Graph embeddings in the plane, space and sphere - source code and results

2018-11-25T04:49:34Z (GMT) by Bartzos, Evangelos Legerský, Jan

This is supplementary material for the paper On the Maximal Number of Real Embeddings of Minimally Rigid Graphs in \(\mathbb{R}^2\), \(\mathbb{R}^3\) and \(S^2\). It includes source code and results. See also the project website.

Construction of Laman and Geiringer graphs

MATLAB and SageMath code for construction of Laman and Geiringer graphs using Henneberg steps.

Computation of mixed volume and complex solutions

Functions that compute the mixed volume and the (complex) embeddings in the plane, space or sphere of a given graph using different algebraic formulations.

Coupler curve visualization and sampling method

Implementation of our method for obtaining edge lengths of a Geiringer graph with many real spatial embeddings.
Visualization of coupler curves of the 7-vertex Geiringer graph with the maximal number of real embeddings (G48).

Maple parametric search

Maple worksheet illustrating a method to improve the number of real embeddings of G48 using RootFinding[Parametric] subpackage.

Results

Mixed volumes and numbers of real and complex embeddings in the plane, space and sphere.
Edge lengths giving high numbers of real embeddings, in the plane, space and sphere. 
Maple script verifying the results.