An analogue of Edmonds' Branching Theorem for infinite digraphs

TL;DR

This paper extends Edmonds' Branching Theorem to infinite digraphs by introducing pseudo-arborescences, enabling a new packing result and exploring their properties beyond traditional arborescences.

Contribution

It introduces pseudo-arborescences as a novel concept to generalize Edmonds' Theorem for locally finite infinite digraphs.

Findings

Pseudo-arborescences enable packing results in infinite digraphs.

They exhibit tree-like properties but are not always topological trees.

The concept broadens the applicability of branching theorems to infinite structures.

Abstract

We extend Edmonds' Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the notion of pseudo-arborescences and prove a corresponding packing result. Finally, we verify some tree-like properties for these objects, but give also an example that their underlying graphs do in general not correspond to topological trees in the Freudenthal compactification of the underlying multigraph of the digraph.