# The Complete Lattice of Erd\H{o}s-Menger Separations

## TL;DR

This paper extends the structure of minimal vertex separations from finite to infinite graphs, showing that Erdős-Menger separations form a complete lattice and can represent any such lattice, generalizing classical results.

## Contribution

It introduces Erdős-Menger separations for infinite graphs and proves they form a complete lattice, generalizing Escalante and Gallai's finite lattice results.

## Key findings

- Erdős-Menger separations always exist in infinite graphs.
- These separations form a complete lattice under a natural partial order.
- Any complete lattice can be represented as such a lattice of separations.

## Abstract

F. Escalante and T. Gallai studied in the seventies the structure of different kind of separations and cuts between a vertex pair in a (possibly infinite) graph. One of their results is that if there is a finite separation, then the optimal (i.e. minimal sized) separations form a finite distributive lattice with respect to a natural partial order. Furthermore, any finite distributive lattice can be represented this way. If there is no finite separation then cardinality is a too rough measure to capture being 'optimal'. Menger's theorem provides a structural characterization of optimality if there is a finite separation. We use this characterization to define Erd\H{o}s-Menger separations even if there is no finite separation. The generalization of Menger's theorem to infinite graphs (which was not available until 2009) ensures that Erd\H{o}s-Menger separations always exist. We show that they form a complete lattice with respect to the partial order given by Escalante and every complete lattice can be represented this way.