Quasi-4-connected components

Grohe, Martin

doi:10.1017/9781139028868.011

Cited by 4 publications

(5 citation statements)

References 23 publications

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“…The block-cutvertex tree of a graph achieves this for k = 2, and Tutte [21] showed for k = 3 that every 2-connected finite graph has a tree-decomposition of adhesion 2 whose torsos are either 3-connected or cycles. Tutte's theorem has recently been extended by Grohe [17] to k = 4, and by Carmesin, Diestel, Hundertmark and Stein [4] to arbitrary k as follows.…”

Section: Introductionmentioning

confidence: 99%

Profiles of Separations: in Graphs, Matroids, and Beyond

2018

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We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combinatorial structures, which has further applications.These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem. arXiv:1110.6207v5 [math.CO] 17 Apr 20171 The order of a separation {A, B} of a graph is the number |A ∩ B|. Separations of order k are k-separations; separations of order < k are (< k)-separations.2 Indeed, in Section 2 we shall introduce more abstract 'k-profiles', not necessarily induced by k-blocks, of which k-tangles -those of order k, in the terminology of [20] -are a prime example. See [1,2] for more on the relationship between profiles, blocks and tangles.

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Section: Introductionmentioning

confidence: 99%

Profiles of Separations: in Graphs, Matroids, and Beyond

2018

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“…We expect that the results of this paper are also true for infinite graphs. A natural next step would be to prove a local version of the Grohe-Decomposition-Theorem, which gives a decomposition of a 3-connected graph into 'quasi 4-connected components' [17]. We hope that with the methods of this paper one should be able to prove the following.…”

Section: Discussionmentioning

confidence: 95%

Local 2-separators

Carmesin

2020

Preprint

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How can sparse graph theory be extended to large networks, where algorithms whose running time is estimated using the number of vertices are not good enough? I address this question by introducing 'Local Separators' of graphs. Applications include:1. A unique decomposition theorem for graphs along their local 2separators analogous to the 2-separator theorem;2. an exact characterisation of graphs with no bounded subdivision of a wheel;

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“…Nevertheless, it is shown in [6] that there is an extension of the theorem to tangles of order 4 if we replace 4-connectivity by the slightly weaker "quasi-4-connectivity": a graph G is quasi-4-connected if it is 3-connected and for all separations (A, B) of order 3, either…”

Section: From Tangles To Componentsmentioning

confidence: 99%

“…The paper provides background material for my talk at LATA. The talk itself will be concerned with more recent results [6] and, in particular, computational aspects and applications of tangles [9,10,11].…”

Section: Introductionmentioning

confidence: 99%