RIGIDITY OF GRAPHS .2.

We regard a graph G as a set {1,..., v} together with a nonempty set E of two-element subsets of {1,..., v}. Let p = (p1,..., pv) be an element of Rnv representing v points in Rn and consider the realization G(p) of G in Rn consisting of the line segments [pi, pj] in Rn for {i, j} ε{lunate} E. The figure G(p) is said to be rigid in Rn if every continuous path in Rnv, beginning at p and preserving the edge lengths of G(p), terminates at a point q ε{lunate} Rnv which is the image (Tp1,..., Tpv) of p under an isometry T of Rn. We here study the rigidity and infinitesimal rigidity of graphs, surfaces, and more general structures. A graph theoretic method for determining the rigidity of graphs in R2 is discussed, followed by an examination of the rigidity of convex polyhedral surfaces in R3. © 1979.