# Minimal Separators in Graphs

## TL;DR

This paper provides a new elementary proof of Menger's theorem by studying minimal separators between non-adjacent vertices in finite graphs, offering insights into graph connectivity and separator structures.

## Contribution

It introduces a novel elementary proof of Menger's theorem focusing on minimal separators, simplifying understanding of graph connectivity.

## Key findings

- New elementary proof of Menger's theorem
- Characterization of minimal separators in finite graphs
- Enhanced understanding of graph connectivity

## Abstract

The Known Menger's theorem states that in a finite graph, the size of a minimum separator set of any pair of vertices is equal to the maximum number of disjoint paths that can be found between these two vertices. In this paper, we study the minimal separators of two non-adjacent vertices in a finite graph, and we give a new elementary proof of Menger's theorem.