On the excluded minors for the matroids of branch-width k

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

doi:10.1016/s0095-8956(02)00046-1

Cited by 48 publications

(52 citation statements)

References 3 publications

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“…The boundary of H ′ 1 is of size k t, and so [34,Lemma 4.1] can be applied to F 1 . Therefore, we have |E(H 1 )| g(k) g(t) and |E(F )| g(t) + t 2 , and there are only nitely many such t-boundaried graphs without isolated vertices in X * t .…”

Section: Fii and Structural Parameterizationmentioning

confidence: 99%

Kernelization using structural parameters on sparse graph classes

Gajarský

Hlinźný

Óbdržálek

et al. 2017

Journal of Computer and System Sciences

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We prove that graph problems with nite integer index have linear kernels on graphs of bounded expansion when parameterized by the size of a modulator to constant-treedepth graphs. For nowhere dense graph classes, our result yields almost-linear kernels. We also argue that such a linear kernelization result with a weaker parameter would fail to include some of the problems covered by our framework. We only require the problems to have FII on graphs of constant treedepth. This allows to prove linear kernels also for problems such as LongestPath/Cycle, Exact-s, t-Path, Treewidth, and Pathwidth, which do not have FII on general graphs.

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Section: Fii and Structural Parameterizationmentioning

confidence: 99%

Kernelization using structural parameters on sparse graph classes

Gajarský

Hlinźný

Óbdržálek

et al. 2017

Journal of Computer and System Sciences

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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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“…A more promising strategy towards detecting constructive fragments of Theorem 3, is to detect parameters -or families of parameters -where O ≤m p,k is recursive. For this one may either prove upper bounds for |O ≤m p,k |, as done in [57] for the case of branchwidth 5 , or provide partial characterizations of O ≤m p,k , as done in [2,19,21,89,94], that permit its recursive computation.…”

Section: Minor-closed Graph Parametersmentioning

confidence: 99%

Graph Minors and Parameterized Algorithm Design

Thilikos

2012

The Multivariate Algorithmic Revolution and Beyond

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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic applications of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique.

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On the excluded minors for the matroids of branch-width k

Cited by 48 publications

References 3 publications

Kernelization using structural parameters on sparse graph classes

Kernelization using structural parameters on sparse graph classes

Outerplanar Obstructions for Matroid Pathwidth

Graph Minors and Parameterized Algorithm Design

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