The rigidity of graphs

Authors:
L. Asimow and B. Roth

Journal:
Trans. Amer. Math. Soc. **245** (1978), 279-289

MSC:
Primary 57M15; Secondary 05C10, 52A40, 53B50, 73K05

MathSciNet review:
511410

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Abstract: We regard a graph *G* as a set together with a nonempty set *E* of two-element subsets of . Let be an element of representing *v* points in . Consider the figure in consisting of the line segments in for . The figure is said to be rigid in if every continuous path in , beginning at *p* and preserving the edge lengths of , terminates at a point which is the image of *p* under an isometry *T* of . Otherwise, is flexible in . Our main result establishes a formula for determining whether is rigid in for almost all locations *p* of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in , and other related matters.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0511410-9

Article copyright:
© Copyright 1978
American Mathematical Society