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**Ends and multi-endings. I.**
*(English)*
Zbl 0855.05051

The main material of this paper comes from several former French-written papers. It has been reorganized, and so doing, the terminology and notation have been unified, some results have been improved, several proofs are new or have been shortened. This first paper deals with the fundamental properties of ends. Three important classes of vertices called concentrated, fragmented and dispersed, are studied. These sets proved to be useful for characterizing the fundamental aspects, “height” and “width”, of infinite graphs. A uniform structure on the vertex set of a graph \(G\) is defined in such way that the associated topological space is compact if and only if \(G\) is finite. More precisely, with each finite subset \(S\) of \(V(G)\), one associates an equivalence relation \(\widetilde S\) on \(V(G)\) by \((x, y)\in \widetilde S\) if and only if \(x\) and \(y\) are not separated by \(S\). The family of all such equivalences \(\widetilde S\) generates a uniformity \({\mathfrak S}_G\) on \(V(G)\). The importance of the uniform space thus obtained comes from the fact that it is complete if and only if \(G\) is rayless, and totally bounded if and only if each cofinite induced subgraph of \(G\) only has finitely many components. Thus, failure of the uniform space to be complete or totally bounded is intimately related to the two essential characteristics of infiniteness of the underlying graph: infinite height, or infinite width.

The completion of this space is obtained by canonically extending the uniformity \({\mathfrak S}_G\) to the set of all vertices and all ends of \(G\). The induced topology on the end set of \(G\) corresponds to the topology originally introduced for locally finite graphs by H. Freudental [Über die Enden diskreter Räume und Gruppen, Comment. Math. Helv. 17, 1-38 (1944)], and later reintroduced by H. A. Jung [Connectivity in infinite graphs, Studies in Pure Mathematics, 137-143 (1971)]. The last section is devoted to the study of particular subgraphs, mainly trees, of a graph, from the viewpoint of the associated end topology. In particular it is proved that a connected graph \(G\) contains an end-faithful tree whose end space is homeomorphic to that of \(G\) if and only if this space is ultrametrizable. Moreover, if \(G\) is in addition countable, then one proves that \(G\) contains a spanning such tree.

The completion of this space is obtained by canonically extending the uniformity \({\mathfrak S}_G\) to the set of all vertices and all ends of \(G\). The induced topology on the end set of \(G\) corresponds to the topology originally introduced for locally finite graphs by H. Freudental [Über die Enden diskreter Räume und Gruppen, Comment. Math. Helv. 17, 1-38 (1944)], and later reintroduced by H. A. Jung [Connectivity in infinite graphs, Studies in Pure Mathematics, 137-143 (1971)]. The last section is devoted to the study of particular subgraphs, mainly trees, of a graph, from the viewpoint of the associated end topology. In particular it is proved that a connected graph \(G\) contains an end-faithful tree whose end space is homeomorphic to that of \(G\) if and only if this space is ultrametrizable. Moreover, if \(G\) is in addition countable, then one proves that \(G\) contains a spanning such tree.

Reviewer: N.Polat (Lyon)

### MSC:

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