The rigidity of graphs

Abstract

We regard a graph G as a set { 1 , … , v } \{ 1, \ldots , v \} together with a nonempty set E of two-element subsets of { 1 , … , v } \{ 1, \ldots , v \} . Let p = ( p 1 , … , p v ) p = ({p_1},\ldots ,{p_v}) be an element of R n v {\textbf {R}^{nv}} representing v points in R n {\textbf {R}^n} . Consider the figure G ( p ) G(p) in R n {\textbf {R}^n} consisting of the line segments [ p i , p j ] [{p_i},{p_j}] in R n {\textbf {R}^n} for { i , j } ∈ E \{ i,j\} \in E . The figure G ( p ) G(p) is said to be rigid in R n {\textbf {R}^n} if every continuous path in R n v {\textbf {R}^{nv}} , beginning at p and preserving the edge lengths of G ( p ) G(p) , terminates at a point q ∈ R n v q \in {\textbf {R}^{nv}} which is the image ( T p 1 , … , T p v ) (T{p_1}, \ldots ,T{p_v}) of p under an isometry T of R n {\textbf {R}^n} . Otherwise, G ( p ) G(p) is flexible in R n {\textbf {R}^n} . Our main result establishes a formula for determining whether G ( p ) G(p) is rigid in R n {\textbf {R}^n} for almost all locations p of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in R 3 {\textbf {R}^3} , and other related matters.