Ends of digraphs III: normal arborescences

TL;DR

This paper introduces normal spanning arborescences in infinite digraphs, establishing their end-faithfulness and a homeomorphism with the digraph's end space, along with a criterion for their existence.

Contribution

It develops the concept of normal spanning arborescences, proving their end-faithfulness, a homeomorphism with the digraph's end space, and provides a Jung-type existence criterion.

Findings

Normal spanning arborescences are end-faithful.

A homeomorphism exists between the end space of a digraph and its normal arborescence.

A Jung-type criterion determines the existence of such arborescences.

Abstract

In a series of three papers we develop an end space theory for digraphs. Here in the third paper we introduce a concept of depth-first search trees in infinite digraphs, which we call normal spanning arborescences. We show that normal spanning arborescences are end-faithful: every end of the digraph is represented by exactly one ray in the normal spanning arborescence that starts from the root. We further show that this bijection extends to a homeomorphism between the end space of a digraph $D$, which may include limit edges between ends, and the end space of any normal arborescence with limit edges induced from $D$. Finally we prove a Jung-type criterion for the existence of normal spanning arborescences.