An upper bound on Euclidean embeddings of rigid graphs with 8 vertices

TL;DR

This paper investigates the maximum number of planar Euclidean embeddings for minimally rigid graphs with 8 vertices, addressing a key unknown case in graph rigidity theory.

Contribution

It provides an upper bound on the number of embeddings for 8-vertex minimally rigid graphs in the plane, advancing understanding of graph rigidity.

Findings

Established an upper bound for embeddings of 8-vertex rigid graphs

Identified the smallest unresolved case in planar graph rigidity

Contributed to the theoretical understanding of graph embedding limits

Abstract

A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining the maximum number of planar Euclidean embeddings of minimally rigid graphs with 8 vertices, because this is the smallest unknown case in the plane.