Sufficient connectivity conditions for rigidity of symmetric frameworks

TL;DR

This paper establishes new connectivity conditions that guarantee the rigidity of symmetric frameworks in the plane, extending classical results to symmetric cases.

Contribution

It introduces sufficient connectivity criteria for symmetric framework rigidity, covering both forced and incidental symmetry cases, advancing understanding beyond non-symmetric graph rigidity.

Findings

Connectivity conditions for symmetric rigidity are established.

Results extend classical rigidity theorems to symmetric frameworks.

Provides criteria applicable to both forced and incidental symmetry cases.

Abstract

It is a famous result of Lovasz and Yemini (1982) that 6-connected graphs are rigid in the plane. This was recently improved by Jackson and Jordan (2009) who showed that 6-mixed connectivity is also sufficient for rigidity. Here we give sufficient graph connectivity conditions for both `forced symmetric' and `incidentally symmetric' infinitesimal rigidity in the plane.