Which graphs are rigid in $$\ell _p^d$$?

We present three results which support the conjecture that a graph is minimally rigid in d-dimensional $$\ell _p$$ ℓ p -space, where $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and $$p\not =2$$ p ≠ 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 $$\ell _p^d$$ ℓ p d to $$\ell _p^{d+1}$$ ℓ p d + 1 . We then prove that every (d, d)-sparse graph with minimum degree at most $$d+1$$ d + 1 and maximum degree at most $$d+2$$ d + 2 is independent in $$\ell _p^d$$ ℓ p d . Finally, we prove that every triangulation of the projective plane is minimally rigid in $$\ell _p^3$$ ℓ 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.

Epsilon local rigidity and numerical algebraic geometry

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek–Geiringer–Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 4.2 and its corresponding Algorithm 1 which decide if a configuration is [Formula: see text]-locally rigid, a notion we define. A configuration which is [Formula: see text]-locally rigid may be locally rigid or flexible, but any continuous deformations remain within a sphere of radius [Formula: see text] in configuration space. Deciding [Formula: see text]-local rigidity is possible for configurations which are smooth or singular, generic or non-generic. We also present Algorithms 2 and 3 which use numerical algebraic geometry to compute a discrete-time sample of a continuous flex, providing useful visual information for the scientist.

Rigidity of Linearly Constrained Frameworks

We consider the problem of characterising the generic rigidity of bar-joint frameworks in $\mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by Streinu and Theran [14] in 2010. We will extend their characterisation to the case when $d\geq 3$ and each vertex is constrained to lie in an affine subspace of dimension $t$, when $t=1,2$ and also when $t\geq 3$ and $d\geq t(t-1)$. We then point out that results on body–bar frameworks obtained by Katoh and Tanigawa [8] in 2013 can be used to characterise when a graph has a rigid realisation as a $d$-dimensional body–bar framework with a given set of linear constraints.

Periodic Rigidity on a Variable Torus Using Inductive Constructions

In this paper we prove a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice. We follow the approach of modelling periodic frameworks as frameworks on a torus and use the language of gain graphs for the finite counterpart of a periodic graph. In this setting we employ variants of the Henneberg operations used frequently in rigidity theory.