Realizations of Rigid Graphs

TL;DR

This paper develops a recursive formula using algebraic and tropical geometry to count the realizations of minimally rigid graphs in the plane, providing new bounds on their maximum number of realizations.

Contribution

It introduces a novel recursive approach and gluing techniques to count and bound the realizations of Laman graphs, advancing understanding of their geometric properties.

Findings

Derived a recursive formula for counting realizations.

Established a new lower bound on the maximum number of realizations.

Combined computational and theoretical methods for analysis.

Abstract

A minimally rigid graph, also called Laman graph, models a planar framework which is rigid for a general choice of distances between its vertices. In other words, there are finitely many ways, up to isometries, to realize such a graph in the plane. Using ideas from algebraic and tropical geometry, we derive a recursive formula for the number of such realizations. Combining computational results with the construction of new rigid graphs via gluing techniques, we can give a new lower bound on the maximal possible number of realizations for graphs with a given number of vertices.