Canonical graph decompositions via coverings

Reinhard Diestel; Raphael W. Jacobs; Paul Knappe; Jan Kurkofka

doi:10.48550/arXiv.2207.04855

Canonical graph decompositions via coverings

Reinhard Diestel, Raphael W. Jacobs, Paul Knappe, Jan Kurkofka

Mathematics

Research output: Working paper/Preprint › Preprint

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Abstract

We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser model. This is a simpler graph determined entirely by the decomposition, not imposed.

The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure while unfolding its global structure. In this way, the tangle theory from graph minors is brought to bear canonically on arbitrary graphs, which need not be tree-like.

Our theorem extends to locally finite quasi-transitive graphs, and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition for finitely generated groups into local parts, whose relative structure is displayed by a graph.

Original language	English
Publisher	arXiv
DOIs	https://doi.org/10.48550/arXiv.2207.04855
Publication status	Published - 1 Feb 2023

Bibliographical note

This is the second main version of this paper. It differs from the first, v.2, by some minor editing and corrections of misprints. An extended ArXiv version corresponding to this v.4 was uploaded yesterday as v.3

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abstract = "We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser model. This is a simpler graph determined entirely by the decomposition, not imposed. The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure while unfolding its global structure. In this way, the tangle theory from graph minors is brought to bear canonically on arbitrary graphs, which need not be tree-like. Our theorem extends to locally finite quasi-transitive graphs, and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition for finitely generated groups into local parts, whose relative structure is displayed by a graph. ",

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AU - Kurkofka, Jan

N1 - This is the second main version of this paper. It differs from the first, v.2, by some minor editing and corrections of misprints. An extended ArXiv version corresponding to this v.4 was uploaded yesterday as v.3

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N2 - We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser model. This is a simpler graph determined entirely by the decomposition, not imposed. The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure while unfolding its global structure. In this way, the tangle theory from graph minors is brought to bear canonically on arbitrary graphs, which need not be tree-like. Our theorem extends to locally finite quasi-transitive graphs, and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition for finitely generated groups into local parts, whose relative structure is displayed by a graph.

AB - We present a canonical way to decompose finite graphs into highly connected local parts. The decomposition depends only on an integer parameter whose choice sets the intended degree of locality. The global structure of the graph, as determined by the relative position of these parts, is described by a coarser model. This is a simpler graph determined entirely by the decomposition, not imposed. The model and decomposition are obtained as projections of the tangle-tree structure of a covering of the given graph that reflects its local structure while unfolding its global structure. In this way, the tangle theory from graph minors is brought to bear canonically on arbitrary graphs, which need not be tree-like. Our theorem extends to locally finite quasi-transitive graphs, and in particular to locally finite Cayley graphs. It thereby offers a canonical decomposition for finitely generated groups into local parts, whose relative structure is displayed by a graph.

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M3 - Preprint

BT - Canonical graph decompositions via coverings

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Canonical graph decompositions via coverings

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