A Splitter Theorem Relative to a Fixed Basis

Brettell, Nick; Semple, Charles

doi:10.1007/s00026-013-0208-3

Cited by 16 publications

(28 citation statements)

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“…In particular, Theorem 2.31 implies Theorem 1.1. Most of the relevant results and terminology on matroid connectivity can either be found in Oxley [12] or in the recent literature on removing elements relative to a fixed basis [3,14,24]. The results and terminology on matroid representation theory can be found in [11,16,17].…”

Section: Preliminaries and The Main Theoremsmentioning

confidence: 99%

“…We need the following, which is one of the main results of [3]. We will also require the following lemma, which can be proved by making routine modifications to [24,Lemma 5.4] For the remainder of this section, we work under the following assumptions.…”

Section: Robust Elementsmentioning

confidence: 99%

“…We also use the following, which is proved in [3]. N ), (M * 0 , N * )} such that M 1 has a pair of elements {a, b} for which M 1 \a, b is 3-connected and has an N 1 -minor.…”

Section: Spike-like 3-separatorsmentioning

confidence: 99%

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Excluded minors are almost fragile

Brettell

Clark

Oxley

et al. 2020

Journal of Combinatorial Theory, Series B

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Let M be an excluded minor for the class of P-representable matroids for some partial field P, and let N be a 3-connected strong Pstabilizer that is non-binary. We prove that either M is bounded relative to N , or, up to replacing M by a ∆-Y -equivalent excluded minor, we can choose a pair of elements {a, b} such that either M \{a, b} is N -fragile, or M * \{a, b} is N * -fragile.Date: September 21, 2018. The first, fourth, and fifth authors were supported by the New Zealand Marsden Fund. 1 Lemma 2.1. Let e be an element of a matroid M , and let (X, {e}, Y ) be a partition of E(M ). Then e ∈ cl(X) if and only if e / ∈ cl * (Y ). Connectivity. The following results are well known. Lemma 2.2. Let M be a 3-connected matroid. If X is a rank-2 subset of E(M ) and |X| ≥ 4, then M \x is 3-connected for all x ∈ X. Lemma 2.3 (Bixby's Lemma [2]). Let M be a 3-connected matroid, and let e ∈ E(M ). Then si(M/e) or co(M \e) is 3-connected.The next three results state elementary properties of 3-separations that we shall use frequently. We use the notation e ∈ cl ( * ) (X) to mean e ∈ cl(X) or e ∈ cl * (X).Lemma 2.4. Let X be an exactly 3-separating set in a 3-connected matroid, and suppose that e ∈ E(M ) − X. Then X ∪ e is 3-separating if and only if e ∈ cl ( * ) (X). 4 N. BRETTELL, B. CLARK, J. OXLEY, C. SEMPLE, AND G. WHITTLE Lemma 2.5. Let (X, Y ) be an exactly 3-separating partition of a 3connected matroid M . Suppose |X| ≥ 3 and x ∈ X. ThenWe say that a partition (X, {z}, Y ) is a vertical 3-separation of M when both (X ∪ {z}, Y ) and (X, Y ∪ {z}) are vertical 3-separations and z ∈ cl(X) ∩ cl(Y ). We will write (X, z, Y ) for (X, {z}, Y ). If (X, z, Y ) is a vertical 3-separation of M , then we say that (X, z, Y ) is a cyclic 3-separation of M * .A path of 3-separations of M is a partition (P 1 , . . . , P n ) of E(M ) such that (P 1 ∪· · ·∪P i , P i+1 ∪· · ·∪P n ) is a 3-separation of M for each i ∈ {1, . . . , n−1}. In particular, a vertical 3-separation (X, z, Y ) is a path of 3-separations. Lemma 2.7 ([23, Lemma 3.5]). Let M be a 3-connected matroid, and e ∈ E(M ). The matroid M has a vertical 3-separation (X, e, Y ) if and only if si(M/e) is not 3-connected.Let k be a positive integer, and let (P, Q) be a k-separation. We call the set cl(P ) ∩ cl(Q) the guts of (P, Q), and cl * (P ) ∩ cl * (Q) the coguts of (P, Q). We also say that an element z ∈ cl(P ) ∩ cl(Q) is a guts element, and z ∈ cl * (P ) ∩ cl * (Q) is a coguts element.We write "by uncrossing" to refer to an application of the next result.Lemma 2.8. Let M be a 3-connected matroid, and let X and Y be 3separating subsets of E(M ). Then the following hold.Series classes. We will use the following two results on series classes. We omit the easy proof of the first lemma.Lemma 2.9. Let M be a matroid such that co(M ) is 3-connected. If S and S ′ are distinct series classes of M , then either S ∪ S ′ is independent, or co(M ) ∼ = U 1,3 .When S is a series class of size two, we say S is a series pair.Lemma 2.10. Let M be a 3-connected matroid, and let u ∈ E(M ) be a...

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Section: Preliminaries and The Main Theoremsmentioning

confidence: 99%

Section: Robust Elementsmentioning

confidence: 99%

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Excluded minors are almost fragile

Brettell

Clark

Oxley

et al. 2020

Journal of Combinatorial Theory, Series B

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“…In one hand, there are results regarding the number of (vertically) contractible elements in matroids and graphs and their structure and distribution [1,8,19,15,14]. In other hand, there are the so-called splitter theorems that asserts that, for 3-connected matroids M > N satisfying certain hypothesis, there is an (vertically) N -contractible or N -deletable element in M. For example, Seymour's Splitter Theorem [17] and many others [2,10,3,9].…”

Section: Introductionmentioning

confidence: 99%

A splitter theorem on 3-connected matroids

Costalonga¹

2018

European Journal of Combinatorics

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ABSTRACT. We establish the following splitter theorem for graphs and its generalization for matroids: Let G and H be 3-connected simple graphs such that G has an H -minor and k := |V (G)| − |V (H )| ≥ 2. Let n := ⌈k/2⌉ + 1. Then there are pairwise disjoint sets X 1 , . . . , X n ⊆ E (G) such that each G/X i is a 3-connected graph with an H -minor, each X i is a singleton set or the edge set of a triangle of G with 3 degree-3 vertices and X 1 ∪ · · · ∪ X n contains no edge sets of circuits of G other than the X i 's. This result extends previous ones of Whittle (for k = 1, 2) and Costalonga (for k = 3).

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“…We define it as follows; see Figure 4 for an illustration. Let M be a matroid with a 6element, rank-4, corank-4, exactly 3-separating set P = {p 1 , p 2 , q 1 , q 2 , s 1 , s 2 } such that ( 2 has an N -minor, and since {x ′ 1 , x 2 } is a parallel pair in this matroid, M \d\x 2 /x 1 has an N -minor too. Now {x ′ 2 , c} is a series pair in this matroid, so M \x 2 /x 1 /c has an N -minor.…”

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N-detachable pairs in 3-connected matroids I: Unveiling X

Brettell¹,

Whittle²,

Williams³

2020

Preprint

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© 2019 Elsevier Inc. Let M be a 3-connected matroid, and let N be a 3-connected minor of M. We say that a pair {x1,x2}⊆E(M) is N-detachable if one of the matroids M/x1/x2 or M\x1\x2 is both 3-connected and has an N-minor. This is the first in a series of three papers where we describe the structures that arise when M has no N-detachable pairs. In this paper, we prove that if no N-detachable pair can be found in M, then either M has a 3-separating set, which we call X, with certain strong structural properties, or M has one of three particular 3-separators that can appear in a matroid with no N-detachable pairs.

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A Splitter Theorem Relative to a Fixed Basis

Cited by 16 publications

References 9 publications

Excluded minors are almost fragile

Excluded minors are almost fragile

A splitter theorem on 3-connected matroids

N-detachable pairs in 3-connected matroids I: Unveiling X

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