Ends in digraphs

AbstractIn this paper a sort of end concept for directed graphs is introduced and examined. Two one-way infinite paths are called equivalent iff there are infinitely many pairwise disjoint paths joining them. An end of an undirected graph is an equivalence class with respect to this relation. For two one-way infinite directed paths U and V define: (a) U⩽V iff there are infinitely many pairwise disjoint directed paths from U to V; (b) U ∼ V iff U ⩽ V and V ⩽ U. The relation ⩽ is a quasiorder, and hence ∼ is an equivalence relation whose classes are called ends. Furthermore, ⩽ induces a partial order on the set of ends of a digraph. In the main section, necessary and sufficient conditions are presented for an abstract order to be representable by the end order of a digraph