\Lambda This paper is dedicated to Don Row who introduced all three authors to matroids. 1

Abstract not found

A 3-separation (A; B), in a matroid M , is called sequential if the elements of A can be ordered (a1 ; \Delta \Delta \Delta ; ak ) such that, for i = 3; \Delta \Delta \Delta ; k, (fa1 ; \Delta \Delta \Delta ; a i g; fa i+1 ; \Delta \Delta \Delta ; akg [ B) is a 3-separation. A matroid M is sequentially 4-connected if, 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 Mnx or M=x is sequentially 4-connected.

An element e of a 3--connected matroid M is essential if neither the deletion Mne nor the contraction M=e is 3--connected. Tutte's Wheels and Whirls Theorem proves that the only 3--connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3--connected matroids that have some non-essential elements showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2k or 2k+1 elements for some k 2, then M can be obtained by sticking together a (k+1)--spoked wheel and a certain 3--connected minor of M . The results proved here will be used elsewhere to ...

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 0 of M by an element x 0 such that fx; x 0 g is independent and M 0 is unaltered by swapping the labels on x and x 0 . When x is xed, a representation of Mnx 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 rst being the free spike. Finally, Seymour's 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 coxed elements occur in the initial segment of the sequence.

Abstract. For all positive integers k, theclassBk 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 seriesparallel 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, theclassB3 can be characterized by a finite list of excluded minors. 1.

An essential element of a 3--connected matroid M is one for which neither the deletion nor the contraction is 3--connected. Tutte's Wheels and Whirls Theorem proves that the only 3--connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3--connected matroid with at least one non-essential element has at least two such elements. This paper completely determines all 3--connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3--connected matroid M for which no single-element contraction is 3--connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion Mne is 3--connected.

Abstract. Let {a, b, c} be a triangle in a 3-connected matroid M. In this paper, we describe the structure of M relative to {a, b, c} when, for all t in {a, b, c}, either M\t is not 3-connected, or M\t has a 3-separation that is not equivalent to one induced by M. 1.

be a 3-connected minor of M that is minimal having N as a minor. This paper commences the study of the problem of finding a best-possible upper bound on jE(M 0) \Gamma E(N)j. The main result solves this problem in the case that N and M have the same rank.

A standard matrix representation A of a matroid M represents M relative to a fixed basis B. Deleting rows and columns of A correspond to contracting elements of B and deleting elements of E(M) −B. If M is 3-connected, it is often desirable to perform such an element removal from M while maintaining 3-connectivity. This paper proves that this is always possible provided M has no 4-element fans. We also show that, subject to a mild essential restriction, this element removal can be done so as to retain a copy of a specified 3-connected minor of M.