM J. Discrete Math., to appear.

[GGK00] J. Geelen, A.M.H. Gerards, A. Kapoor. The Excluded Minors for GF(4)-Representable Matroids. J. Combin. Theory Ser. B, 79, 247-299, 2000.

[MWZ12] D. Mayhew, G. Whittle, S.H.M. van Zwam. Stability, Fragility, and Rota’s Conjecture. J. Combin. Theory Ser. B, 102(3):760-783, 2012.

[PZ10] R.A. Pendavingh, S.H.M. van Zwam. Confinement of matroid representations to subsets of partial fields. J. Combin. Theory Ser. B, 100(6):510-545, 2010.

[S99] C. Semple. On maximum-sized k-regular matroids. Graphs Combin., 15, 441-462, 1999.

[W15] A. Williams. Detachable pairs in 3-connected matroids.. PhD thesis, Victoria University of Wellington, 2015.

# Infinite trees of matroids

Welcome back to this series of posts about infinite matroids. Up to this point we’ve seen what infinite matroids are, and we’ve started to understand the notion a little by looking at a variety of examples. Next we will look at how to stick together infinite matroids to build bigger ones. We’ll see that it is possible to stick infinitely many matroids together at once, and this will give us a flexible way to build new infinite matroids by sticking together infinitely many finite matroids.

Let’s start with a familiar way to stick matroids together: via 2-sums. Suppose that we have matroids $M_1$ and $M_2$ with ground sets $E_1$ and $E_2$, and suppose that these ground sets meet in only a single edge $e$. For the sake of simplicity, we’ll assume that $e$ isn’t a loop or a coloop in either matroid. Now we can build a new matroid $M_1 \oplus_2 M_2$ on the ground set $E_1 \triangle E_2$, by taking the circuits to be sets of the following 3 kinds:

• circuits of $M_1$ not containing $e$
• circuits of $M_2$ not containing $e$
• sets of the form $C_1 \cup C_2 -e$, where $C_1$ and $C_2$ are circuits of $M_1$ and $M_2$ respectively, both containing $e$.

These three kinds of circuit are illustrated below:

This operation is associative, in the sense we might expect: suppose we have 3 matroids $M_1$, $M_2$ and $M_3$ on ground sets $E_1$, $E_2$ and $E_3$. Suppose that $E_1 \cap E_2$ and $E_2 \cap E_3$ each contain just a single edge, and that $E_1 \cap E_3$ is empty. Then the two matroids $M_1 \oplus_2 (M_2 \oplus_2 M_3)$ and $(M_1 \oplus_2 M_2) \oplus_2 M_3$ are equal. This is easy to see by considering the six kinds of circuits which can arise in the sum, as shown here:

Usually associativity means that it doesn’t matter how we bracket a list of things that we want to add together, but in the case of 2-sums the geometry is a little different. The kind of configuration we might want to stick together with 2-sums in general is not a list of matroids, but rather a tree of matroids. More precisely, suppose we have a finite tree $T$, and that for each node $t$ of $T$ we have a matroid $M_t$ with ground set $E_t$. Suppose further that for distinct nodes $t$ and $t’$ of $T$ the intersection $E_t \cap E_{t’}$ either consists of a single edge $e_{tt’}$, if $t$ and $t’$ are neighbours in $T$, or else is empty. Let’s call a structure like this a finite tree of matroids. Then there is an unambiguously defined 2-sum of our tree of matroids, whose ground set is the union of all the sets $E_t$ but without the gluing edges’ $e_{tt’}$. A typical circuit in such a 2-sum of a tree is shown here:

More precisely, each circuit of the 2-sum is supported on some subtree $S$ of $T$. It is determined by a choice, for each node $t$ of $S$, of a circuit $C_t$ of $M_t$. This circuit $C_t$ should contain precisely those gluing edges $e_{tt’}$ of $M_t$ such that $t’$ is also in $S$. Given such a subtree $S$ and such a family of circuits $C_t$, we get a circuit of the 2-sum whose edges are the non-gluing edges of the circuits $C_t$.

All of this works just as well for sticking together infinite matroids as for finite ones. But the real power of this technique for building new infinite matroids comes when we consider 2-sums of infinitely many matroids at once.

So what happens if we try to stick together an infinite tree of matroids in the way outlined above? To understand the kinds of difficulty which can arise, let’s first of all consider the simplest possible infinite tree: an infinite ray. We certainly want to allow circuits of the following kind:

But what about ones which go all the way to the end of the ray:

Should we allow them?

It turns out that if we allow them, then we get an infinite matroid. But we also get a different infinite matroid by banning them: by only allowing those circuits whose underlying tree $S$ is finite. So we have two different options for how to build a matroid by sticking together an infinite ray of matroids. We say that the circuits are allowed to use the end of the ray in the first of these matroids, but not in the second.

We could just decide to fix one of these two as the correct’ 2-sum of the ray of matroids, but it turns out that that would be a bad idea. To see why, we should consider the interaction of 2-sums with duality. If we take the dual of the 2-sum of 2 matroids, then it turns out to be the same as the 2-sum of their duals. It follows that the same is true for finite trees of matroids: the dual of the 2-sum of a tree of matroids is the 2-sum of the duals of those matroids. This is certainly a property which we would like to keep for infinite 2-sums.

Now suppose that we have a ray of matroids, and we take the variant of the 2-sum where the circuits are not allowed to use the end, then take the dual of that 2-sum. What we get is the 2-sum of the duals of the matroids, but this time in the version where the circuits are allowed to use the end. So these two different constructions of the 2-sum of a ray are dual to each other, and we shouldn’t privilege either of them.

Formally, we take the decision about whether the circuits should be allowed to use the end of the ray to be part of the data used to specify the sum, in addition to the choices of the matroids lying along the ray. More generally, suppose that we have an infinite tree of matroids. Then in order to specify a 2-sum for this tree of matroids, we must decide, for each end of the tree, whether the circuits are allowed to use that end. Here the ends of the tree can be identified with the rays from a fixed root.

More precisely, suppose that we have a (possibly infinite) tree $T$, and that for each node $t$ of $T$ we have a matroid $M_t$ with ground set $E_t$. Suppose further that for distinct nodes $t$ and $t’$ of $T$ the intersection $E_t \cap E_t’$ either consists of a single edge $e_{tt’}$, if $t$ and $t’$ are neighbours in $T$, or else is empty. Suppose further that we have a subset $\Psi$ of the set of ends of $T$, which we intend to be the set of ends which we will allow circuits to use. Then we might hope to define the 2-sum of this tree of matroids as follows:

Each circuit of the 2-sum will be supported on some subtree $S$ of $T$, such that all rays in $S$ go to ends in $\Psi$. The circuit is determined by a choice, for each node $t$ of $S$, of a circuit $C_t$ of $M_t$. This circuit $C_t$ should contain precisely those gluing edges $e_tt’$ of $M_t$ such that $t’$ is also in $S$. Given such a subtree $S$ and such a family of circuits $C_t$, we get a circuit of the 2-sum whose edges are the non-gluing edges of the circuits $C_t$.

So the circuits would look something like this:

Unfortunately, this construction does not always give a matroid. The reason why not is related to some tricky set-theoretical issues, so we will not discuss it here. But due to a very deep and beautiful set-theoretic result called Borel Determinacy, we know that this construction will give a matroid if the set $\Psi$ is topologically nice enough (if it is a Borel set) [BC15].

There are also certain infinite trees of matroids such that the above construction will give a matroid for any set $\Psi$ of ends. This has the immediate consequence that there are a lot of infinite matroids – as many as there could possibly be. A matroid is determined by a set of subsets of its ground set, so there could be no more than $2^{2^{\aleph_0}}$ matroids on a countable set. And in fact, there are that many non-isomorphic matroids on a countable set, because there are that many possibilities for the set $\Psi$.

This means that when an infinite matroid is constructed from a tree of matroids together with a set $\Psi$ of ends, most of the information is encoded in the set $\Psi$ – if the matroids in the tree are finite, then there are only $2^{\aleph_0}$ possibilities for the tree of matroids, but there are $2^{2^{\aleph_0}}$ possibilities for $\Psi$. This huge reservoir of extra information hidden at the ends means that countable matroids behave in many ways like uncountable graphs. For example, the constructions which show that infinite graphs in general are not well-quasi-ordered under the minor relation already work for countable matroids, even though they do not work for countable graphs [BC15].

The construction we are considering also plays a key role for the reconstruction of infinite matroids from their decompositions into 3-connected parts. It is known that any finite matroid is canonically expressible as a 2-sum of a finite tree of matroids, each of which is either 3-connected or else consists of a single circuit or cocircuit [CE80, S80]. This result extends directly to infinite matroids: any matroid can be canonically expressed as a 2-sum (in the above sense) of a tree of matroids, each of which is either 3-connected or else consists of a single circuit or cocircuit [ADP15, BC16]. As with the finite result, this implies that it often suffices to prove a result only for 3-connected matroids in order to be able to deduce it for all matroids.

The construction we have considered was based on the 2-sum, one of the simplest possible ways to stick matroids together. Next time, we shall see that similar ideas work to give infinitary versions of other more complicated ways of sticking matroids together, and the surprisingly close connections between these infinitary sums and the matroids naturally arising in infinite graphs.

[ADP15] E. Aigner-Horev, R. Diestel, L. Postle. The structure of 2-separations of infinite matroids, to appear in JCTB, available here.
[BC15] N. Bowler and J. Carmesin, Infinite Matroids and Determinacy of Games, submitted to LMS, available here.
[BC16] N. Bowler and J. Carmesin, The ubiquity of Psi-matroids, preprint, available here.
[CE80] W. H. Cunningham and J. Edmonds, A combinatorial decomposition theory, Canad. J. Math. 32 (1980), no. 3, 734–765.
[S80] P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), no. 3, 305–359.