A necessary condition for generic rigidity of bar-and-joint frameworks in $d$-space

TL;DR

This paper extends a known necessary condition for the rigidity of bar-and-joint frameworks in Euclidean space from dimensions 3 to 11, providing a broader understanding of rigidity criteria in higher dimensions.

Contribution

The authors generalize Cheng and Sitharam's result, proving that the strengthened necessary condition holds for all dimensions up to 11.

Findings

The necessary condition is valid for all d ≤ 11.

Maximal d-sparse subgraphs must have specific edge counts.

The result broadens the applicability of rigidity conditions in higher dimensions.

Abstract

A graph $G=(V,E)$ is $d$-sparse if each subset $X\subseteq V$ with $|X|\geq d$ induces at most $d|X|-{{d+1}\choose{2}}$ edges in $G$. Maxwell showed in 1864 that a necessary condition for a generic bar-and-joint framework with at least $d+1$ vertices to be rigid in ${\mathbb R}^d$ is that $G$ should have a $d$-sparse subgraph with $d|X|-{{d+1}\choose{2}}$ edges. This necessary condition is also sufficient when $d=1,2$ but not when $d\geq 3$. Cheng and Sitharam strengthened Maxwell's condition by showing that every maximal $d$-sparse subgraph of $G$ should have $d|X|-{{d+1}\choose{2}}$ edges when $d=3$. We extend their result to all $d\leq 11$.