Which graphs are rigid in $\ell_p^d$?

TL;DR

This paper investigates the conditions under which graphs are minimally rigid in $ ext{ell}_p^d$ spaces, proposing new operations and characterizations that support a conjecture relating rigidity to $(d,d)$-tightness.

Contribution

It introduces a graph bracing operation that preserves independence across dimensions, and proves rigidity for certain sparse graphs and triangulations in $ ext{ell}_p^d$ spaces.

Findings

Graph bracing operation preserves independence in $ ext{ell}_p^d$

Certain sparse graphs are independent in $ ext{ell}_p^d$

Triangulations of the projective plane are minimally rigid in $ ext{ell}_p^3$

Abstract

We present three results which support the conjecture that a graph is minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and $p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $\ell_p^d$ to $\ell_p^{d+1}$. We then prove that every $(d,d)$-sparse graph with minimum degree at most $d+1$ and maximum degree at most $d+2$ is independent in $\ell_p^d$. Finally, we prove that every triangulation of the projective plane is minimally rigid in $\ell_p^3$. A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.