# SummarIes

In the following you find summaries of some of my past projects.

## Linked & lean tree-decompositions

Kříž and Thomas showed that every (finite or infinite) graph of tree-width < k has a lean rooted tree-decomposition of width < k. Does their result generalise to graphs of finite tree-width, that is k = ℵ₀? What are the frontiers of such generalisation? We present two such generalisation which lie at the frontier of what it is still true and have several applications: Short, unified proofs of Robertson, Seymour and Thomas's characterisations of graphs with no half grid minor and of graphs without binary tree subdivision, resolving a question of Halin, and extending Robertson and Seymour's tangle-tree duality theorem to infinite graphs. Click here for a summary of my results.

## Typical graphs

What are the typical graphs which are infinitely edge-connected? Given a containment relation ≤, such as the (topological-)minor- or the weak\strong-immersion-relation, these are the ≤-minimal infinitely edge-connected graphs (up to ≤-equivalence). But which ones are they? Perhaps surprisingly, the Farey graph turns out to be quite important. Click here for a summary of my results.

## Trees of tangles in locally finite graphs

Robertson and Seymour's trees-of-tangles theorem states that every finite graph admits a tree-decomposition which efficiently distinguishes its tangles. When does a locally finite graph admit such a tree-decomposition? While previous work achieved extensions of Robertson and Seymour's trees-of-tangles theorem to infinite graphs through relaxation of the conclusion, we identified the obstructions directly and obtain a full-strength extension when these are absent. Click here for a summary of my results.

## Common cyclic structure

In light of the Lovasz-Woodall-Conjecture one may in general ask: When do k prescribed edges of a graph lie on a common cyclic structure? We show that any k edges in a graph whose edge-connectivity is at least 2·⌈(k+1)/2⌉ lie on a common circuit. Click here for a summary of my results.

## A Cantor-Bernstein Result for Trees

If the edge set of a graph contains λ edge-disjoint spanning trees and is also covered by λ spanning trees, does it admit a λ-partition into spanning trees? An easy counting argument answers this question in the affirmative. We extend it to infinite graphs. Click here for a summary of my results.

Do you miss a summary of some article of mine? Drop me a mail and come back later, as my aim is summarise all of them.