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Geoff Whittle

Publications and source records attributed to Geoff Whittle.

At least 19 recordsLinked to original sources
arXiv2024

Clonal cores and flexipaths in matroids

A partitioned matroid $(M, \{X_1,X_2,\dots,X_n\})$ consists of a matroid $M$ and a partition $\{X_1,X_2,\dots,X_n\}$ of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general technique for analyzing those special properties of partitioned matroids that depend solely on the values of the connectivities $\lambda(X_i)$, the local connectivities $\sqcap(\cup_{j\in J}X_j, \cup_{k\in K}X_k,)$, and the dual local connectivities $\sqcap^*(\cup_{h\in H}X_h, \cup_{g\in G}X_g)$. In particular, we consider those partitioned matroids in which each $X_i$ is an independent, coindependent set of clones of cardinality $\lambda(X_i)$. Calling such partitioned matroids clonal-core matroids, we show that special results of the above type for partitioned matroids can be verified in general by proving them just for clonal-core matroids. Aiming at the long-term goal of finding the unavoidable minors of $4$-connected matroids, we illustrate this technique by studying $4$-paths. These are sequences $(L,P_1,P_2,\ldots, P_n,R)$ of sets that partition the ground set of a matroid so that the union of any proper initial segment of parts is $4$-separating. Viewing the ends $L$ and $R$ as fixed, we call such a partition a $4$-flexipath if $(L,Q_1,Q_2,\ldots, Q_n,R)$ is a $4$-path for all permutations $(Q_1,Q_2,\ldots, Q_n)$ of $(P_1,P_2,\ldots, P_n)$. A straightforward simplification enables us to focus on $(4,c)$-flexipaths for some $c$ in $\{1,2,3\}$, that is, those $4$-flexipaths for which $\lambda(Q_i) = c$ and $\lambda(Q_i \cup Q_j) > c$ for all distinct $i$ and $j$. Our main result for $4$-paths is that the only non-trivial case that arises here is when $c=2$. In that case, there are essentially only two possible dual pairs of $(4,c)$-flexipaths when $n \ge 5$.

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arXiv2023

What is a 4-connected matroid?

The {\em breadth} of a tangle $\mathcal{T}$ in a matroid is the size of the largest spanning uniform submatroid of the tangle matroid of $\mathcal{T}$. A matroid $M$ is {\em weakly $4$-connected} if it is 3-connected and whenever $(X,Y)$ is a partition of $E(M)$ with $|X|,|Y|>4$, then $\lambda(X)\geq 3$. We prove that if $\mathcal{T}$ is a tangle of order $k\geq 4$ and breadth $l$ in a matroid $M$, then $M$ has a weakly 4-connected minor $N$ with a tangle $\mathcal{T}$ of order $k$, breadth $l$ and has the property that $\mathcal{T}$ is the tangle in $M$ induced by $\mathcal{T}_N$. A set $Z$ of elements of a matroid $M$ is $4$-{\em connected} if $\lambda(A)\geq\min\{|A\cap Z|,|Z-A|,3\}$ for all $A\subseteq E(M)$. As a corollary of our theorems on tangles we prove that if $M$ contains an $n$-element $4$-connected set where $n\geq 7$, then $M$ has a weakly $4$-connected minor that contains an $n$-element $4$-connected set.

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arXiv2022

The excluded minors for 2- and 3-regular matroids

The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of matroids representable over the Hydra-5 partial field, and the 3-connected matroids in the class with a $U_{2,5}$- or $U_{3,5}$-minor are precisely those with six inequivalent representations over GF(5). We also prove that an excluded minor for this class has at most 15 elements.

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arXiv2022

Excluded minors are almost fragile II: essential elements

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb{P}$, let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary, and suppose $M$ has a pair of elements $\{a,b\}$ such that $M\backslash a,b$ is $3$-connected with an $N$-minor. Suppose also that $|E(M)| \geq |E(N)|+11$ and $M \backslash a,b$ is not $N$-fragile. In the prequel to this paper, we proved that $M \backslash a,b$ is at most five elements away from an $N$-fragile minor. An element $e$ in a matroid $M'$ is $N$-essential if neither $M'/e$ nor $M' \backslash e$ has an $N$-minor. In this paper, we prove that, under mild assumptions, $M \backslash a,b$ is one element away from a minor having at least $r(M)-2$ elements that are $N$-essential.

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arXiv2021

$N$-detachable pairs in 3-connected matroids III: the theorem

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the third and final paper in a series where we prove that if $|E(M)|-|E(N)| \ge 10$, then either $M$ has an $N$-detachable pair after possibly performing a single $Δ$-$Y$ or $Y$-$Δ$ exchange, or $M$ is essentially $N$ with a spike attached. Moreover, we describe the additional structures that arise if we require only that $|E(M)|-|E(N)| \ge 5$.

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arXiv2020

$N$-detachable pairs in 3-connected matroids II: life in $X$

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. A pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the second in a series of three papers where we describe the structures that arise when it is not possible to find an $N$-detachable pair in $M$. In the first paper in the series, we showed that, under mild assumptions, either $M$ has an $N$-detachable pair, $M$ has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs, or there is a 3-separating set $X$ with certain strong structural properties. In this paper, we analyse matroids with such a structured set $X$, and prove that they have either an $N$-detachable pair, or one of five particular 3-separators that can appear in a matroid with no $N$-detachable pairs.

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arXiv2020

$N$-detachable pairs in 3-connected matroids I: unveiling $X$

Let $M$ be a 3-connected matroid, and let $N$ be a 3-connected minor of $M$. We say that a pair $\{x_1,x_2\} \subseteq E(M)$ is $N$-detachable if one of the matroids $M/x_1/x_2$ or $M \backslash x_1 \backslash x_2$ is both 3-connected and has an $N$-minor. This is the first in a series of three papers where we describe the structures that arise when it is not possible to find an $N$-detachable pair in $M$. In this paper, we prove that if $M$ has no $N$-detachable pairs, then either $M$ has a 3-separating set, which we call $X$, with certain strong structural properties, or $M$ has one of three particular 3-separators that can appear in a matroid with no $N$-detachable pairs.

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arXiv2019

Fractal classes of matroids

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 number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size.

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arXiv2019

On Density-Critical Matroids

For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by $k$ independent sets is density-critical. It is straightforward to show that $U_{1,k+1}$ is the only minor-minimal loopless matroid with no covering by $k$ independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the density-critical matroids $M$ such that $d(M) > 2$ but $d(N) \le 2$ for all proper minors $N$ of $M$. All density-critical matroids of density less than $2$ are series-parallel networks. For $k \ge 2$, although finding all density-critical matroids of density at most $k$ does not seem straightforward, we do solve this problem for $k=\tfrac{9}{4}$.

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arXiv2018

Excluded minors are almost fragile

Let $M$ be an excluded minor for the class of $\mathbb{P}$-representable matroids for some partial field $\mathbb P$, and let $N$ be a $3$-connected strong $\mathbb{P}$-stabilizer that is non-binary. We prove that either $M$ is bounded relative to $N$, or, up to replacing $M$ by a $Δ$-$Y$-equivalent excluded minor, we can choose a pair of elements $\{a,b\}$ such that either $M\backslash \{a,b\}$ is $N$-fragile, or $M^* \backslash \{a,b\}$ is $N^*$-fragile.

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arXiv2018

On a generalisation of spikes

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the $(t,2t)$-property, then it is a $t$-spike. Finally, we present some properties of $t$-spikes.

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arXiv2017

Tangles and the Mona Lisa

We show how an image can, in principle, be described by the tangles of the graph of its pixels. The tangle-tree theorem provides a nested set of separations that efficiently distinguish all the distinguishable tangles in a graph. This translates to a small data set from which the image can be reconstructed. The tangle duality theorem says that a graph either has a certain-order tangle or a tree-structure witnessing that this cannot exist. This tells us the maximum resolution at which the image contains meaningful information.

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arXiv2017

A splitter theorem for 3-connected 2-polymatroids

Seymour's Splitter Theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. A series compression in such a structure is an analogue of contracting an edge of a graph that is in a series pair. A $2$-polymatroid $N$ is an s-minor of a $2$-polymatroid $M$ if $N$ can be obtained from $M$ by a sequence of contractions, series compressions, and dual-contractions, where the last are modified deletions. The main result proves that if $M$ and $N$ are $3$-connected $2$-polymatroids such that $N$ is an s-minor of $M$, then $M$ has a $3$-connected s-minor $M'$ that has an s-minor isomorphic to $N$ and has $|E(M)| - 1$ elements unless $M$ is a whirl or the cycle matroid of a wheel. In the exceptional case, such an $M'$ can be found with $|E(M)| - 2$ elements.

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arXiv2017

Quasi-graphic matroids

Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, {\em quasi-graphic matroids}, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every $3$-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a frame matroid.

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arXiv2017

Excluding Kuratowski graphs and their duals from binary matroids

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.

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arXiv2017

Yes, the "missing axiom" of matroid theory is lost forever

We prove there is no sentence in the monadic second-order language MS0 that characterises when a matroid is representable over at least one field, and no sentence that characterises when a matroid is K-representable, for any infinite field K. By way of contrast, because Rota's Conjecture is true, there is a sentence that characterises F-representable matroids, for any finite field F.

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arXiv2016

Connectivity Functions and Polymatroids

A {\em connectivity function on} a set $E$ is a function $λ:2^E\rightarrow \mathbb R$ such that $λ(\emptyset)=0$, that $λ(X)=λ(E-X)$ for all $X\subseteq E$ and that $λ(X\cap Y)+λ(X\cup Y)\leq λ(X)+λ(Y)$ for all $X,Y \subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity function of a half-integral self-dual polymatroid.

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arXiv2016

Is the missing axiom of matroid theory lost forever?

We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in monadic second-order logic and show that it is possible to finitely axiomatize matroids using only sentences in this collection. Moreover, we can also axiomatize representability over any fixed finite field (assuming Rota's conjecture holds). We prove that it is not possible to finitely axiomatize representability, or representability over any fixed infinite field, using sentences from the collection.

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