Dillon Mayhew, Gordon Royle, Geoff Whittle
We give a characterization of the internally 4-connected binary matroids that have no minor isomorphic to M(K3,3). Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a cubic or quartic Mobius ladder, or is isomorphic to one of eighteen sporadic matroids.
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Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The analysis in the splitter theorem requires the constraint that |E(M)|>15. In this article, we complement that analysis by using an exhaustive computer search to find all such pairs satisfying |E(M)|<16.
Let $M$ be an internally $4$-connected binary matroid with every element in three triangles. Then $M$ has at least four elements $e$ such that si$(M/e)$ is internally 4-connected.
In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restri…
Let M and N be internally 4-connected binary matroids such that M has a proper N-minor, and |E(N)| is at least seven. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N-minor whenever e is in a triangle of M, and M/e has no N-minor whenever e is in a triad of M, then M has a minor, M', such that M' is internally 4-connected with an N-minor, and 0 < |E(M)|-|E(M')| < 3.
Let AG(3,2)xU(1,1) denote the binary matroid obtained from the direct sum of AG(3,2) and a coloop by completing the 3-point lines between every element in AG(3,2) and the coloop. We prove that every internally 4-connected binary matroid that does not have a minor isomorphic to M(K5\e) is isomorphic to a minor of (AG(3,2)xU(1,1))*.
Let $M$ be a $3$-connected binary matroid; $M$ is internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a $4$-element fan. Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. Our aim is to show that $M$ has a proper internally $4$-connected minor with an $N$-minor that can be obtained from $M$ either by removing at most f…
We present infinite sequences of excluded minors for both the class of lifted-graphic matroids and the class of frame matroids.
Let $M$ be a $3$-connected binary matroid; $M$ is called internally $4$-connected if one side of every $3$-separation is a triangle or a triad, and $M$ is $(4,4,S)$-connected if one side of every $3$-separation is a triangle, a triad, or a $4$-element fan. Assume $M$ is internally $4$-connected and that neither $M$ nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let $N$ be an internally $4$-connected proper minor of $M$. Our aim is to show that $M$ has a proper internally $4$-connected minor with an $N$-minor that can be obtained from $M$ either by removing at…
We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in some subset of {M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in the subset. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.
We use the Strong Splitter Theorem to decompose the excluded minor class of binary matroids with no $E_4$-minor. Using this theorem we can get the 3-decomposers and the extremal internally 4-connected matroids as well as any other important matroids in the class. The matroid $E_4$ is a self-dual 10-element binary 3-connected matroid that plays a useful role in structural results. It is a single-element coextension of $P_9$, which is a single-element extension of the 4-wheel. We show that the extremal matroids in this class are the binary rank-$r$ spikes $Z_r$, the rank 3 and 4 projective geome…
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both the class of lifted-graphic matroids and the class of frame matroids, each of which generalises the class of graphic matroids. In this paper, we prove that the matroids $U_{3,7}$ and $U_{4,7}$ are the only $9$-connected excluded minors for the class of quasi-graphic matroids.
The element splitting operation on a graphic matroid, in general may not yield a cographic matroid. In this paper, we give a necessary and sufficient condition for the graphic matroid to yield cographic matroid under the element splitting operation.
We prove that the non-regular binary matroids with no $P_9^*$-minor have linear growth rate and the maximum size binary matroids with no $P_9^*$-minor are graphic. The main technique in the proof is the Strong Splitter Theorem using which we find the precise infinite families of 3-connected binary matroids with no $P_9^*$-minor.
A new minor-closed class of transversal matroids
本文刻画了横截拟阵单元素收缩仍为横截拟阵的条件，据此定义了新的次要闭横截拟阵类——路径-循环拟阵，其推广了双循环拟阵和多路径拟阵。
Split matroids form a minor-closed class of matroids, and are defined by placing conditions on the system of split hyperplanes in the matroid base polytope. They can equivalently be defined in terms of structural properties involving cyclic flats. We confirm a conjecture of Joswig and Schröter by proving an excluded-minor characterisation of the class of split matroids.
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $\mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $\mathcal{M}$ has only a finite number of excluded minors of rank $r$.
Lattice path matroids: the excluded minors
本文通过确定被排除的未成年，完整刻画了格路径拟阵的未成年封闭类。
In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.
We show that almost every matroid contains the rank-3 whirl $\mathcal{W}^3$ or the complete-graphic matroid $M(K_4)$ as a minor.
We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $\mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N \in \mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid.
The excluded minors for the intersection of bicircular and lattice path matroids
本文针对双圈拟阵与格路拟阵的交集类别，确立了其完整的排除子式列表，从而解决了一个最近提出的开放问题。
We use counting arguments to show that asymptotically almost all sparse paving matroids contain an $H$-minor, where $H$ falls into one of several simple classes of matroids. Furthermore the result holds for all $H$ in a larger class of matroids, if we restrict to asymptotically almost all sparse paving matroids of fixed rank $r$ (where $r$ is necessarily no smaller than the rank of $H$).
We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no AG(3,2)-minor is $\binom{r+1}{2}$ and characterise those matroids achieving this bound. When $r \geq 6$, the graphic matroid $M(K_{r+1})$ is the unique matroid meeting the bound, but there are a handful of smaller examples. In addition, we determine the size function for non-regular simple binary matroids with no AG(3,2)-minor and characterise the matroids of maximum size for each rank.
For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the structural characterization.
Computing excluded minors for classes of matroids representable over partial fields
本文实现了一种计算机搜索方法，用于寻找可在部分域上表示的拟阵类的“小”排除子式，枚举了二元和2-正则拟阵类至多15个元素的排除子式，并对两者的排除子式提出了猜想与证明。
Excluded minors of interval positroids that are paving matroids
本文研究区间正拟阵的排除子式，证明铺砌情形可通过松弛相关超平面归约为三个基本族，并据此完成非正拟阵情形的分类，同时给出铺砌正拟阵情形的判定准则。
We construct, for every $r \ge 3$ and every prime power $q > 10$, a rank-$r$ matroid with no $U_{2,q+2}$-minor, having more hyperplanes than the rank-$r$ projective geometry over $\mathrm{GF}(q)$.
The notion of thin sums matroids was invented to extend the notion of representability to non-finitary matroids. A matroid is tame if every circuit-cocircuit intersection is finite. We prove that a tame matroid is a thin sums matroid over a finite field k if and only if all its finite minors are representable over k. We expect that the method we use to prove this will make it possible to lift many theorems about finite matroids representable over a finite field to theorems about tame thin sums matroids over these fields. We give three examples of this: various characterisations of binary tame …
Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid $L(G)$ are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. For each cycle $C$ of $G$, since $L(G)/C$ is graphic and most graphic matroids are not bicircular lift, the class of bicircular lift matroids is not minor-closed. In this paper, we prove that the class of matroids that are graphic or bicircular lift has a finite list of excluded minors.
The excluded minors for GF(5)-representable matroids on ten elements
针对GF(5)可表示拟阵的排除子拟阵问题，本文枚举得出10个元素的该类拟阵有2128个排除子拟阵，还同时得到了至多10个元素的ℍ₂、ℍ₃、ℍ₄可表示拟阵及至多13个元素的ℍ₅可表示拟阵的排除子拟阵。
We introduce the matroid-minor coalgebra C, which has labeled matroids as distinguished basis, and coproduct given by splitting a matroid into a submatroid and complementary contraction all possible ways. We introduce two new bases for C; the first of these is is related to the distinguished basis by Mobius inversion over the rank-preserving weak order on matroids, the second by Mobius inversion over the suborder excluding matroids that are irreducible with respect to the free product operation. We show that the subset of each of these bases corresponding to the set of irreducible matroids is …
The excluded minors for 2- and 3-regular matroids
本文刻画了2-正则拟阵类的排除子式，并证明3-正则拟阵类与Hydra-5部分域可表示类一致，其排除子式至多含15个元素。
In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special 16-element matroid $Y_{16}$, a 3-connected regular matroid, a binary spike with rank at least four, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid $M^*(K_{3, n})$, $M^*(K_{3, n}^{\prime})$, $M^*(K_{3, n}^{\prime\prime})$, or $M^*(K_{3, n}^{\prime\prime\prime})$ ($n\ge 2$). Here the simple graphs $K_{3, n}^{\prime}, K_…
Let ${\bf A}={\bf A}_{n,m,k}$ be a random $n\times m$ matrix over $\mathbf{GF}_2$ wher each column consists of $k$ randomly chosen ones. Let $M$ be an arbirary fixed binary matroid. We show that if $m/n$ and $k$ are sufficiently large then as $n\to\infty$ the binary matroid induced by {\bf A} contains $M$ as a minor.
For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M \backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we show that $U_{1,2}$ and $M(\mathcal{W}_2)$ are the only connected matroids $N$ such that, whenever a matroid has an $N$-minor using $\{e,f\}$ and an $N$-minor using $\{f,g\}$, it also has an $N$-minor using $\{e…
A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is described by a given frame template over a prime field. Subject to the matroid minors hypothesis of Geelen, Gerards, and Whittle, we use our result to determine, for each projective or affine geometry $N$ over a prime field $\mathbb F$, a best-possible upper bound on the number of elements in a simple $\mathbb F$-representable matroid $M$ of sufficiently large r…
Classes of binary matroids with small lists of excluded induced minors
该研究在前期拟阵诱导子式相关刻画工作的基础上，确定了不含指定二元拟阵对作为诱导子式的拟阵类，并证明了用于刻画不含$M(K_4)$诱导子式拟阵类的结构引理。
Prohibited Minors For Graphic Matroids That Gives A Binary Gammoid After Splitting
针对拟阵分裂运算无法保持图性、二元gammoid分裂后未必仍为二元gammoid的问题，研究刻画了分裂后可得二元gammoid的图拟阵类，并得到对应图拟阵与余图拟阵的禁用子式。
The foundation of a matroid is a canonical algebraic invariant which classifies representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures, a simultaneous generalization of partial fields and hyperfields. Using deep results due to Tutte, Dress-Wenzel, and Gelfand-Rybnikov-Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain "cross-ratios" generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank co…
A Sharp Basis-Mass Inequality for Binary Matroids without a Fano Minor
针对不含F₇子式的二元拟阵，证明了基生成多项式最大值的严格上界2^r/(r+1)，明确等号成立条件及该类拟阵的最大子式闭性质。
We show that, if $q$ is a prime power at most 5, then every rank-$r$ matroid with no $U_{2,q+2}$-minor has no more lines than a rank-$r$ projective geometry over GF$(q)$. We also give examples showing that for every other prime power this bound does not hold.
In this paper, we give a cancellation-free antipode formula for the matroid-minor Hopf algebra. We then explore applications of this formula. For example, the cancellation-free formula expresses the antipode of uniform matroids as a sum over certain ordered set partitions. We also prove that all matroids over any hyperfield (in the sense of Baker and Bowler) have cancellation-free antipode formulas; furthermore, the cancellations in the antipode are independent of the hyperfield structure and only depend on the underlying matroid.
A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. …
We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for $\ell >= 2$, if $M$ is a $3$-connected, representable matroid of sufficiently high rank with no $U_{2,\ell+2}$-minor and $|E(M)| \geq (4\ell)^{r(M)/2}$, then $M$ is representable over a field of order at most $\ell$.
In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the matroid obtained from it with a Delta-Y operation; and P8. We provide a proof of this characterization.
On the density of matroids omitting a complete-graphic minor
本文证明了排除特定线子式和完全图圈拟阵子式的简单拟阵的密度受单指数函数上界控制，并给出了 frame matroid 特例下接近最优的界。
We show that, if $k$ and $\ell$ are positive integers and $r$ is sufficiently large, then the number of rank-$k$ flats in a rank-$r$ matroid $M$ with no $U_{2,\ell+2}$-minor is less than or equal to number of rank-$k$ flats in a rank-$r$ projective geometry over GF$(q)$, where $q$ is the largest prime power not exceeding $\ell$.
The excluded minors for three classes of 2-polymatroids having special types of natural matroids
本文针对自然拟阵分别属于二元拟阵类、不含M(K4)-子式的拟阵类以及串并联网络圈拟阵连通分支类的三类2-多拟阵，求出了它们的排除子式，并引入k-自然拟阵证明这些类在2-对偶下封闭。
There are only a finite number of excluded minors for the class of bicircular matroids
本文证明了双圈拟阵类仅有有限个排除子式，核心方法是利用偏置图和拟图拟阵表示，证明秩至少为十的排除子式必为拟图拟阵并包含已知小排除子式，从而得出结论。
Let $s,n \ge 2$ be integers. We give a qualitative structural description of every matroid $M$ that is spanned by a frame matroid of a complete graph and has no $U_{s,2s}$-minor and no rank-$n$ projective geometry minor, showing that every such matroid is `close' to a frame matroid. We also give a similar description of every matroid $M$ with a spanning projective geometry over a field GF$(q)$ as a restriction and with no $U_{s,2s}$-minor and no PG$(n,q')$-minor for any $q' > q$, showing that such an $M$ is `close' to a GF$(q)$-representable matroid.
A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k>4) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the …
A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We strengthen this result by dropping the condition that such element $e$…
We show that any infinite matroid can be reconstructed from the torsos of a tree-decomposition over its 2-separations, together with local information at the ends of the tree. We show that if the matroid is tame then this local information is simply a choice of whether circuits are permitted to use that end. The same is true if each torso is planar, with all gluing elements on a common face.
Almost generalized uniform matroids and excluded minors
本文通过Tutte多项式系数刻画了几乎广义均匀拟阵，并构造了其排除子式的无限族，推广了已知结果。
For any positive integer $l$ we prove that if $M$ is a simple matroid with no $(l+2)$-point line as a minor and with sufficiently large rank, then $|E(M)|\le \frac{q^{r(M)}-1}{q-1}$, where $q$ is the largest prime power less than or equal to $l$. Equality is attained by projective geometries over GF$(q)$.
In 1984, Kelly and Oxley introduced the model of a random representable matroid $M[A_n]$ corresponding to a random matrix $A_n \in \mathbb{F}_q^{m(n) \times n}$, whose entries are drawn independently and uniformly from $\mathbb{F}_q$. Whereas properties such as rank, connectivity, and circuit size have been well-studied, forbidden minors have not yet been analyzed. Here, we investigate the asymptotic probability as $n \to \infty$ that a fixed $\mathbb{F}_q$-representable matroid $M$ is a minor of $M[A_n]$. (We always assume $m(n) \geq \text{rank}(M)$ for all sufficiently large $n$, otherwise $…
The excluded 3-minors for vf-safe delta-matroids
本文通过寻找最小障碍即排除3-minors来刻画vf-safe delta-matroids和ribbon-graphic delta-matroids，并解释了binary delta-matroids中3-minors与图vertex minors的等价性。
Unavoidable Minors of Matroids with Minimum Cocircuit Size Four
本文推广了Halin-Jung及Mills-Turner的图与二元拟阵子式定理，证明了所有余圈至少有四个元素的简单拟阵必包含九个秩至多五、元素至多十二的特定拟阵之一作为子式。
The minors of matroids with an adjoint
本研究围绕带伴随的拟阵展开，明确了拟阵伴随的定义，证明了带伴随的拟阵类具有子式封闭性，为拟阵结构理论研究提供了新的结论。
Let $\mathcal{N}$ be a set of matroids. A matroid $M$ is strictly $\mathcal{N}$-fragile if $M$ has a member of $\mathcal{N}$ as minor and, for all $e \in E(M)$, at least one of $M\backslash e$ and $M/e$ has no minor in $\mathcal{N}$. In this paper we give a structural description of the strictly $\{U_{2,5},U_{3,5}\}$-fragile matroids that have six inequivalent representations over $\mathrm{GF}(5)$. Roughly speaking, these matroids fall into two classes. The matroids without an $\{X_8, Y_8, Y_8^{*}\}$-minor are constructed, up to duality, from one of two matroids by gluing wheels onto specified…
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