Edge-connectivity and tree-structure in finite and infinite graphs

TL;DR

This paper demonstrates that any graph can be uniquely decomposed into tree-like structures based on its edge connectivity for all levels, finite and infinite, providing a unified framework for understanding graph connectivity.

Contribution

It introduces a canonical decomposition of graphs into edge-connected components for all connectivity levels simultaneously, extending previous work to both finite and infinite graphs.

Findings

Universal tree-like decomposition for all edge-connectivity levels.

Applicable to both finite and infinite graphs.

Provides a unified framework for graph connectivity analysis.

Abstract

We show that every graph admits a canonical tree-like decomposition into its $k$-edge-connected pieces for all $k\in\mathbb{N}\cup\{\infty\}$ simultaneously.