Vertex Cuts

TL;DR

This paper extends structure tree theory from edge removal to vertex removal, broadening the understanding of graph decompositions and group structures, applicable to both finite and infinite graphs with multiple ends.

Contribution

It introduces a generalized structure tree theory based on vertex removal, extending Tutte's and Stallings' theorems to more complex graph and group structures.

Findings

Generalizes Tutte's tree decomposition to vertex removal

Applies to infinite graphs with multiple ends

Provides a unified framework for graph and group decompositions

Abstract

We generalise structure tree theory, which is based on removing finitely many edges, to removing finitely many vertices. This gives a significant generalization of Tutte's tree decomposition of 2-connected graphs into 3-connected blocks. For a finite graph there is a structure tree that contains information about $k$-connectivity for any $k$. The theory can also be applied to infinite graphs that have more than one vertex end, i.e. ends that can be separated by removing a finite number of vertices. This gives a generalization of Stallings' structure theorem for groups with more than one end.