91 research outputs found
End-faithful spanning trees in graphs without normal spanning trees
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.Comment: 9 pages, no figure
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