# A Stallings' type theorem for quasi-transitive graphs

## TL;DR

This paper extends Stallings' theorem to infinite quasi-transitive graphs, showing such graphs with multiple ends decompose into tree amalgamations, with applications to hyperbolic, planar, and end-structured graphs.

## Contribution

It provides a graph-theoretical analogue of Stallings' splitting theorem, characterizing multi-ended quasi-transitive graphs via tree amalgamations and answering a question about planar graphs.

## Key findings

- Multi-ended quasi-transitive graphs are tree amalgamations.
- The results imply Stallings' theorem for finitely generated groups.
- Application to planar graphs confirms a prior open question.

## Abstract

We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings' splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. It will also lead to a characterisation of accessible graphs in terms of tree amalgamations. We obtain applications of our results for hyperbolic graphs, planar graphs and graphs without any thick end. The application for planar graphs answers a question of Mohar in the affirmative.