Cocompact unfolding trees

TL;DR

This paper characterizes when rooted path trees of finite directed graphs have finitely many automorphism orbits, providing an algorithm to identify cocompact trees and their near-isomorphisms.

Contribution

It offers a characterization and an algorithm to determine cocompactness of rooted path trees in finite directed graphs.

Findings

Identifies conditions for cocompactness of rooted path trees.

Provides an algorithm to test cocompactness.

Classifies trees almost isomorphic to cocompact trees.

Abstract

This paper will show when a rooted path tree of a finite directed rooted graph has only finitely many orbits under the action of its undirected automorphism group (i.e. when it is cocompact). This will allow us to specify which trees are almost isomorphic to cocompact trees. We will provide an algorithm that will determine this, thus mostly answering question (1) from arXiv:2212.07205.