A decomposition of locally finite graphs

We prove that every infinite, connected, locally finite graph G can be expressed as an edge-disjoint union of a leafless tree T, rooted at an arbitrarily chosen vertex of G, and a collection of finite graphs H1, H2, H3,...such that, for all i less than j, the vertices common to Hi and Hj lie in T, and no vertex of Hj lies on T between a vertex of Hi∩T and the root. © 1993

Canonical graph decompositions via coverings

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 \emph{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.Comment: This is the journal version of this paper. An extended version is available in this ArXiv thread as v

Entanglements

Robertson and Seymour constructed for every graph G a tree-decomposition that efficiently distinguishes all the tangles in G. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical.The key ingredient is an axiomatisation of ‘local properties’ of tangles. Generalisations to locally finite graphs and matroids are also discusse

Entanglements

Robertson and Seymour constructed for every graph $G$ a tree-decomposition that efficiently distinguishes all the tangles in $G$. While all previous constructions of these decompositions are either iterative in nature or not canonical, we give an explicit one-step construction that is canonical. The key ingredient is an axiomatisation of 'local properties' of tangles. Generalisations to locally finite graphs and matroids are also discussed.Comment: 8 pages, 4 figure

Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.Comment: 36 pages, 10 figure

Decomposition spaces in combinatorics

A decomposition space (also called unital 2-Segal space) is a simplicial object satisfying an exactness condition weaker than the Segal condition: just as the Segal condition expresses (up to homotopy) composition, the new condition expresses decomposition. It is a general framework for incidence (co)algebras. In the present contribution, after establishing a formula for the section coefficients, we survey a large supply of examples, emphasising the notion's firm roots in classical combinatorics. The first batch of examples, similar to binomial posets, serves to illustrate two key points: (1) the incidence algebra in question is realised directly from a decomposition space, without a reduction step, and reductions are often given by CULF functors; (2) at the objective level, the convolution algebra is a monoidal structure of species. Specifically, we encounter the usual Cauchy product of species, the shuffle product of L-species, the Dirichlet product of arithmetic species, the Joyal-Street external product of q-species and the Morrison `Cauchy' product of q-species, and in each case a power series representation results from taking cardinality. The external product of q-species exemplifies the fact that Waldhausen's S-construction on an abelian category is a decomposition space, yielding Hall algebras. The next class of examples includes Schmitt's chromatic Hopf algebra, the Fa\`a di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf algebra of trees and several variations from operad theory. Similar structures on posets and directed graphs exemplify a general construction of decomposition spaces from directed restriction species. We finish by computing the M\Preprin

Local Certification of Graph Decompositions and Applications to Minor-Free Classes

Local certification consists in assigning labels to the nodes of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received a considerable attention. The goal is to certify that a graph G belongs to a given graph class ?. Such certifications with labels of size O(log n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors. In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log n). We also show matching lower bounds using a new proof technique