# Representations of Infinite Tree Sets

@article{Gollin2019RepresentationsOI, title={Representations of Infinite Tree Sets}, author={James Gollin and Jakob Kneip}, journal={Order}, year={2019}, volume={38}, pages={79-96} }

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ω + 1. Then we introduce and study a topological generalisation of infinite trees which can…

## 16 Citations

### Characterising path-, ray- and branch spaces of order trees, and end spaces of infinite graphs

- Mathematics
- 2023

We investigate path-, ray- and branch spaces of trees, certain topological spaces naturally associated with order theoretic trees, and provide topological characterisations for these spaces in terms…

### Connectivity and tree structure in infinite graphs and digraphs

- Mathematics
- 2019

This dissertation deals with different aspects of connectivity and tree structure
in infinite graphs, which make it part of the research area of structural infinite
graph theory. It consists of two…

### Canonical graph decompositions via coverings

- Mathematics
- 2022

. 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…

### Trees of tangles in infinite separation systems

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2021

Abstract We present infinite analogues of our splinter lemma for constructing nested sets of separations. From these we derive several tree-of-tangles-type theorems for infinite graphs and infinite…

### Edge-connectivity and tree-structure in finite and infinite graphs

- MathematicsArXiv
- 2020

It is shown that every graph admits a canonical tree-like decomposition into its edge-connected pieces for all $k$-edge-connected Pieces for all N simultaneously.

### Duality theorems for stars and combs IV: Undominating stars

- MathematicsJ. Graph Theory
- 2022

In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs from the well‐known star—comb lemma for infinite graphs.…

### ON A LINKING PROPERTY OF INFINITE MATROIDS

- Mathematics
- 2020

Let M0 and M1 be matroids on E having only finitary and cofinitary components and let Xi ⊆ E for i ∈ {0, 1}. We show that if Xi can be spanned in Mi by an M1−i-independent set for i ∈ {0, 1}, then…

### A tree-of-tangles theorem for infinite-order tangles

- Mathematics
- 2020

Carmesin has extended Robertson and Seymour's tree-of-tangles theorem to the infinite-order tangles of locally finite infinite graphs. We extend it further to the infinite-order tangles of all…

## References

SHOWING 1-10 OF 10 REFERENCES

### Tree Sets

- Computer Science, MathematicsOrder
- 2018

We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a…

### Graph Searching and a Min-Max Theorem for Tree-Width

- MathematicsJ. Comb. Theory, Ser. B
- 1993

The tree-width of a graph G is the minimum k such that G may be decomposed into a "tree-structure" of pieces each with at most k + l vertices. We prove that this equals the maximum k such that there…

### Graph-like continua, augmenting arcs, and Menger’s theorem

- MathematicsComb.
- 2008

It is shown that an adaptation of the augmenting path method for graphs proves Menger’s Theorem for wide classes of topological spaces, including locally compact, locally connected, metric spaces.

### Infinite Graphic Matroids

- MathematicsComb.
- 2018

We introduce a class of infinite graphic matroids that contains all the motivating examples and satisfies an extension of Tutte’s excluded minors characterisation of finite graphic matroids.We prove…

### Tangles and the Mona Lisa

- Mathematics, Computer ScienceArXiv
- 2016

It is shown how an image can, in principle, be described by the tangles of the graph of its pixels, and the nested set of separations that efficiently distinguish all the distinguishable tangles in a graph.

### Tangle-Tree Duality: In Graphs, Matroids and Beyond

- MathematicsCombinatorica
- 2019

A recent duality theorem for tangles in abstract separation systems is applied to derive tangle-type duality theorems for width-parameters in graphs and matroids and it is shown that carving width is dual to edge-tangles.

### Abstract Separation Systems

- Computer ScienceOrder
- 2018

This paper is intended as a concise common reference for the basic definitions and facts about abstract separation systems in these and any future papers using this framework.