Abstract
A bar framework (G, p) in dimension r is a graph G whose nodes are points \(p^1,\ldots ,p^n\) in \(\mathbb {R}^r\) and whose edges are line segments between pairs of these points. Two frameworks (G, p) and (G, q) are equivalent if each edge of (G, p) has the same (Euclidean) length as the corresponding edge of (G, q). A pair of non-adjacent vertices i and j of (G, p) is universally linked if \(||p^i-p^j||\) = \(||q^i-q^j||\) in every framework (G, q) that is equivalent to (G, p). Framework (G, p) is universally rigid iff every pair of non-adjacent vertices of (G, p) is universally linked. In this paper, we present a unified treatment of the universal rigidity problem based on the geometry of spectrahedra. A spectrahedron is the intersection of the positive semidefinite cone with an affine space. This treatment makes it possible to tie together some known, yet scattered, results and to derive new ones. Among the new results presented in this paper are: (1) The first sufficient condition for a given pair of non-adjacent vertices of (G, p) to be universally linked. (2) A new, weaker, sufficient condition for a framework (G, p) to be universally rigid thus strengthening the existing known condition. An interpretation of this new condition in terms of the Strong Arnold Property is also presented.
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The author would like to thank the referees for their comments which improved the presentation of the paper. In particular we would like to thank referee 2 for useful insights and for suggesting an alternative proof of Theorem 4.
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Research supported by the Natural Sciences and Engineering Research Council of Canada.
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Alfakih, A.Y. Universal rigidity of bar frameworks via the geometry of spectrahedra. J Glob Optim 67, 909–924 (2017). https://doi.org/10.1007/s10898-016-0448-y
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DOI: https://doi.org/10.1007/s10898-016-0448-y
Keywords
- Semidefinite programming
- Universal rigidity
- Bar frameworks
- Spectrahedra
- Stress matrices
- Gale transform
- Cayley configuration space
- Distance geometry
- Strong Arnold Property