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

Abstract

A (k - 1, k)-graph is a multi-graph satisfying e′ ≤ (k - 1)v′ - k for every non-empty subset of e′ edges on v′ vertices, with equality when e′ = |E(G)|. A (k - 1, k)-frame is a structure generalizing an (n - 2, 2)-framework in n-space, a structure consisting of a set of (n - 2)-dimensional bodies in n-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. © 1991 Springer-Verlag.