The Highly Connected Matroids in Minor-Closed Classes

Geelen, Jim; Gerards, Bert; Whittle, Geoff

doi:10.1007/s00026-015-0251-3

Cited by 35 publications

(94 citation statements)

References 24 publications

(30 reference statements)

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“…, Y m are unbalanced, then all of the unbalanced cycles in G − C are either in L or in F. In the latter case, the rank of G − C is still |V (G)|, a contradiction. Thus all of the unbalanced cycles in G − C are in L and by minimality C has the form described in statement (3). So now assume that not all of X, Y 1 , .…”

Section: 2mentioning

confidence: 99%

Describing quasi-graphic matroids

Bowler

Funk

Slilaty

2020

European Journal of Combinatorics

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The class of quasi-graphic matroids recently introduced by Geelen, Gerards, and Whittle generalises each of the classes of frame matroids and liftedgraphic matroids introduced earlier by Zaslavsky. For each biased graph (G, B) Zaslavsky defined a unique lift matroid L(G, B) and a unique frame matroid F (G, B), each on ground set E(G). We show that in general there may be many quasi-graphic matroids on E(G) and describe them all: for each graph G and partition (B, L, F) of its cycles such that B satisfies the theta property and each cycle in L meets each cycle in F, there is a quasi-graphic matroid M (G, B, L, F) on E(G). Moreover, every quasi-graphic matroid arises in this way. We provide cryptomorphic descriptions in terms of subgraphs corresponding to circuits, cocircuits, independent sets, and bases. Equipped with these descriptions, we prove some results about quasi-graphic matroids. In particular, we provide alternate proofs that do not require 3-connectivity of two results of Geelen, Gerards, and Whittle for 3-connected matroids from their introductory paper: namely, that every quasi-graphic matroid linearly representable over a field is either liftedgraphic or frame, and that if a matroid M has a framework with a loop that is not a loop of M then M is either lifted-graphic or frame. We also provide sufficient conditions for a quasi-graphic matroid to have a unique framework.Zaslavsky has asked for those matroids whose independent sets are contained in the collection of independent sets of F (G, B) while containing those of L(G, B), for some biased graph (G, B). Adding a natural (and necessary) non-degeneracy condition defines a class of matroids, which we call biased graphic. We show that the class of biased graphic matroids almost coincides with the class of quasigraphic matroids: every quasi-graphic matroid is biased graphic, and if M is a biased graphic matroid that is not quasi-graphic then M is a 2-sum of a frame matroid with one or more lifted-graphic matroids.

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Section: 2mentioning

confidence: 99%

Describing quasi-graphic matroids

Bowler

Funk

Slilaty

2020

European Journal of Combinatorics

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“…One natural question motivated by our work is to find out which other matroids have only polynomially many near-minimum circuits. Analogous to Seymour's decomposition theorem, Geelen, Gerards, and Whittle [11] have proposed a structure theorem for any proper minor-closed class of matroids representable over a finite field. Can we use this structure theorem to upper bound the number of near-minimum circuits in these matroids.…”

Section: Future Directionsmentioning

confidence: 99%

On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

Gurjar¹,

Vishnoi²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We show that for any regular matroid on m elements and any α ≥ 1, the number of α-minimum circuits, or circuits whose size is at most an α-multiple of the minimum size of a circuit in the matroid is bounded by m O(α 2 ) . This generalizes a result of Karger for the number of α-minimum cuts in a graph. As a consequence, we obtain similar bounds on the number of α-shortest vectors in "totally unimodular" lattices and on the number of α-minimum weight codewords in "regular" codes. Max-flow min-cut matroids 32A Bounding the number of circuits in a graphic or cographic matroid 38

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“…. a r+t,t Lemma 2.6 appears in [3] as Lemma 2.1; however, no proof was given in [3]. We will need it to prove our main result, so we give a proof here.…”

Section: Introductionmentioning

confidence: 99%

“…We use this and Lemma 2.9 to prove Lemma 2.10 below. Although [14] was based on [3], the previous lemma was proved independently of [3] and remains accurate for that reason.…”

Section: Introductionmentioning

confidence: 99%

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On Perturbations of Highly Connected Dyadic Matroids

Grace

Zwam

2018

Ann. Comb.

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Geelen, Gerards, and Whittle [3] announced the following result: let q = p k be a prime power, and let M be a proper minor-closed class of GF(q)-representable matroids, which does not contain PG(r − 1, p) for sufficiently high r. There exist integers k, t such that every vertically k-connected matroid in M is a rank-(≤ t) perturbation of a frame matroid or the dual of a frame matroid over GF(q). They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates.We show a family of dyadic matroids that form a counterexample to this result. We offer several weaker conjectures to replace the ones in [3], discuss consequences for some published papers, and discuss the impact of these new conjectures on the structure of frame templates. 1 arXiv:1712.07702v3 [math.CO]In this paper we present a counterexample to this conjecture. In particular, we build a family of dyadic matroids that are vertically kconnected for any desired k, and not a bounded-rank perturbation of either a represented frame matroid or the dual of a represented frame matroid. The construction starts with a cyclically k-connected graph G, modifies it at a number of vertices that grows with |V (G)|, and dualizes the resulting matroid. We detail the construction, and prove its key properties, in Section 3.Vertical connectivity and cographic matroids are not very compatible notions. Because of this, our examples, which are very sparse, arise only in situations where one might expect the second outcome of the conjecture to hold. For this reason, the forthcoming proof of the Matroid Structure Theorem itself is not jeopardized, and versions of Conjecture 1.2 can be recovered. In Section 4, we provide several such conjectures. Included in Section 4 is Section 4.3, where we discuss consequences for [10] and [15]. In Section 5, we discuss consequences for the notion of frame templates, introduced in [3] to describe the perturbations in more detail. PreliminariesUnexplained notation and terminology will generally follow Oxley [16]. One exception is that we denote the vector matroid of a matrix A by M (A), rather than M [A]. The following characterization of the dyadic matroids was shown by Whittle in [23].Theorem 2.1. A matroid is dyadic if and only if it is representable over both GF(3) and GF(5).We will need some definitions and results related to bounded-rank perturbations of represented matroids. The next three definitions are from [3].Definition 2.2. Let M 1 and M 2 be F-represented matroids on a common ground set. Then M 2 is a rank-(≤ t) perturbation of M 1 if there exist matrices A 1 and P such that M (A 1 ) = M 1 , the rank of P is at most t, and M (A 1 + P ) = M 2 . Definition 2.3. Let M 1 and M 2 be F-represented matroids on ground set E. If there is some F-represented matroid M on ground set E ∪ {e} such that M 1 = M \e and M 2 = M/e, then M 1 is an elementary lift of M 2 , and M 2 is an elementary projection of M 1 . Note that an elementary lift of a represented matroid M (A) can be ob...

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The Highly Connected Matroids in Minor-Closed Classes

Abstract: Abstract. For any minor-closed class of matroids over a fixed finite field, we state an exact structural characterization for the sufficiently connected matroids in the class. We also state a number of conjectures that might be approachable using the structural characterization.

Cited by 35 publications

References 24 publications

Describing quasi-graphic matroids

Describing quasi-graphic matroids

On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

On Perturbations of Highly Connected Dyadic Matroids

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