Geoffrey P. Whittle

Matroid 4-Connectivity

A 3-separation (A, B), in a matroid M, is called sequential if the elements of A can be ordered (a1, ?, ak) such that, for i=3, ?, k, ({a1, ?, ai}, {ai+1, ?, ak}?B) is a 3-separation. A matroid M is sequentially 4-connected if M is 3-connected and, for every 3-separation (A, B) of M, either (A, B) or (B, A) is sequential. We prove that, if M is a sequentially 4-connected matroid that is neither a wheel nor a whirl, then there exists an element x of M such that either M\x or M/x is sequentially 4-connected.

Totally Free Expansions of Matroids

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M? of M by an element x? such that {x, x?} is independent and M? is unaltered by swapping the labels on x and x?. When x is fixed, a representation of M\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F -representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r?4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymours Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence.

Branch-width and Rota's conjecture

For a fixed finite field F and an integer k there are a finite number of matroids of branch-width k that are excluded minors for F-representability.

Cliques in dense GF( q )-representable matroids

We prove that, for any finite field F and positive integer n, there exists an integer λ such that if M is a simple F-representable matroid with no M(Kn)-minor, then |E(M)|≥λr(M).

Disjoint cocircuits in matroids with large rank

We prove that, for any positive integers n, k and q, there exists an integer R such that, if M is a matroid with no M(Kn)- or U2,q+2-minor, then either M has a collection of k disjoint cocircuits or M has rank at most R. Applied to the class of cographic matroids, this result implies the edge-disjoint version of the Erdos-Posa Theorem.

On matroids of branch-width three

For all positive integers k, the class Bk of matroids of branch-width at most k is minor-closed. When k is 1 or 2, the class Bk is, respectively, the class of direct sums of loops and coloops, and the class of direct sums of series-parallel networks. B3 is a much richer class as it contains infinite antichains of matroids and is thus not well-quasi-ordered under the minor order. In this paper, it is shown that, like B1 and B2, the class B3 can be characterized by a finite list of excluded minors.

On Weak Maps of Ternary Matroids

LetMandNbe ternary matroids having the same rank and the same ground set, and assume that every independent set inNis also independent inM. The main result of this paper proves that ifMis3-connected andNis connected and non-binary, thenM=N. A related result characterizes precisely when a matroid that is obtained by relaxing a circuit-hyperplane of a ternary matroid is also ternary.

A 2-Isomorphism Theorem for Hypergraphs

One can associate a polymatroid with a hypergraph that naturally generalises the cycle matroid of a graph. Whitneys 2-isomorphism theorem characterises when two graphs have isomorphic cycle matroids. In this paper Whitneys theorem is generalised to hypergraphs and polymatroids by characterising when two hypergraphs have isomorphic associated polymatroids.

Bridging Separations in Matroids

Let

On the non-uniqueness of q -cones of matriod

(X_1,X_2)