Explorations on the number of realizations of minimally rigid graphs

TL;DR

This paper analyzes computational data on rigid graphs to establish bounds and properties of their realizations, advancing understanding of their combinatorial and geometric complexity.

Contribution

It provides new bounds on the number of realizations of rigid graphs and explores effects of construction rules, supported by certificate graphs.

Findings

New lower bounds on maximal realization counts

Upper bounds for minimal realization counts in higher dimensions

Effects of rigidity-preserving constructions on realization numbers

Abstract

Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also possible for the first time to do computations on large sets of graphs. In this paper we show what we can conclude from the data we got from these computations. This includes new lower bounds on the maximal realization count for a given number of vertices, upper bounds for the minimal realization count in higher dimensions and effects of rigidity preserving construction rules on the realization number. In all cases we give certificate graphs which prove the respective results.