End-faithful spanning trees in graphs without normal spanning trees

TL;DR

This paper extends the concept of ordinal rank functions to characterize normally traceable graphs, a larger class than those with normal spanning trees, and links the existence of rayless spanning trees to end domination.

Contribution

It generalizes Schmidt's ordinal rank function to normally traceable graphs and characterizes end-faithful spanning trees within this broader class.

Findings

Normally traceable graphs are characterized by an ordinal rank function.

Having a rayless spanning tree is equivalent to all ends being dominated in normally traceable graphs.

The paper provides a new characterization of normally traceable graphs using transfinite induction.

Abstract

Schmidt characterised the class of rayless graphs by an ordinal rank function, which makes it possible to prove statements about rayless graphs by transfinite induction. Halin asked whether Schmidt's rank function can be generalised to characterise other important classes of graphs. We answer Halin's question in the affirmative. Another largely open problem raised by Halin asks for a characterisation of the class of graphs with an end-faithful spanning tree. A well-studied subclass is formed by the graphs with a normal spanning tree. We determine a larger subclass, the class of normally traceable graphs, which consists of the connected graphs with a rayless tree-decomposition into normally spanned parts. Investigating the class of normally traceable graphs further we prove that, for every normally traceable graph, having a rayless spanning tree is equivalent to all its ends being dominated. Our proofs rely on a characterisation of the class of normally traceable graphs by an ordinal rank function that we provide.