Robust Tensegrity Polygons

A tensegrity polygon is a planar cable-strut tensegrity framework in which the cables form a convex polygon containing all vertices. The underlying edgelabeled graph, in which the cable edges form a Hamilton cycle, is an abstract tensegrity polygon. It is said to be robust if every convex realization as a tensegrity polygon has an equilibrium stress which is positive on the cables and negative on the struts. It is called stable if every convex realization is infinitesimally rigid. We characterize the robust as well as the stable abstract tensegrity polygons on n vertices with n−2 struts, answering a question of B. Roth and W. Whiteley from 1981 and solving an open problem of R. Connelly from 2008.