The generic minimal rigidity of a partially triangulated torus

TL;DR

This paper establishes necessary and sufficient conditions for the minimal 3-rigidity of certain graphs derived from triangulated tori, advancing understanding of rigidity in geometric graph theory.

Contribution

It provides a complete characterization of minimal 3-rigidity for graphs formed from triangulated tori with specific edge deletions, a novel result in rigidity theory.

Findings

Characterization of minimal 3-rigidity conditions

Conditions applicable to graphs from triangulated tori

Advancement in geometric rigidity theory

Abstract

A simple graph is $3$-rigid if its generic bar-joint frameworks in $R^3$ are infinitesimally rigid. Necessary and sufficient conditions are obtained for the minimal $3$-rigidity of a simple graph which is obtained from the $1$-skeleton of a triangulated torus by the deletion of edges interior to a triangulated disc.