# Small unit-distance graphs in the plane

## TL;DR

This paper characterizes all small unit-distance graphs in the plane with up to 9 vertices by identifying 74 minimal forbidden configurations, extending previous classifications for graphs with up to 7 vertices.

## Contribution

The authors extend the classification of unit-distance graphs to 9 vertices, identifying 74 minimal forbidden graphs, thus advancing the understanding of graph structures in geometric graph theory.

## Key findings

- Complete characterization of unit-distance graphs up to 9 vertices
- Identification of 74 minimal forbidden graphs
- Extension of previous classifications for smaller graphs

## Abstract

We prove that a graph on up to 9 vertices is a unit-distance graph if and only if it does not contain one of 74 so-called minimal forbidden graphs. This extends the work of Chilakamarri and Mahoney (1995), who provide a similar classification for unit-distance graphs on up to 7 vertices.