Inductive constructions for frameworks on a two-dimensional fixed torus

An infinite periodic framework in the plane can be represented as a framework on a torus, using a ${\mathbb{Z}}^2$-labelled gain graph. We find necessary and sufficient conditions for the generic minimal rigidity of frameworks on the two-dimensional fixed torus ${\mathcal{T}}_0^2$. It is also shown that every minimally rigid periodic orbit framework on ${\mathcal{T}}_0^2$ can be constructed from smaller frameworks through a series of inductive constructions. These are fixed torus adapted versions of the results of Laman and Henneberg respectively for finite frameworks in the plane. The proofs involve the development of inductive constructions for ${\mathbb{Z}}^2$-labelled graphs.