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New Upper Bounds for the Number of Embeddings of Minimally Rigid Graphs


By definition, a rigid graph in \(\mathbb {R}^d\) (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 \(\mathbb {C}^d\). A major open problem has been to obtain tight upper bounds on the number of embeddings in \(\mathbb {C}^d\), for a given number |V| of vertices, which obviously also bound their number in \(\mathbb {R}^d\). Moreover, in most known cases, the maximal numbers of embeddings in \(\mathbb {C}^d\) and \(\mathbb {R}^d\) coincide. For decades, only the trivial bound of \(O(2^{d|V|})\) was known on the number of embeddings. Recently, matrix permanent bounds have led to a small improvement for \(d\ge 5\). This work improves upon the existing upper bounds for the number of embeddings in \(\mathbb {R}^d\) and \(S^d\), 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, we improve the exponent basis in the asymptotic behavior with respect to the number of vertices of the recent bound mentioned above by the factor of \(\sqrt{2}\). 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.

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  1. In [4], indegree constraints were used but here, following [35], we use (equivalently) outdegrees.

  2. A similar discussion in [4, Sect. 2.2] offers details on the algebraic systems.

  3. Hanging edges are reminiscent of "directed loops" in hypergraphs [35]; "half-edges" also have a single endpoint.

  4. In [19, Chap. 3] these subgraphs are called blocks; "biconnected component" is used equivalently, e.g. [22, Chap. 8].


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EB was fully supported and IZE was partially supported by project ARCADES which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No 675789. EB and IZE are members of team AROMATH, joint between INRIA Sophia-Antipolis, France, and NKUA.

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Correspondence to Evangelos Bartzos.

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Bartzos, E., Emiris, I.Z. & Vidunas, R. New Upper Bounds for the Number of Embeddings of Minimally Rigid Graphs. Discrete Comput Geom 68, 796–816 (2022).

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  • Distance geometry
  • Minimally rigid graph
  • Rigid embedding
  • Upper bound
  • Laman graph
  • Oriented graph

Mathematics Subject Classification

  • 52C25
  • 14N10