Lower bounds on the number of realizations of rigid graphs

TL;DR

This paper establishes new lower bounds on the maximum number of complex realizations of minimally rigid graphs in two and three dimensions, using computational and theoretical methods.

Contribution

It introduces novel lower bounds for realizations of rigid graphs, combining computational algorithms with graph construction theory in 2D and 3D.

Findings

New lower bounds for 2D minimally rigid graphs

Extension of bounds to 3D rigid graphs

Use of computational algebra for bound estimation

Abstract

Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Towards this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gr\"obner basis computations.