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. 
Anthony Nixon and Elissa Ross
math.CO math.MG