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

Oxley, James; Semple, Charles; Whittle, Geoff

doi:10.1016/j.ejc.2006.01.007

Cited by 18 publications

(19 citation statements)

References 14 publications

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“…Then Theorem 1 tells us that G is connected and n ≥ 5, as every connected graph with n ≤ 4 has a pendant vertex or a pair of twins. It turns out that splits of G are related to a certain type of ordinary 3-separation, which has been studied by Oxley, Semple and Whittle [16,17]. We begin with a couple of lemmas.…”

Section: Splits Of G and Separations Of M [Ias(g)]mentioning

confidence: 99%

“…Most of the balance of the paper is devoted to proving the results already mentioned. In Section 5 we observe that there is a connection between splits of G and the analysis of 3-separations of matroids due to Oxley, Semple and Whittle [16,17]. In Section 8 we deduce a characterization of circle graphs from the above results and Bouchet's circle graph obstructions theorem [8].…”

Section: Some Relevant Examples Appear Inmentioning

confidence: 99%

“…The theory of non-sequential ordinary 3-separations presented by Oxley, Semple and Whittle [16,17] includes the following notion of equivalence. (G)] is 3-connected.…”

Section: Corollaries 22 and 25 Tell Us That If G Is A Connected Graphmentioning

confidence: 99%

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Isotropic Matroids III: Connectivity

Brijder

2017

Electron. J. Combin.

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The isotropic matroid M [IAS(G)] of a graph G is a binary matroid, which is equivalent to the isotropic system introduced by Bouchet. In this paper we discuss four notions of connectivity related to isotropic matroids and isotropic systems. We show that the isotropic system connectivity defined by Bouchet is equivalent to vertical connectivity of M [IAS(G)], and if G has at least four vertices, then M [IAS(G)] is vertically 5-connected if and only if G is prime (in the sense of Cunningham's split decomposition). We also show that M [IAS(G)] is 3-connected if and only if G is connected and has neither a pendant vertex nor a pair of twin vertices. Our most interesting theorem is that if G has n ≥ 7 vertices then M [IAS(G)] is not vertically n-connected. This abstract-seeming result is equivalent to the more concrete assertion that G is locally equivalent to a graph with a vertex of degree < n−1 2 .

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Section: Splits Of G and Separations Of M [Ias(g)]mentioning

confidence: 99%

Section: Some Relevant Examples Appear Inmentioning

confidence: 99%

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Isotropic Matroids III: Connectivity

Brijder

2017

Electron. J. Combin.

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“…A quad in a matroid is a 4-element set that is both a circuit and a cocircuit. This paper will use some results and terminology from our papers describing the structure of 3-separations in 3-connected matroids [10,11]. In this section, we introduce the relevant definitions.…”

Section: Preliminariesmentioning

confidence: 99%

A chain theorem for matroids

Oxley

Semple

Whittle

2008

Journal of Combinatorial Theory, Series B

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Tutte's Wheels-and-Whirls Theorem proves that if M is a 3-connected matroid other than a wheel or a whirl, then M has a 3-connected minor N such that |E(M)| − |E(N)| = 1. Geelen and Whittle extended this theorem by showing that when M is sequentially 4-connected, the minor N can also be guaranteed to be sequentially 4-connected, that is, for every 3-separation (X, Y ) of N , the set E(N) can be obtained from X or Y by successively applying the operations of closure and coclosure. Hall proved a chain theorem for a different class of 4-connected matroids, those for which every 3-separation has at most five elements on one side. This paper proves a chain theorem for those sequentially 4-connected matroids that also obey this size condition.

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“…. , P n ) of the ground set of a 3-connected matroid M is a flower [6,7] if λ M (P i ) = 2 = λ M (P i ∪ P i+1 ) for all i in {1, 2, . .…”

Section: Introductionmentioning

confidence: 99%