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James Oxley

Publications and source records attributed to James Oxley.

At least 37 records · Page 2Linked to original sources
arXiv2022

Small cocircuits in minimally vertically $4$-connected matroids

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when $k \le 3$. We show that every minimally vertically $4$-connected matroid with at least six elements has a $4$-element cocircuit, or a $5$-element cocircuit that contains a triangle, with the exception of a specific non-binary $9$-element matroid. Consequently, every minimally vertically $4$-connected binary matroid with at least six elements has a $4$-element cocircuit.

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arXiv2022

Generalizing Cographs to 2-Cographs

A graph in which every connected induced subgraph has a disconnected complement is called a cograph. Such graphs are precisely the graphs that do not have the 4-vertex path as an induced subgraph. We define a $2$-cograph to be a graph in which the complement of every $2$-connected induced subgraph is not $2$-connected. We show that, like cographs, $2$-cographs can be recursively defined. But, unlike cographs, $2$-cographs are closed under induced minors. We characterize the class of non-$2$-cographs for which every proper induced minor is a $2$-cograph. We further find the finitely many members of this class whose complements are also induced-minor-minimal non-$2$-cographs.

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arXiv2021

The smallest class of binary matroids closed under direct sums and complements

The class of cographs or complement-reducible graphs is the class of graphs that can be generated from $K_1$ using the operations of disjoint union and complementation. By analogy, this paper introduces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary non-comatroids for which every proper flat is a binary comatroid. The paper also proves the corresponding results for ternary matroids.

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arXiv2021

On the Cogirth of Binary Matroids

The cogirth, $g^\ast(M)$, of a matroid $M$ is the size of a smallest cocircuit of $M$. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that $g^\ast(M)\leq \frac{1}{2}\vert E(M)\vert$ when $M$ is binary, unless $M$ simplifies to a projective geometry. We also show that, when equality holds, $M$ simplifies to a Bose-Burton geometry, that is, a matroid of the form $PG(r-1,2)-PG(k-1,2)$. These results extend to matroids representable over arbitrary finite fields.

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arXiv2020

Complementation, Local Complementation, and Switching in Binary Matroids

In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set $V$ can be obtained from a complete graph on $V$ via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations for binary matroids and explore their behaviour. We characterize all binary matroids obtainable from the binary projective geometry of rank $r$ under the operations of complementation and switching. Moreover, we show that not all binary matroids of rank at most $r$ can be obtained from a projective geometry of rank $r$ via a sequence of the three generalized operations. We introduce a fourth operation and show that, with this additional operation, we are able to obtain all binary matroids.

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arXiv2020

Circuit-Difference Matroids

One characterization of binary matroids is that the symmetric difference of every pair of intersecting circuits is a disjoint union of circuits. This paper considers circuit-difference matroids, that is, those matroids in which the symmetric difference of every pair of intersecting circuits is a single circuit. Our main result shows that a connected regular matroid is circuit-difference if and only if it contains no pair of skew circuits. Using a result of Pfeil, this enables us to explicitly determine all regular circuit-difference matroids. The class of circuit-difference matroids is not closed under minors, but it is closed under series minors. We characterize the infinitely many excluded series minors for the class.

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arXiv2019

A note on the connectivity of 2-polymatroid minors

Brylawski and Seymour independently proved that if $M$ is a connected matroid with a connected minor $N$, and $e \in E(M) - E(N)$, then $M \backslash e$ or $M / e$ is connected having $N$ as a minor. This paper proves an analogous but somewhat weaker result for $2$-polymatroids. Specifically, if $M$ is a connected $2$-polymatroid with a proper connected minor $N$, then there is an element $e$ of $E(M) - E(N)$ such that $M \backslash e$ or $M / e$ is connected having $N$ as a minor. We also consider what can be said about the uniqueness of the way in which the elements of $E(M) - E(N)$ can be removed so that connectedness is always maintained.

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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

A notion of minor-based matroid connectivity

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 $\{e,g\}$. Finally, we show that $M$ is $U_{0,1} \oplus U_{1,1}$-connected if and only if every clonal class of $M$ is trivial.

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arXiv2018

Relaxations of GF$(4)$-representable matroids

We consider the GF$(4)$-representable matroids with a circuit-hyperplane such that the matroid obtained by relaxing the circuit-hyperplane is also GF$(4)$-representable. We characterize the structure of these matroids as an application of structure theorems for the classes of $U_{2,4}$-fragile and $\{U_{2,5},U_{3,5}\}$-fragile matroids. In addition, we characterize the forbidden submatrices in GF$(4)$-representations of these matroids.

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arXiv2018

A matroid extension result

Adding elements to matroids can be fraught with difficulty. In the Vámos matroid $V_8$, there are four independent sets $X_1,X_2, X_3,$ and $X_4$ such that $(X_1 \cup X_2,X_3 \cup X_4)$ is a $3$-separation while exactly three of the local connectivities $\sqcap(X_1,X_{3})$, $\sqcap(X_1,X_{4})$, $\sqcap(X_2,X_{3})$, and $\sqcap(X_2,X_{4})$ are one, with the fourth being zero. As is well known, there is no extension of $V_8$ by a non-loop element $p$ such that $X_j \cup p$ is a circuit for all $j$. This paper proves that a matroid can be extended by a fixed element in the guts of a $3$-separation provided no Vámos-like structure is present.

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arXiv2018

Generalized Laminar Matroids

Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid $M$ is Hamiltonian if it has a spanning circuit. A matroid $M$ is nested if and only if its Hamiltonian flats form a chain under inclusion; $M$ is laminar if and only if, for every $1$-element independent set $X$, the Hamiltonian flats of $M$ containing $X$ form a chain under inclusion. We generalize these notions to define the classes of $k$-closure-laminar and $k$-laminar matroids. This paper focuses on structural properties of these classes noting that, while the second class is always minor-closed, the first is if and only if $k \le 3$. The main results are excluded-minor characterizations for the classes of 2-laminar and 2-closure-laminar matroids.

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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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arXiv2016

Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VI

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 most four elements, or by removing elements in an easily described way from a special substructure of $M$. When this aim cannot be met, the earlier papers in this series showed that, up to duality, $M$ has a good bowtie, that is, a pair, $\{x_1,x_2,x_3\}$ and $\{x_4,x_5,x_6\}$, of disjoint triangles and a cocircuit, $\{x_2,x_3,x_4,x_5\}$, where $M\backslash x_3$ has an $N$-minor and is $(4,4,S)$-connected. We also showed that, when $M$ has a good bowtie, either $M\backslash x_3,x_6$ has an $N$-minor; or $M\backslash x_3/x_2$ has an $N$-minor and is $(4,4,S)$-connected. In this paper, we show that, when $M\backslash x_3,x_6$ has an $N$-minor but is not $(4,4,S)$-connected, $M$ has an internally $4$-connected proper minor with an $N$-minor that can be obtained from $M$ by removing at most three elements, or by removing elements in a well-described way from one of several special substructures of $M$. This is a significant step towards obtaining a splitter theorem for the class of internally $4$-connected binary matroids.

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arXiv2016

Towards a Splitter Theorem for Internally $4$-connected Binary Matroids VII

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 four elements, or by removing elements in an easily described way from a special substructure of $M$. When this aim cannot be met, the earlier papers in this series showed that, up to duality, $M$ has a good bowtie, that is, a pair, $\{x_1,x_2,x_3\}$ and $\{x_4,x_5,x_6\}$, of disjoint triangles and a cocircuit, $\{x_2,x_3,x_4,x_5\}$, where $M\backslash x_3$ has an $N$-minor and is \ffsc. We also showed that, when $M$ has a good bowtie, either $M\backslash x_3,x_6$ has an $N$-minor and $M\backslash x_6$ is $(4,4,S)$-connected; or $M\backslash x_3/x_2$ has an $N$-minor and is \ffsc. In this paper, we show that, when $M\backslash x_3,x_6$ has no $N$-minor, $M$ has an internally $4$-connected proper minor with an $N$-minor that can be obtained from $M$ by removing at most three elements, or by removing elements in a well-described way from a special substructure of $M$. This is the penultimate step towards obtaining a splitter theorem for the class of internally $4$-connected binary matroids.

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