Totally Free Expansions of Matroids

Abstract

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 MOE of M by an element xOE such that {x, xOE} is independent and MOE is unaltered by swapping the labels on x and xOE. When x is fixed, a representation of M 0 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, 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 cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science

Representable orientations of the free spikes

“…In 2002, Geelen et al strengthened Seymour's Splitter Theorem [5]. One consequence of this paper is that the free spikes are building blocks for non-binary matroids that are representable over non-prime fields.…”

“…The free spikes were introduced in [12] and are also found in [5,4]. The rank-k (k 3) free spike, denoted k is a matroid on the ground set {a 1 , b 1 , a 2 , b 2 , .…”

Generalized Δ–Y Exchange and k-Regular Matroids

“…Furthermore, if x is freer than y, but y is not freer than x, then x is strictly freer than y. The next, and last, result of these preliminaries is a combination of Proposition 4.4(i) and Proposition 4.5(iv) of [9]. Proposition 5.6.…”

A note on clone sets in representable matroids

“…Recently clones have become important in the study of matroid representability [3][4][5][6]. We show that a sufficiently connected matroid that is representable over a small field does not have a large clone set.…”

“…The terminology follows [3,4,7]. Elements e and f in a matroid M are clones if interchanging e and f and fixing all other elements is an automorphism of M. A clonal class of M is a maximal set X ⊆ E(M) such that each pair of elements of X are clones.…”