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#### Infinite antichains of matroids with characteristic set {p}

(University of Canterbury, 1999)

For each prime p, we construct an infinite antichain of matroids in
which each matroid has characteristic set {p}. For p=2, each of the matroids
in our antichain is an excluded minor for the class of matroids representable ...

#### The structure of 3-connected matroids of path width three

(University of Canterbury, 2005)

A 3-connected matroid M is sequential or has path width 3 if its ground set E(M) has a sequential ordering, that is, an ordering (e₁, e₂, ... , ek) such that ({e₁,e₂, .. ,,ek}, {ek+₁,ek+₂, .. ,,en}) is a 3-separation for ...

#### A chain theorem for matroids

(University of Canterbury, 2006)

Tutte's Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)| - |E(N)| = 1. Geelen and Whittle extended this theorem by showing ...

#### The structure of equivalent 3-separations in a 3-connected matroid

(University of Canterbury, 2004)

Let M be a matroid. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. This result was extended by Oxley, Semple, and Whittle, who showed that, when M is ...

#### In search of 4 - (12,6,4) designs. Part I

(University of Canterbury, 1993)

As a first step towards finding all 4-(12, 6, 4) designs which are not 5-(12, 6, 1)
designs, it is shown that if such a design has a pair of blocks with five points in common,
then there is a unique way of assigning the ...

#### An upgraded wheels-and-whirls theorem for 3-connected matroids

(University of Canterbury. Dept. of Mathematics and Statistics, 2009)

Let M be a 3-connected matroid that is not a wheel or a
whirl. In this paper, we prove that M has an element e such that M\e
or M/e is 3-connected and has no 3-separation that is not equivalent to
one induced by M.

#### Wild triangles in 3-connected matroids

(University of Canterbury, 2006)

Tutte's Triangle Lemma proves that if {a, b, c} is a triangle in a 3-connected matroid and neither M\a nor M\b is 3-connected, then M has a triad that contains a and exactly one of b and c. Hence {a, b, c} is contained in ...

#### Fork-decompositions of matroids

(University of Canterbury, 2002)

One of the central problems in matroid theory is Rota's conjecture
that, for all prime powers q, the class of GF(q)-representable matroids has a
finite set of excluded minors. This conjecture has been settled for q ≤ 4 ...

#### The structure of the 3-separations of 3-connected matroids II

(University of Canterbury. Dept. of Mathematics and Statistics, 2005)

The authors showed in an earlier paper that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid. The purpose of this paper is to show that if certain natural ...

#### Four characters suffice to convexly define a phylogenetic tree

(University of Canterbury, 2002)

It was recently shown that just five characters (functions on a finite set X) suffice to convexly define a trivalent tree with leaf set X. Here we show that four characters suffice which, since three characters is not ...