Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

Abstract

A (k − 1,k)-graph is a multi-graph satisfyinge′ ≤ (k − 1)v′ − k for every non-empty subset ofe′ edges onv′ vertices, with equality whene′ = |E(G)|. A (k − 1,k)-frame is a structure generalizing an (n − 2, 2)-framework inn-space, a structure consisting of a set of (n − 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k − 1,k)-frame if and only if it is a (k − 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.