The generic rigidity of triangulated spheres with blocks and holes

TL;DR

This paper provides combinatorial characterizations of minimal 3-rigidity for block and hole graphs derived from triangulated spheres, confirming longstanding conjectures in the field of rigidity theory.

Contribution

It establishes new combinatorial criteria for minimal 3-rigidity in specific block and hole graph configurations, extending previous conjectures.

Findings

Confirmed Whiteley's 1988 conjecture for these graphs.

Validated special cases of Finbow-Singh and Whiteley's 2013 conjecture.

Provided a framework for understanding rigidity in complex triangulated structures.

Abstract

A simple graph G=(V,E) is 3-rigid if its generic bar-joint frameworks in R3 are infinitesimally rigid. Block and hole graphs are derived from triangulated spheres by the removal of edges and the addition of minimally rigid subgraphs, known as blocks, in some of the resulting holes. Combinatorial characterisations of minimal $3$-rigidity are obtained for these graphs in the case of a single block and finitely many holes or a single hole and finitely many blocks. These results confirm a conjecture of Whiteley from 1988 and special cases of a stronger conjecture of Finbow-Singh and Whiteley from 2013.