Guest post by Jim Geelen.
Bert Gerards, Geoff Whittle and I have deleloped a structure theory for minor-closed classes of matroids representable over a given finite field; our results are of a similar flavour to the Graph Minors Structure Theorem of Robertson and Seymour [RS03]. I believe that a similar structure theory should emerge for any minor-closed class that omits a uniform matroid; see [Gee08]. However, the tools and intuition that we have developed for addressing matroids over finite fields fail when we consider minor-closed classes that contain all uniform matroids. Is this barrier the natural end of structure theory?
This question has led me to the following natural problem.
Problem 1: Describe the class of matroids with no $M(K_4)$-minor.
What kind of description do I want?
This is a complicated question for which I have no good answer. Whitney, himself, got away with posing the vague problem of characterizing the representable matroids, and I’m hoping for similar leniency here.
Pendavingh and van der Pol [PvdP] give a lower bound on the number of $n$-element matroids with no $M(K_4)$-minor that is asymptotic to Knuth’s lower bound [Knu74] on the number $n$-element matroids. In particular, the bound grows faster than $2^{p(n)}$ for any polynomial $p(n)$, making it quite difficult to give a rigorous complexity theoretic definition of a description.
What do we know?
Let $EX(M(K_4))$ denote the class of matroids with no $M(K_4)$-minor. Obviously $EX(M(K_4))$ is closed under minors, duality, direct sums, and $2$-sums.
Let $e$ be an element of a matroid $M$ and let $F$ be a flat of $M\backslash e$. If $F\cup \{e\}$ is a cyclic flat of $M$ and each cyclic flat of $M$ that contains $e$ also contains $F$, then we call $M$ a principal extension of $M\backslash e$. The following result implies that $EX(M(K_4))$ contains all transversal matroids and, hence, all gammoids.
Lemma 1: $EX(M(K_4))$ is closed under principal extension.
Proof.
Suppose that $M$ is obtained from $M\backslash e$ by principal extension into the flat $F$ and that $M\backslash e$ has no $M(K_4)$-minor. Toward a contradiction suppose that $D$ and $C$ are disjoint sets such that $M\backslash D/ C$ is isomorphic to $M(K_4)$. By possibly removing elements we may assume that $C\subseteq \{e\}$ and that $D\subseteq F$. Since one cannot obtain an $M(K_4)$-minor by parallel-extension, $r_M(F)\ge 2$. Each element of $M(K_4)$ is on the intersection of two triangles but $e$ is not, so $C=\{e\}$.
Consider a triangle $T$ of $M(K_4)$. If $T$ is independent in $M$, then $F$ is contained in the closure of $T\cup \{e\}$. Since $r(F)\ge 2$, at most two of the triangles of $M(K_4)$ become independent in $M$. Thus at least two of the triangles remain. Since $M(K_4)$ is not a restriction of $M\backslash e$, exactly two of the triangles remain. Let $f$ be the unique point not on one of those two triangles. Now it is easy to check that $\{e,f\}$ spans $F$ and $M/ f\backslash e$ has an $M(K_4)$-minor. $\Box$
The constructions described so far preserve representability, here is a construction that does not; the proof is obvious and omitted.
Lemma 2: $EX(M(K_4))$ is closed under the relaxation of circuit hyperplanes.
This is close to all that I know about the class.
Connectivity does not help
Matroid structure theory is built upon the founding principal that structure smooths out with increased connectivity. Unfortunately, this is just not the case for the class of matroids with no $M(K_4)$-minor. For each matroid $M$ we can construct, via principal extension, a round matroid $M’$ such that $M$ has an $M(K_4)$-minor if and only if $M’$ does. (A round matroid is one with infinite vertical connectivity.)
I don’t really know where to go from here, but we may learn something by retreating into some easier classes.
Sparse paving matroids
Motivated by the conjecture that almost all matroids are sparse paving and by the result in [PvdP] that there are many sparse paving matroids with no $M(K_4)$-minor, this seems to be a natural class to consider.
Problem 2: Describe the class of sparse paving matroids with no $M(K_4)$-minor.
This problem looks complicated, even in rank 3. In higher rank, there are quite subtle constructions. Let $k$, $n$, and $m$ be positive integers with $k\le m$ and let $S(k,m,n)$ be the collection of all $k$-element subsets of $\{1,\ldots,n\}$ whose elements sum to $m$ modulo $n$. Now let $M(k,m,n)$ be the rank-$k$ matroid on $\{1,\ldots,n\}$ whose nonspanning circuits are $S(k,m,n)$. Now $M(k,m,n)$ is clearly a sparse paving matroid. Pendavingh and van der Pol prove that, if $n$ is odd, then
$M(k,m,n)$ has no $M(K_4)$-minor. This construction leads to an enormous number of $n$-element sparse paving matroids with no $M(K_4)$-minor since we can relax any subset of the circuit-hyperplanes $S(k,m,n)$ in $M(k,m,n)$.
This diversion into the class of sparse paving matroids does not look that promising. However, sparse paving matroids are, themselves, highly structured. They are both a single lift of a uniform matroid and a single projection of a uniform matroid. It might be necessary to take the class of sparse paving matroids with no $M(K_4)$-minor as one of the building blocks for $EX(M(K_4))$.
Back to finite fields
Having failed to come up with any significant general insights into matroids with no $M(K_4)$-minor, I will return to the comfort of finite fields, where we know considerably more. This is not likely to shed much light on the general problem, but interesting questions remain nonetheless.
For binary and ternary matroids we have exact characterizations due to Tutte and Oxley, respectively. Note that it suffices to characterize the $3$-connected matroids in the class.
Theorem 1: Every $3$-connected binary matroid with at least $4$ elements has an $M(K_4)$-minor.
Theorem 2: If $M$ is a $3$-connected ternary matroid with no $M(K_4)$-minor, then either $M$ is a whirl or $M$ is isomorphic to a minor of a particular $12$-element matroid $S(5,6,12)$.
For arbitrary finite fields we have a bound on the branch-width [GGW07].
Theorem 3: For each finite field $\mathbb{F}$ there is an integer $k$ such that, if $M$ is an $\mathbb{F}$-representable matroid with no $M(K_4)$-minor, then $M$ has branch-width at most $k$.
That theorem holds more generally with $K_4$ replaced with any planar graph. One would hope to be able to say more for $K_4$.
Conjecture: For each finite field $\mathbb{F}$ there are only finitely many vertically $4$-connected $\mathbb{F}$-representable matroids with no $M(K_4)$-minor.
In fact, I hope for even more, one should be able to describe how to construct all $3$-connected $\mathbb{F}$-representable matroids with no $M(K_4)$-minor. The construction should involve some thin infinite classes, a finite set of exceptional matroids, and a $3$-sum operation. One important thin class is the set of free spikes, and the matroids obtained from these by adding other points freely to the legs. Another thin class is constructed in a similar way from whirls. The $3$-sum operation needs to be carefully defined so that it does not construct $M(K_4)$-minors. For example, suppose that $M_1$ and $M_2$ are matroids, $E(M_1)\cap E(M_2) = L$, and $a,b\in L$ such that $L$ is a line of both $M_1$ and $M_2$, $L$ is modular in $M_2$, each point of $M_2$ in $L-\{a,b\}$ is freely placed on $L$, and neither $M_1$ nor $M_2\backslash (L-\{a,b\})$ has an $M(K_4)$-minor. Now let $M$ be obtained from the modular sum of $M_1$ and $M_2$ on $L$ by deleting $L-\{a,b\}$. Then $M$ has no $M(K_4)$-minor.
References
[Gee08]
J. Geelen,
Some open problems on excluding a uniform matroid,
Adv. in Appl. Math. 41 (2008), 628-637.
[GGW07]
J. Geelen, B. Gerards, G. Whittle,
Excluding a planar graph from GF$(q)$-representable matroids,
J. Combin. Theory, Ser. B 97 (2007), 971-998.
[Knu74]
D.E. Knuth,
The asymptitic number of geometries,
J. Combin. Theory, Ser. A 17 (1974), 398-400.
[RS03]
N. Robertson, P.D. Seymour,
Graph Minors. XVI. Excluding a non-planar graph,
J. Combin. Theory, Ser. B 89 (2003), 43-76.
[PvdP]
R.A. Pendavingh, J.G. van der Pol,
Counting matroids in minor-closed classes,
arXiv:1302.1315v3 [math.CO].