# Minimal bricks

## TL;DR

This paper introduces minimal bricks, a special class of 3-connected graphs with unique matching properties, and proves a generation theorem along with bounds on edges and degree conditions.

## Contribution

It provides a generation theorem for minimal bricks and establishes bounds on edges and degree properties, advancing understanding of their structure.

## Key findings

- Minimal bricks have at most 5n-7 edges for n>4.
- Every minimal brick has at least three vertices of degree three.
- A generation theorem for minimal bricks is proved.

## Abstract

A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is minimal if for every edge e the deletion of e results in a graph that is not a brick. We prove a generation theorem for minimal bricks and two corollaries: (1) for n>4, every minimal brick on 2n vertices has at most 5n-7 edges, and (2) every minimal brick has at least three vertices of degree three.