Infinitesimal rigidity for non-Euclidean bar-joint frameworks

Abstract

The minimal infinitesimal rigidity of bar-joint frameworks in the non-Euclidean spaces ( R 2 , ∥ · ∥ q ) , for 1 ⩽ q ⩽ ∞ , q ≠ 2 , is characterized in terms of ( 2 , 2 ) -tight graphs. Specifically, a generically placed bar-joint framework ( G , p ) in the plane is minimally infinitesimally rigid with respect to a non-Euclidean ℓ q norm if and only if the underlying graph G = ( V , E ) contains 2 | V | - 2 edges and every subgraph H = ( V ( H ) , E ( H ) ) contains at most 2 | V ( H ) | - 2 edges.