On Extremal Connectivity Properties of Unavoidable Matroids

Wu, Zhaoyang

doi:10.1006/jctb.1998.1856

Cited by 3 publications

(2 citation statements)

References 14 publications

(21 reference statements)

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“…If two points are chosen from each of the lines in L so that each chosen point is on no other line in L; then the matroid induced by this set of points is an example of a spike. Spikes turn out to be a fundamental class of matroids (see, for example, [2,3,4,9,12]). …”

Section: Article In Pressmentioning

confidence: 99%

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

Oxley

Semple

Whittle

2004

Journal of Combinatorial Theory, Series B

121

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Tutte defined a k-separation of a matroid M to be a partition ðA; BÞ of the ground set of M such that jAj; jBjXk and rðAÞ þ rðBÞ À rðMÞok: If, for all mon; the matroid M has no mseparations, then M is n-connected. Earlier, Whitney showed that ðA; BÞ is a 1-separation of M if and only if A is a union of 2-connected components of M: When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M: r

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Section: Article In Pressmentioning

confidence: 99%

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

Oxley

Semple

Whittle

2004

Journal of Combinatorial Theory, Series B

121

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“…Ding, Oporowski, Oxley, and Vertigan [2] showed that every sufficiently large 3-connected matroid has, as a minor, U 2,n+2 , U n,n+2 , a wheel or whirl of rank n, M (K 3,n ), M * (K 3,n ), or an n-spike. Moreover, Wu [12] showed that spikes, like wheels and whirls, can be characterized in terms of a natural extremal connectivity condition. Wu [13] discussed the representability of spikes over finite fields, and found the exact numbers of n-spikes over fields with at most seven elements, and the asymptotic values for larger finite fields .…”

Section: Introductionmentioning

confidence: 99%

On the unique representability of spikes over prime fields

Wu¹,

Sun²

2006

Discrete Mathematics

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For an integer n ≥ 3, a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k ∈ {1, 2, . . . , n − 1}, the union of every set of k of these lines has rank k + 1. Spikes are very special and important in matroid theory. Wu [13] found the exact numbers of n-spikes over fields with 2, 3, 4, 5, 7 elements, and the asymptotic values for larger finite fields. In this paper, we prove that, for each prime number p, a GF (p) representable n-spike M is only representable on fields with characteristic p provided that n ≥ 2p − 1. Moreover, M is uniquely representable over GF (p).

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