Mixed Volume Techniques for Embeddings of Laman Graphs

TL;DR

This paper explores the use of mixed volume techniques and Bernstein's Theorem to analyze the number of embeddings of Laman graphs, providing bounds that are sometimes tight for specific graph classes.

Contribution

It introduces methods to compute mixed volumes of polynomial systems derived from Laman graph constraints, offering new bounds on the number of embeddings.

Findings

Bounds obtained from mixed volume techniques are sometimes tight for certain graph classes.

Most bounds are weaker than existing known bounds on embeddings.

The paper provides a framework for studying polynomial systems from graph constraints.

Abstract

Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is to provide the methods to study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. While in most cases the resulting bounds are weaker than the best known bounds on the number of embeddings, for some classes of graphs the bounds are tight.