Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach

TL;DR

This paper explores the rigidity of bar-and-joint frameworks using matroid theory, highlighting key combinatorial results and their applications across engineering, CAD, molecular biology, and data science.

Contribution

It provides a comprehensive overview of combinatorial rigidity theory with a focus on matroid-theoretic approaches and their diverse applications.

Findings

Key rigidity conditions characterized by matroids

Connections between rigidity and combinatorial structures

Applications to engineering and data science

Abstract

A $d$-dimensional (bar-and-joint) framework $(G,p)$ consists of a graph $G=(V,E)$ and a realisation $p:V\to \mathbb{R}^d$. It is rigid if every continuous motion of the vertices which preserves the lengths of the edges is induced by an isometry of $\mathbb{R}^d$. The study of rigid frameworks has increased rapidly since the 1970s stimulated by numerous applications in areas such as civil and mechanical engineering, CAD, molecular conformation, sensor network localisation and low rank matrix completion. We will describe some of the main results in combinatorial rigidity theory and their applications to other areas of combinatorics, putting an emphasis on links to matroid theory.