A cactus theorem for end cuts

TL;DR

This paper extends the cactus encoding of minimal edge cuts from vertices to ends of graphs, providing a unified framework for various cut types in finite and infinite graphs.

Contribution

It introduces a cactus theorem for end cuts, generalizing previous vertex cut encodings to include cuts separating ends of graphs.

Findings

Minimal end cuts can be encoded by cacti.

The method applies to both finite and infinite graphs.

Multiple types of cuts are representable by cacti.

Abstract

Dinits-Karzanov-Lomonosov showed that it is possible to encode all minimal edge cuts of a graph by a tree-like structure called a cactus. We show here that minimal edge cuts separating ends of the graph rather than vertices can be `encoded' also by a cactus. We apply our methods to finite graphs as well and we show that several types of cuts can be encoded by cacti.