On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs

Abstract

Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R d or on a sphere and which in addition satisfy certain edge length constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings.In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in R 3 . One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration.Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in R 2 and R 3 , while in the case of S 2 we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 [[1]].

Counting Realizations of Laman Graphs on the Sphere

“…Remark 6.1. The paper [BELT18] shows that the number of real spherical realizations matches the number of complex ones for some graphs in Table 1 (all graphs with 6 and 7 vertices, and one of the graphs with 8 vertices).…”

“…We hope that, although we still work on a surface, moving from the plane to the sphere could be a first step towards determining the number of realizations for generically minimally rigid graphs in three dimensions. For a related work on this topic, discussing real realizations of graphs on the sphere (in addition to the plane and the space), see the recent paper by Bartzos et al [BELT18]. Among other things, the latter paper proves that for some graphs one can achieve all possible complex realizations via real instances.…”

Ratio-of-Distance Rigidity Theory With Application to Similar Formation Control

“…Before that, we make some necessary definitions. Denote by G G(N ) = (V N , E G G(N ) ), N ≥ 4 the N -agent trilaterated Geiringer graphs, where we follow [35] to call these graphs Geiringer graphs for honoring Hilda Pollaczek-Geiringer who worked on rigid graphs in R 2 and R 3 [36], [37].…”

Solving Polynomial Systems with phcpy

“…The conformations of proteins [LML14], molecules [EM99], and robotic mechanisms (discussed further below) can be studied by counting and classifying unique mechanisms, i.e. real embeddings of graphs with fixed edge lengths, modulo rigid motions, per Bartzos et al [BELT18]. Consider a graph G whose edges e ∈ E G each have a given length d e .…”