38 citations · 106 across the 18 of their papers we have counts for
20 papers · 1 filter
On the Highly Connected Dyadic, Near-Regular, and Sixth-Root-of-Unity Matroids
Ben Clark, Kevin Grace, James Oxley +1
Subject to announced results by Geelen, Gerards, and Whittle, we completely characterize the highly connected members of the classes of dyadic, near-regular, and sixth-root-of-unit…
On perturbations of highly connected dyadic matroids
Kevin Grace, Stefan H. M. van Zwam
Geelen, Gerards, and Whittle [3] announced the following result: let be a prime power, and let be a proper minor-closed class of -representa…
Relaxations of GF-representable matroids
Ben Clark, James Oxley, Stefan H. M. van Zwam
We consider the GF-representable matroids with a circuit-hyperplane such that the matroid obtained by relaxing the circuit-hyperplane is also GF-representable. We charact…
The highly connected even-cycle and even-cut matroids
Kevin Grace, Stefan H. M. van Zwam
The classes of even-cycle matroids, even-cycle matroids with a blocking pair, and even-cut matroids each have hundreds of excluded minors. We show that the number of excluded minor…
Templates for Binary Matroids
Kevin Grace, Stefan H. M. van Zwam
A binary frame template is a device for creating binary matroids from graphic or cographic matroids. Such matroids are said to conform or coconform to the template. We introduce a…
The structure of -fragile matroids
Ben Clark, Dillon Mayhew, Stefan van Zwam +1
Let be a set of matroids. A matroid is strictly -fragile if has a member of as minor and, for all , at least one of $M\…
On the existence of asymptotically good linear codes in minor-closed classes
Peter Nelson, Stefan H. M. van Zwam
Let be a sequence of codes such that each is a linear -code over some fixed finite field , where is the le…
Matroids representable over fields with a common subfield
Peter Nelson, Stefan H. M. van Zwam
A matroid is -regular if it is representable over all proper superfields of the field . We show that, for highly connected matroids having a large proje…
Fan-extensions in fragile matroids
Carolyn Chun, Deborah Chun, Dillon Mayhew +1
If S is a set of matroids, then the matroid M is S-fragile if, for every element e in E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizat…
Computer-verification of the structure of some classes of fragile matroids
Carolyn Chun, Deborah Chun, Benjamin Clark +3
This technical report accompanies the following three papers. It contains the computations necessary to verify some of the results claimed in those papers. [1] Carolyn Chun, Debora…
Intertwining connectivities in representable matroids
Tony Huynh, Stefan van Zwam
Let be a representable matroid, and subsets of the ground set. We prove that, if is sufficiently large, then there is an element such that deleting or cont…
On the relative importance of excluded minors
Rhiannon Hall, Dillon Mayhew, Stefan H. M. van Zwam
If EE is a set of matroids, then ex(EE) denotes the set of matroids that have no minor isomorphic to a member of EE. If EE' is a subset of EE, we say that EE' is /superfluous/ if e…
Matroid 3-connectivity and branch width
Jim Geelen, Stefan H. M. van Zwam
We prove that, for each nonnegative integer k and each matroid N, if M is a 3-connected matroid containing N as a minor, and the the branch width of M is sufficiently large, then t…
Representing some non-representable matroids
R. A. Pendavingh, S. H. M. van Zwam
We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, usi…
The structure of graphs with a vital linkage of order 2
Dillon Mayhew, Geoff Whittle, Stefan H. M. van Zwam
A linkage of order k of a graph G is a subgraph with k components, each of which is a path. A linkage is vital if it spans all vertices, and no other linkage connects the same pair…
Stability, fragility, and Rota's Conjecture
Dillon Mayhew, Geoff Whittle, Stefan H. M. van Zwam
Fix a matroid N. A matroid M is N-fragile if, for each element e of M, at least one of M\e and M/e has no N-minor. The Bounded Canopy Conjecture is that all GF(q)-representable mat…
The Hydra-k partial fields
R. A. Pendavingh, S. H. M. van Zwam
In the paper "Confinement of matroid representations to subsets of partial fields" (arXiv:0806.4487) we introduced the Hydra-k partial fields to study quinary matroids with inequiv…
An obstacle to a decomposition theorem for near-regular matroids
Dillon Mayhew, Geoff Whittle, Stefan H. M. van Zwam
Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or i…
Confinement of matroid representations to subsets of partial fields
R. A. Pendavingh, S. H. M. van Zwam
Let M be a matroid representable over a (partial) field P and B a matrix representable over a sub-partial field P' of P. We say that B confines M to P' if, whenever a P-representat…
Lifts of matroid representations over partial fields
R. A. Pendavingh, S. H. M. van Zwam
There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the…