Abstract
In this chapter we discuss rigidity and flexibility of graph frameworks
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Notes
- 1.
The operator \(\mathsf{dim\, aff}\) returns the dimensionality of the affine hull of the argument, see Appendix A.4.3.
- 2.
Gluck’s theorem actually predates, and indeed inspired, Asimow and Roth’s.
- 3.
Also see the footnote on page 28.
- 4.
In the world of human manufacturing, because humans appears to like symmetry and special relationships, realizations may be more likely to be singular than one would believe reasonable.
- 5.
In most of the existing literature (with the notable exception of [57]), generic rigidity is defined differently: a framework (G, x) is generically rigid in \(\mathbb {R}^K\) if it is rigid and there is no single rational polynomial having all the components \(x_{ij}\) as roots, for all \(i\le K, j\le n=|V(G)|\); or, in other words, x is algebraically independent. If this is the case, then of course the rows of the rigidity matrix could never be linearly dependent, which makes x regular. But since the converse is not true (i.e., there are algebraically dependent but regular realizations), and regularity is really all that is required, we feel, along with [57], that this traditional genericity notion is too strong. By contrast, with our definition a generically rigid framework is simply a rigid framework with a regular realization, so any infinitesimally rigid framework is generic (this is false using the definition based on algebraic independence). This definition is similar to the one given by Graver in [56]: the framework (G, x) is generic if all the nontrivial minors of the rigidity matrix have nonzero value (a minor depending on some symbol x is trivial if it is identically zero independently of the value assigned to x); and generically rigid if it is generic and rigid. Graver’s definition is actually stricter than ours, in the sense that all nontrivial minors of the rigidity matrix are required to take nonzero value at x. The framework at \(x_1=(0,0), x_2=(1,0), x_3=(0,1)\) of a 3-clique provides an example of a generic framework which is not generic in the sense of Graver (its rigidity matrix has maximum rank 3 but the \(3\times 3\) minor given by columns 1, 3, 6 is nontrivial yet has value 0).
- 6.
By contrast, any realization which is generic in the sense of Graver is also general [56].
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Liberti, L., Lavor, C. (2017). Flexibility and rigidity. In: Euclidean Distance Geometry. Springer Undergraduate Texts in Mathematics and Technology. Springer, Cham. https://doi.org/10.1007/978-3-319-60792-4_7
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DOI: https://doi.org/10.1007/978-3-319-60792-4_7
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