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

Sean Dewar, Derek Kitson, Anthony Nixon

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.

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

Abstract

We present three results which support the conjecture that a graph is minimally rigid in -dimensional -space, where and , if and only if it is -tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from to . We then prove that every -sparse graph with minimum degree at most and maximum degree at most is independent in . Finally, we prove that every triangulation of the projective plane is minimally rigid in . 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.

Paper Structure

This paper contains 6 sections, 2 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Rigidity in lp
  3. Configuration spaces
  4. The rigidity matrix
  5. Framework rigidity
  6. Frameworks in $\ell^d_q$

Key Result

Lemma 2.1

Let $X$ be a finite dimensional normed linear space and let $\mathcal{S}(X)$ denote the set of all smooth points in $X$ together with the point $0 \in X$. Define $\Gamma : \mathcal{S}(X) \rightarrow X^*$ by setting $\Gamma(x) = \varphi_x$ and $\Gamma(0)=0$. Then,

Theorems & Definitions (6)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Example 2.6