# The number of realizations of a Laman graph

## TL;DR

This paper develops a recursive algebraic and tropical geometric method to count the number of complex realizations of Laman graphs, which model rigid planar frameworks, up to isometries.

## Contribution

It introduces a novel recursive formula for counting complex solutions of the distance systems defining Laman graphs using algebraic and tropical geometry techniques.

## Key findings

- Provides a recursive formula for the number of realizations
- Connects algebraic and tropical geometry methods
- Advances understanding of Laman graph realizations

## Abstract

Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursive formula for the number of complex solutions of such systems.