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**Ends and multi-endings. II.**
*(English)*
Zbl 0855.05052

In order to study certain problems pertaining to the ends of a graph, and in particular for certain proofs which proceed by induction on the “complexity” of its end space, a notion of expansion of an infinite graph into special subgraphs is introduced. The properties of these subgraphs, called endings, or more generally, multi-endings, are modeled on certain properties of rays, or, in the case of multi-endings, on properties of these subtrees of a tree which have only free ends. Using some of the results of part I, it is shown that certain expansions of the end space of a graph \(G\) into an increasing sequence of closed sets induce expansions of \(G\) and conversely. This enables to prove that every connected graph \(G\) admits an expansion, and that the minimal length of such expansions (the end-degree of \(G\)) is at most \(\omega_1\). This paper deals largely with the existence and the fundamental properties of these concepts which have been used to study various combinatorial problems in different papers. A special attention is given to the graphs that have an end-degree \(\leq \omega+1\). In particular it is shown that the end-degree of a graph \(G\) is \(\leq \omega\) if and only if the end space of \(G\) is scattered, i.e., if \(G\) contains no end-respecting subdivison of the dyadic tree; and that trees and countable graphs have an end-degree \(\leq \omega+ 1\). In the last section these concepts are used to give new characterizations of the connected graphs that have uniformly compatible end-faithful spanning trees.

Reviewer: N.Polat (Lyon)

### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |