On the structure of vertex cuts separating the ends of a graph

TL;DR

This paper extends the structural understanding of minimal vertex cuts separating ends of graphs, showing they can be represented by a succulent, a generalization of a cactus, unifying previous edge and vertex cut results.

Contribution

It introduces the concept of succulents to encode minimal vertex cuts separating ends, generalizing cactus structures and unifying prior results.

Findings

Vertex cuts can be represented by succulents.

The cactus structure results are special cases of the new framework.

Unified approach for edge and vertex cut structures.

Abstract

Dinits, Karzanov and Lomonosov showed that the minimal edge cuts of a finite graph have the structure of a cactus, a tree-like graph constructed from cycles. Evangelidou and Papasoglu extended this to minimal cuts separating the ends of an infinite graph. In this paper we show that minimal vertex cuts separating the ends of a graph can be encoded by a succulent, a mild generalization of a cactus that is still tree-like. We go on to show that the earlier cactus results follow from our work.