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.

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.

Guest post by Yohann Benchetrit.

Introduction

The topic of this post is to introduce a new parameter $\beta$ measuring the complexity of the facets of the matching polytope of a graph, which also extends to binary matroids. We give a simple efficient algorithm deciding whether a binary matroid satisfies $\beta\leq 1$. This is the first non-trivial case after bipartiteness, which is equivalent to $\beta=0$. The results presented here are joint work with András Sebő, and appeared in [1].

The matching polytope of a graph plays an important role in Graph Theory and Combinatorial Optimization. The methods involved in the study of its structure and properties initiated many relations between polyhedra, min-max theorems and polynomial-time algorithms.

A matching of a graph is a set of pairwise non-incident edges. The incidence vector of a matching $M$ of a graph $G$ is the 0-1 vector $\chi^{M}$ of $\mathbb{R}^{E(G)}$ defined for every $e\in E(G)$ by: $\chi^{M}(e)=1$ if and only if $e\in M$. The matching polytope of $G$, denoted by $\mathsf{MATCH}(G)$, is the convex hull of the incidence vectors of the matchings of $G$. As a polyhedron, it is the set of solutions of a finite number of linear inequalities over $\mathbb{R}^{E(G)}$. Basically, information on the inequalities describing this polytope is useful as it allows applying linear programming tools (for example the duality theorem, the Ellipsoid method, etc.).

The following result, due to Edmonds and Pulleyblank [5], gives a complete description of the matching polytope using linear inequalities. A graph $H$ is factor-critical if for every $v\in V(H)$, the graph obtained from $H$ by deleting $v$ and every edge incident to $v$ has a perfect matching (a matching covering every vertex).

Theorem 1.  For every graph $G$:
\begin{equation*}\label{fonda-eqn-MATCH}
\mathsf{MATCH}(G):=\left\{x\in\mathbb{R}^{E(G)}\colon\begin{array}{cc}
x\geq 0, & \\
\displaystyle \sum_{\text{e incident to $v$}}x_e\leq 1 & \forall v\in V(G), \\
\displaystyle \sum_{e\in E(H)}x_e\leq \frac{|V(H)|-1}{2} & \text{$\forall H$ 2-connected factor-critical } \\
& \text{induced subgraph of $G$}
\end{array}\right\}.
\end{equation*}

Furthermore, Edmonds [4] shows that each inequality associated with a 2-connected factor-critical induced subgraph $H$ appears in every description of $ \mathsf{MATCH}(G)$.

An ear-decomposition of a graph $G$ is a sequence $(P_0,\ldots,P_k)$ of a circuit $P_0$ and paths (with two distinct ends) $P_1,\ldots,P_k$ of $G$ such that $E(G)=E(P_0)\cup\ldots\cup E(P_k)$ and for every $i\in\left\{
1,\ldots,k\right\}$, $P_i$ meets $P_0\cup\ldots\cup P_{i-1}$ exactly on its two ends; the $P_i$ are the ears of the decomposition (notice that we do not allow an ear to attach on a single vertex). The decomposition is odd if all ears have an odd number of edges. Lovász proved:

Theorem 2.  A graph is 2-connected and factor-critical if and only if it has an odd ear-decomposition.

It follows from Menger’s theorem that a graph admits an ear-decomposition if and only if it is 2-connected. In addition, all ear-decompositions of a 2-connected graph $G$ have the same number $|E(G)|-|V(G)|+1$ of ears (indeed, deleting a single edge from each ear of an arbitrary ear-decomposition yields a spanning tree), hence the number of ears of $G$ is well-defined.

This and Lovász’s result suggest the following measure of complexity of the matching polytope:

Definition 3.  For each graph $G$, let $\beta(G)$ denote the largest number of ears of a 2-connected factor-critical subgraph of $G$.

For example, $\beta(G)=0$ if and only if $G$ is a bipartite graph, and $\beta(G)\leq 1$ if and only if $G$ does not contain three pairwise internally-disjoint paths with the same ends, such that exactly two of them have an odd number of edges.

The Parity Minor Algorithm [6] implies that for each fixed $k$, determining whether a graph $G$ satisfies $\beta(G)\geq k$ can be done in polynomial-time.  However, the proof in [6] is built upon elaborate techniques of the Graph Minor Project, and is oriented towards generality. This suggests searching for more adapted algorithms. In this direction, Bruhn and Schaudt [2] provided a direct solution to test $\beta\leq 1$ efficiently for graphs with maximum degree 3.

In the rest of this post, we extend $\beta$ to binary matroids and characterize those which satisfy $\beta\leq 1$. As a consequence, we obtain an elementary polynomial-time algorithm for testing whether a binary matroid $M$ satisfies $\beta(M)\leq 1$ or finding an obstruction, only using ear-decompositions and basic computations in the cycle space (mod 2). This completely avoids Graph Minors and implies a rather simple algorithm deciding $\beta\leq 1$ in graphs, with no restriction on the degree.

Also, applying our result to the co-graphic case yields an unexpected consequence: determining whether a graph has two odd bonds meeting on an even number of edges can be carried out efficiently.

Our motivation in studying $\beta$ comes from the recognition problem for $h$-perfect graphs. A graph is $h$-perfect if the convex hull of the incidence vectors of its stable sets (sets of pairwise non-adjacent vertices) is completely described by non-negativity and rank-inequalities of cliques and odd circuits.  The class of $h$-perfect graphs are a superclass of perfect graphs, and share a number of similar properties with the latter (see the second volume of Schrijver’s Combinatorial Optimization for further details).

Whereas perfection can be checked in polynomial-time, it is still open whether the same holds for $h$-perfection. Testing the $h$-perfection of a line graph $L(G)$ is essentially equivalent to checking $\beta(G)\leq 1$, and thus our results on $\beta$ directly imply a simple algorithm recognizing $h$-perfect line graphs.

Testing $\beta\leq 1$ in Binary Matroids

An ear-decomposition of a matroid $M$ is a sequence $(C_1,\ldots,C_k)$ of circuits of $M$ such that: $C_1\cup \ldots \cup C_k=E(M)$ and for each $i\in\left\{2,\ldots,k \right\}$, $C_i$ meets both $\cup_{j<i}C_j$ and its complement, and $C_i\setminus (\cup_{j<i}C_j)\neq\emptyset$ is a circuit of the contraction $M/(\cup_{j<i}C_j)$. The ear-decomposition is odd if the sets $C_i\setminus (\cup_{j<i}C_j)$ (the ears) all have odd cardinality (for $i\in\left\{1,\ldots,k\right\}$).

A matroid is factor-critical if it has an odd ear-decomposition.  It is easy to check that a matroid has an ear-decomposition if and only if it is connected, and that all the ear-decompositions of a connected binary matroid have the same number $|E(M)|-\mathsf{rk}(M)$ of ears. Hence, for each binary matroid $M$,  we may define $\beta(M)$ as the largest number of ears of a factor-critical restriction of $M$. It is straightforward to check that the values of $\beta$ for a graph and its circuit matroid are equal.

For algorithmic considerations, we assume that binary matroids are given by a linear representation or an independence oracle; complexity refers to the number of required calls to this oracle.

The first important observation is the following (see [1] for a proof).

Lemma 4.  A connected binary matroid $M$ satisfies $\beta(M)\geq 2$ if and only if it has two odd circuits which meet in an even number of elements.  Furthermore, a factor-critical restriction of $M$ with two ears can be constructed from two such odd circuits in polynomial-time.

This states that we have to check the parity of the intersection of every pair of odd circuits of $M$ to certify that $\beta(M)\leq 1$. In fact, we need only to check it for pairs of a certain basis of the cycle space of $M$.

A cycle of a matroid is a union of disjoint circuits, and it is odd if it has odd cardinality.  The set of (incidence vectors of) cycles of a binary matroid $M$ is a subspace of $\mathbb{F}_2^{E(M)}$ (this characterizes binary matroids), called the cycle space of $M$ and denoted by $\mathcal{C}(M)$. Clearly, $\mathcal{C}(M)$ is spanned by the circuits of $M$ and $\dim \mathcal{C}(M)=|E(M)|-\mathsf{rk}(M)$ (consider the set of fundamental circuits of a basis). A set $S$ of cycles of $M$ is a cycle basis of $M$ if the incidence vectors of the elements of $S$ form a basis of $\mathcal{C}(M)$.
A cycle basis $B$ is odd if all its cycles are odd, and it is totally odd if moreover all the cycles of $B$ pairwise-intersect on an odd number of elements.

We now prove the main ingredient of our characterization. The fact that we have only circuits in the basis obtained is crucial.

Lemma 5.  Each connected non-bipartite binary matroid has an odd cycle basis formed by circuits only. It can be constructed in polynomial time.

Proof. Let $M$ be a connected and non-bipartite binary matroid. Let $M_p$ be the binary matroid obtained by adding successively an all-0 column and an all-1 line to a matrix representation of $M$, and let $p$ be the new element of $M_p$.
Lehman’s matroid-port theorem easily implies that each element of $M$ belongs to an odd circuit (see [8]). It follows straightforwardly that $M_p$ is a connected matroid.
Furthermore, a theorem of Coullard and Hellerstein states that for each connected matroid $N$ and every $e\in E(N)$, there exists an ear-decomposition of $N$ whose circuits all contain $e$ and which can be found in polynomial-time [3].

Now, consider an ear-decomposition $\left\{C_1,\ldots,C_k \right\}$ of $M_p$ such that all the $C_i$ contain $p$. It is easy to check that $\left\{C_1\setminus \left\{p\right\},\ldots,C_k\setminus \left\{p\right\}\right\}$ is an odd cycle basis of $M$, formed by circuits only. $\blacksquare$

The following statement is a direct consequence of the well-known fact that, for two subsets $S_1,S_2$ of a set $S$: the sum of the incidence vectors of $S_1$ and $S_2$ in $\mathbb{F}_2^{S}$ is the incidence vector of $S_1\Delta S_2$ and their product is the parity of $ |S_1\cap S_2|$ (with respect to the standard bilinear form on $\mathbb{F}_2^{S}$).

Lemma 6. If a binary matroid has a totally odd cycle basis, then all its odd cycles pairwise-intersect in an odd number of elements.

We can finally prove our characterization.

Theorem 7.  Let $M$ be a connected non-bipartite binary matroid. The following statements are equivalent.

1. $\beta(M)\leq 1$,
2. $M$ has a totally odd cycle basis formed by circuits only,
3. each odd cycle basis of $M$ is totally odd.

Proof.  We first prove that $ (1)\Rightarrow (2) $. Since $\beta(M)\leq 1$ and as $M$ is connected and non-bipartite, Lemma 5 shows that $M$ has an odd cycle basis $B$ whose elements are circuits. Lemma 4 implies that the elements of $B$ pairwise-intersect in an odd number of elements. That is, $B$ is totally odd.

The implication $ (2)\Rightarrow (3) $ straightforwardly follows from Lemma 6.

We finally prove $ (3)\Rightarrow (1) $. Suppose that each odd cycle basis of $M$ is totally odd.  Since $M$ is connected and non-bipartite, Lemma 5 shows that $M$ has an odd cycle basis $B$. Since $B$ is totally odd, Lemma 6 implies that all odd cycles, and in particular all odd circuits of $M$, pairwise-intersect in an odd number of elements. By Lemma 4, we get $\beta(M)\leq 1$. $\blacksquare$

Now, testing whether a connected non-bipartite binary matroid $M$ satisfies $\beta(M)\leq 1$ can be done as follows: build an odd cycle basis $B$ of $M$ formed by circuits only (with Lemma 5), and compute the parities of the intersections of pairs of elements of $B$; there is only a polynomial number of such pairs, since $|B|=\dim \mathcal{C}(M)=|E(M)|-\mathsf{rk}(M)$.
If two elements of $B$ meet in an even number of elements, then Lemma 4 shows $\beta(M)\geq 2$ and a factor-critical restriction of $M$ with exactly two ears. Otherwise, $B$ is totally odd and Theorem 7 implies that $\beta(M)\leq 1$.

Clearly, a binary matroid $M$ satisfies $\beta(M)\leq 1$ if and only if every non-bipartite block of $M$ satisfies this condition. The blocks of $M$ can be easily found in polynomial-time, and hence we need only one more subroutine to finish the algorithm: deciding in polynomial-time whether a connected binary matroid is bipartite. This can be carried out using the following straightforward proposition, which generalizes the bipartiteness test of graphs.

Proposition 8.  Let $M$ be a connected binary matroid. The following statements are equivalent.

1.  $M$ is bipartite,
2. There exists a cycle basis of $M$ containing only even circuits,
3. Each cycle basis of $M$ contains only even cycles.

Hence, testing bipartiteness only requires building a cycle basis formed by circuits (from the fundamental circuits of a basis of $M$, for example) and checking whether all its circuits are even. If so, the matroid is bipartite and otherwise we find an odd circuit.

Open Problems

$\beta$ can be used as a parameter to separate on, for properties of the matching polytope (for example, the integer decomposition property [1]).  The complexity of computing $\beta$ for graphs is apparently not known.

Clearly, the property $\beta(G)\geq k$ is in $\mathsf{NP}$ (as factor-criticality can be tested using a maximum-matching algorithm), but we do not know if it belongs to $\mathsf{co}$-$\mathsf{NP}$. Also, it is not clear whether the Parity Minor algorithm can be circumvented to check efficiently, for fixed $k\geq 3$, whether a graph $G$ satisfies $\beta(G)\geq k$.

For binary matroids, the situation is even less clear: results of Szegedy and Szegedy [9] show that testing whether a binary matroid is factor-critical can be done in randomized polynomial-time, but a deterministic algorithm is still missing, even for co-graphic matroids. Finally, it is not clear whether $\beta\leq 1$ can be decided efficiently for other interesting classes of matroids (our approach does not directly extend to anything else).

References

[1] Y. Benchetrit and A. Sebő. Ear-decompositions and the complexity of the matching polytope. arXiv preprint, 2015.

[2] H. Bruhn and O. Schaudt. Claw-free t-perfect graphs can be recognised in polynomial time. In Jon Lee and Jens Vygen, editors, IPCO, volume 8494 of LNCS, pages 404–415. 2014.

[3] C. Coullard and L. Hellerstein. Independence and port oracles for matroids, with an application to computational learning theory. Combinatorica, 16(2):189–208, 1996.

[4] J. Edmonds. Maximum matching and a polyhedron with 0, 1 vertices. J. of Res. the Nat. Bureau of Standards, 69 B:125–130, 1965.

[5] J. Edmonds and W. Pulleyblank. Facets of 1-matching polyhedra. In Hypergraph Seminar of Columbus, 1974.

[6] K. Kawarabayashi, B. Reed, and P. Wollan. The graph minor algorithm with parity conditions. In FOCS, 2011 IEEE 52nd Annual Symposium, pages 27–36. IEEE, 2011.

[7] L. Lovász. A note on factor-critical graphs. Studia Sci. Math. Hungar., 7:279–280, 1972.

[8] James Oxley. Matroid Theory. 1992.

[9] B. Szegedy and C. Szegedy. Symplectic spaces and ear-decomposition of matroids. Combinatorica, 26(3):353–377, 2006.