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Journal Articles Discrete and Computational Geometry Year : 2022

New upper bounds for the number of embeddings of minimally rigid graphs

Abstract

By definition, a rigid graph in Rd (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system. Naturally, the complex solutions of such systems extend the notion of rigidity to Cd. A major open problem has been to obtain tight upper bounds on the number of embeddings in Cd, for a given number |V | of vertices, which obviously also bound their number in Rd. Moreover, in most known cases, the maximal numbers of embeddings in Cd and Rd coincide. For decades, only the trivial bound of O(2d·|V |) was known on the number of embeddings. Recently, matrix permanent bounds have led to a small improvement for d ≥ 5. This work improves upon the existing upper bounds for the number of embeddings in Rd and Sd, by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively eliminates vertices or vertex paths. For the most important cases of d = 2 and d = 3, the new bounds are O(3.7764|V |) and O(6.8399|V |), respectively. In general, the recent asymptotic bound mentioned above is improved by a factor of 1/√2. Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic root counts.

Dates and versions

hal-03895585 , version 1 (12-12-2022)

Identifiers

Cite

Evangelos Bartzos, Ioannis Z. Emiris, Raimundas Vidunas. New upper bounds for the number of embeddings of minimally rigid graphs. Discrete and Computational Geometry, 2022, 68 (3), pp.796. ⟨10.1007/s00454-022-00370-3⟩. ⟨hal-03895585⟩
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