Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

Geelen, Jim; Gerards, A.M.H.; Whittle, Geoff

doi:10.1006/jctb.2001.2082

Cited by 90 publications

(121 citation statements)

References 8 publications

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“…• Isotropic system [Bouchet, 1987] and Scraps • Extension of Menger's theorem on scraps • If rank-width of G is n, then there is a linked rank-decompositon of width n. [Geelen et al, 2002] cf. [Thomas, 1990] For any e, f in the rank-decomposition T , any vertex partition separating e, f has cut-rank ≥ min cut-rank of an edge in the path from e to f in T .…”

Section: Toolsmentioning

confidence: 99%

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Rank-Width and Well-Quasi-Ordering

Oum¹

2008

SIAM J. Discrete Math.

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Section: Toolsmentioning

confidence: 99%

“…• Robertson and Seymour's "Lemma on trees" [Robertson and Seymour, 1990] Binary matroids and wqo Thm (Geelen, Gerards, Whilttle [Geelen et al, 2002]). If {M 1 , M 2 , .…”

Section: Toolsmentioning

confidence: 99%

Rank-Width and Well-Quasi-Ordering

Oum¹

2008

SIAM J. Discrete Math.

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“…[4,5] in past several years extended significant portion of the Graph Minors theory to matroids representable over finite fields. Questions of matroid representability, and the notion of branchwidth, turned out to be the key ingredients in that research.…”

Section: Matroid Minorsmentioning

confidence: 99%

“…Another motivation for our research lies in a current hot trend in structural matroid theory; work of Geelen, Gerards and Whittle, e.g. [4,5] extending significant portion of the Robertson-Seymour's Graph Minors project [15] to matroids. It turns out that matroids represented over finite fields play a crucial role in that research, analogous to the role played by graphs embedded on surfaces in Graph Minors.…”

mentioning

confidence: 99%

On Matroid Representability and Minor Problems

Hliněný

2006

Lecture Notes in Computer Science

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Abstract. In this paper we look at complexity aspects of the following problem (matroid representability) which seems to play an important role in structural matroid theory: Given a rational matrix representing the matroid M , the question is whether M can be represented also over another specific finite field. We prove this problem is hard, and so is the related problem of minor testing in rational matroids. The results hold even if we restrict to matroids of branch-width three.Keywords: matroid representability, minor, finite field, spike, swirl. 2000 Math subject classification: 05B35, 68Q17, 68R05. IntroductionWe postpone necessary formal definitions until later sections. Matroids present a wide combinatorial generalization of graphs. A useful geometric essence of a matroid is shown in its vector representation over a field ; the elements-vectors of the representation can be viewed as points in the projective geometry over . Not all matroids, however, have vector representations. That is why the question of -representability of a matroid is important to solve. Another motivation for our research lies in a current hot trend in structural matroid theory; work of Geelen, Gerards and Whittle, e.g. [4,5] extending significant portion of the Robertson-Seymour's Graph Minors project [15] to matroids. It turns out that matroids represented over finite fields play a crucial role in that research, analogous to the role played by graphs embedded on surfaces in Graph Minors. Such a role is further justified by related works concerning logic and complexity aspects of matroids, e.g. our [8,10], and by a somehow surprising connection of binary matroids with graph rank-width [1] of Courcelle and Oum.In this paper we prove that it is hard to decide whether a matroid given by a vector representation over the rational numbers, has a vector representation over a specific finite field (Theorems 3.1 and 4.1). In particular this result implies that also the problem of minor testing in rational matroids is generally hard. We moreover prove that the minor testing problem is hard even for a certain small planar minor (Theorem 5.6). Matroids and Vector RepresentationsWe refer to Oxley's book [12]. Since matroid theory seems not widely known among computer scientists, we should briefly review some basic terms here: A matroid is a pair M = (E, B) where E = E(M ) is the finite ground set of M (elements of M ), and B ⊆ 2 E is a nonempty collection of bases of M , no two of which are in an inclusion. Moreover, matroid bases satisfy the "exchange axiom": if B 1 , B 2 ∈ B and x ∈ B 1 \ B 2 , then there is y ∈ B 2 \ B 1 such that (B 1 \ {x}) ∪ {y} ∈ B. Subsets of bases are called independent sets, and the remaining sets are dependent. Minimal dependent sets are called circuits. All bases have the same cardinality called the rank r(M ) of the matroid. The rank function r M (X) in M assigns the maximal cardinality of an independent subset of a set X ⊆ E(M ). A set X is spanning if r M (X) = r(M ), and maximal non-spanning sets are called hyperp...

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“…The counterpart of branchwidth on matroids has been introduced by Geelen and Whittle in [6] and was extensively studied in [6,15,10,14,11,5]. However, not much is known for the counterpart of pathwidth on matroids.…”

Section: Introductionmentioning

confidence: 99%

Outerplanar Obstructions for Matroid Pathwidth

Koutsonas

Thilikos

Yamazaki

2011

Electronic Notes in Discrete Mathematics

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To cite this version:Athanassios Koutsonas, Dimitrios M. Thilikos, Koichi Yamazaki. Outerplanar obstructions for matroid pathwidth. Discrete Mathematics, Elsevier, 2014, 315, pp.95-101 For each non-negative integer k, we provide all outerplanar obstructions for the class of graphs whose cycle matroid has pathwidth at most k. Our proof combines a decomposition lemma for proving lower bounds on matroid pathwidth and a relation between matroid pathwidth and linear width. Our results imply the existence of a linear algorithm that, given an outerplanar graph, outputs its matroid pathwidth.

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Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

Cited by 90 publications

References 8 publications

Rank-Width and Well-Quasi-Ordering

Rank-Width and Well-Quasi-Ordering

On Matroid Representability and Minor Problems

Outerplanar Obstructions for Matroid Pathwidth

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