The structure of rayless graphs (Q1969639)

A ray is a one-way infinite path. The author first presents the main concepts in investigating rayless graphs, called order and kernel, developed by \textit{R. Schmidt} [Ein Reduktionsverfahren für weg-endliche Graphen, Dissertation, Univ. Hamburg (1982; Zbl 0526.05059); see also: Ein Ordnungsbegriff für Graphen ohne unendliche Wege mit Anwendungen auf \(n\)-fach zusammenhängende Graphen, Arch. Math. 40, 283-288 (1983; Zbl 0526.05058)]. The author then investigates the relationship between order and kernel and tree-decompositons, and finally generalizes an interesting theorem of \textit{N. Polat} and \textit{G. Sabidussi} [Fixed elements of infinite trees, Discrete Math. 130, No. 1-3, 97-102 (1994; Zbl 0817.05065)], saying that in every rayless tree there is a vertex or an edge left fixed by every automorphism of the tree to connected rayless graphs.