One brick at a time: a survey of inductive constructions in rigidity theory

TL;DR

This survey reviews how inductive graph moves are used to analyze and characterize the rigidity of various frameworks, including symmetric, periodic, and surface frameworks, highlighting key results and open problems.

Contribution

It compiles and discusses the use of inductive constructions in rigidity theory across multiple framework types, emphasizing recent advances and open challenges.

Findings

Inductive constructions can characterize rigidity in various frameworks.

Recursive graph moves preserve framework rigidity.

Open problems remain in extending inductive methods to new framework classes.

Abstract

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. We summarize the key outstanding open problems related to inductions.