Obstructions for normally spanned sets of vertices
Nicola Lorenz, Max Pitz
TL;DR
The paper proves a local Halin-type characterisation: a set $U$ is normally spanned in a connected graph $G$ iff every $U$-rooted minor of $G$ has countable colouring number. It develops a structural framework based on well-connected adhesion sets and normal semi-partition trees, then proves a decomposition lemma to reduce to smaller subgraphs and applies a transfinite inductive construction to obtain a normal spanning tree containing $U$. The key methodological innovations are the closure argument for expanding vertex sets to achieve well-connected adhesions and the integration of normal partition trees with finite-adhesion decompositions, yielding a concise, countable-branching proof of the local characterisation. This strengthens Halin’s global result by providing a precise local criterion and broadens the toolkit for handling infinite graphs with prescribed vertex sets. The findings have implications for understanding how forbidden minors interact with prescribed vertex sets and for applications relying on normal spanning trees in infinite graph theory.
Abstract
Halin conjectured that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. This has recently been proven by the second author. In this paper, we strengthen this result by establishing the following local version of it: Given a prescribed set of vertices $U$ in a connected graph $G$, there is a normal tree in $G$ that includes $U$ if and only if every $U$-rooted minor of $G$ (i.e. a minor every branch set of which meets $U$) has countable colouring number. Our proof relies on a novel approach that combines normal partition trees as introduced by Brochet and Diestel with a suitable closure argument developed by Robertson, Seymour and Thomas in their discussion of infinite graphs of finite tree width.