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

Brettell, Nick; Whittle, Geoff; Williams, Aaron

doi:10.26686/wgtn.11904942.v1

Cited by 1 publication

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“…Brettell, Whittle, and Williams [4][5][6] proved that either M has a spikelike 3-separator, or, after performing at most one ∆-Y or Y -∆ exchange on M , we obtain a matroid with a detachable pair. More specifically:…”

Section: Definition Let B Be a Basis Of M And Letmentioning

confidence: 97%

“…This situation is a more general version of the one that arises in the proof of the excluded-minor characterisation of GF(4)-representable matroids [8]. There, the partial field is GF(4) and the strong stabilizer is U 2,4 . (See also the introduction to [4] for more detail on this strategy.)…”

Section: Introductionmentioning

confidence: 96%

“…There, the partial field is GF(4) and the strong stabilizer is U 2,4 . (See also the introduction to [4] for more detail on this strategy.) Ideally, we would develop techniques that would reduce problems of the above type to routine computation.…”

Section: Introductionmentioning

confidence: 99%

“…The first is the partial field H 5 which was introduced by Pendavingh and van Zwam [16]. The class of matroids representable over this field is the class obtained by taking the 3connected matroids that have exactly six inequivalent representations over GF (5) and closing the class under minors. This class forms the bottom layer of Pendavingh and van Zwam's hierarchy of GF(5)-representable matroids.…”

Section: Introductionmentioning

confidence: 99%

“…This class forms the bottom layer of Pendavingh and van Zwam's hierarchy of GF(5)-representable matroids. Finding excluded minors for this class would be a key first step towards finding the excluded minors for matroids representable over GF (5).…”

Section: Introductionmentioning

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: Definition Let B Be a Basis Of M And Letmentioning

confidence: 97%