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The Rigidity of Graphs

L. Asimow and B. Roth
Transactions of the American Mathematical Society
Vol. 245 (Nov., 1978), pp. 279-289
DOI: 10.2307/1998867
Stable URL: http://www.jstor.org/stable/1998867
Page Count: 11
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The Rigidity of Graphs
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Abstract

We regard a graph $G$ as a set $\{1,\ldots,v\}$ together with a nonempty set $E$ of two-element subsets of $\{1,\ldots,v\}$. Let $p = (p_1,\ldots,p_v)$ be an element of $R^{nv}$ representing $v$ points in $R^n$. Consider the figure $G(p)$ in $R^n$ consisting of the line segments $\lbrack p_i,p_j\rbrack$ in $R^n$ for $\{i,j\} \in E$. The figure $G(p)$ is said to be rigid in $R^n$ if every continuous path in $R^{nv}, beginning at $p$ and preserving the edge lengths of $G(p)$, terminates at a point $q \in R^{nv}$ which is the image $(Tp_1,\ldots, Tp_v)$ of $p$ under an isometry $T$ of $R^n$. Otherwise, $G(p)$ is flexible in $R^n$. Our main result establishes a formula for determining whether $G(p)$ is rigid in $R^n$ for almost all locations $p$ of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in $R^3$, and other related matters.

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