Generalized Δ–Y Exchange and k-Regular Matroids

Oxley, James; Semple, Charles; Vertigan, Dirk

doi:10.1006/jctb.1999.1947

Cited by 39 publications

(65 citation statements)

References 25 publications

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“…A further weakening, fork connectivity, was introduced by Oxley et al [21]. This notion of connectivity is related to a generalization of the familiar -Y exchange introduced by Oxley et al [20]. It is shown in [21] that Conjecture 9.1 holds if and only if it holds for fork-connected matroids.…”

Section: Connectivitymentioning

confidence: 96%

Recent work in matroid representation theory

Whittle¹

2005

Discrete Mathematics

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Section: Connectivitymentioning

confidence: 96%

Recent work in matroid representation theory

Whittle¹

2005

Discrete Mathematics

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“…Such is indeed the case for ternary matroids: U 2, 3 is a universal stabilizer for the class of ternary matroids with no U 2, 4 -minor and U 2, 4 is a universal stabilizer for the class of all ternary matroids. Also, universal stabilizers have recently proved a very useful tool in the characterizations of [19]. Unfortunately, an example in [10] shows, it seems, that, for classes beyond subclasses of binary and ternary matroids, it is often too much to ask for a reasonable set of universal stabilizers.…”

Section: Overviewmentioning

confidence: 97%

Totally Free Expansions of Matroids

Geelen

Oxley

Vertigan

et al. 2002

Journal of Combinatorial Theory, Series B

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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

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“…To prove Theorem 1.1, we use the theory of totally free matroids developed in [2] and the theory of segment-cosegment exchanges introduced by Oxley, Semple and Vertigan [5]. While we restate enough material from [2,5] to make this paper essentially self-contained, familiarity with these papers would be an advantage.…”

Section: Introductionmentioning

confidence: 98%

“…While we restate enough material from [2,5] to make this paper essentially self-contained, familiarity with these papers would be an advantage.…”

Section: Introductionmentioning

confidence: 99%