On the Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidian space are embedded graphs having a unique local realization (up to Euclidian motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.88^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley-Menger variety $CM^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n-4$ hyperplanes yields at most $deg(CM^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}={1/2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular we show that it leads to an upper bound of $2 D^{3,n}= {\frac{2^{n-3}}{n-2}}{{n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to rigid motions

On the maximal number of real embeddings of spatial minimally rigid graphs

The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in $\mathbb{R}^3$. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the {\em a priori} number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in $\mathbb{R}^3$, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in $\mathbb{R}^3$

On the maximal number of real embeddings of minimally rigid graphs in R2\mathbb{R}^2, R3\mathbb{R}^3 and S2S^2

Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space $\mathbb{R}^d$ 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 $\mathbb{R}^3$. 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 $\mathbb{R}^2$ and $\mathbb{R}^3$, 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 (Bartzos et al, 2018)

On the Number of Embeddings of Minimally Rigid Graphs

Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with $n$ vertices. We show that, modulo planar rigid motions, this number is at most ${{2n-4}\choose {n-2}} \approx 4^n$. We also exhibit several families which realize lower bounds of the order of $2^n$, $2.21^n$ and $2.28^n$. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley--Menger variety ${\it CM}^{2,n}(C)\subset P_{{{n}\choose {2}}-1}(C)$ over the complex numbers $C$. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with $2n-4$ hyperplanes yields at most $deg({\it CM}^{2,n})$ zero-dimensional components, and one finds this degree to be $D^{2,n}=\frac{1}{2}{{2n-4}\choose {n-2}}$. The lower bounds are related to inductive constructions of minimally rigid graphs via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of $2 D^{3,n}= {({2^{n-3}}/({n-2}})){{2n-6}\choose{n-3}}$ for the number of spatial embeddings with generic edge lengths of the $1$-skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the non-Euclidean case

Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs

A graph G is called generically minimally rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a function of the number of vertices. The study of rigid graphs is motivated by numerous applications, mostly in robotics, bioinformatics, sensor networks and architecture. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, yields interesting upper bounds on the number of embeddings. We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots, and by applying the theory of distance geometry. We focus on R2 and R3, where Laman graphs and 1-skeleta (or edge graphs) of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for n ≤ 10 in R2 and R3, which reduce the existing gaps and lead to tight bounds for n ≤ 7 in both R2 and R3; in particular, we describe the recent settlement of the case of Laman graphs with 7 vertices. Our approach also yields a new upper bound for Laman graphs with 8 vertices, which is conjectured to be tight. We also establish the first lower bound in R3 of about 2.52n, where n denotes the number of vertices

Upper Bounds on the number of embeddings of minimally rigid graphs

Στη θεωρία γραφημάτων (γράφων), ένα άκαμπτο γράφημα είναι ένα γράφημα που έχει πεπερασμένο αριθμό εμβυθίσεων στο $\mathbb{R}^d$, ως προς τις Ευκλείδιες κινήσεις, για δεδομένα μήκη ακμών. Η εμβύθιση γραφήματος στο $\mathbb{R}^d$ είναι μια ανάθεση των κορυφών σε σημεία στο $\mathbb{R}^d$, η οποία δημιουργεί ένα σύνολο με μήκη ακμών που αντιστοιχούν στις αποστάσεις μεταξύ των συνδεδεμένων κορυφών. Μια σημαντική κλάση άκαμπτων γραφημάτων είναι η κλάση των ελαχιστικώς άκαμπτων γραφημάτων. Ένα ελαχιστικώς άκαμπτο γράφημα, είναι ένα γράφημα που είναι άκαμπτο και έχει την ιδιότητα ότι η αφαίρεση οποιασδήποτε ακμής του, δίνει ένα γράφημα που δεν είναι άκαμπτο. Ένα σημαντικό ανοιχτό πρόβλημα είναι η εύρεση άνω φραγμάτων στον αριθμό τον εμβυθίσεων στο $\mathbb{R}^d$. Για ένα μεγάλο χρονικό διάστημα, μόνο το άνω φράγμα $\mathcal{O}(2^{d\cdot|V|})$ ήταν γνωστό στον αριθμό των εμβυθίσεων, που προέρχεται από την άμεση εφαρμογή του θεωρήματος του B\'ezout. Στο [Bartzos et al., 2020], το φράγμα βελτιώθηκε για $d\geq5$, χρησιμοποιώντας τους permanent πίνακες. Πρόσφατα στο [Bartzos et al., 2021], το ασυμπτωτικό άνω φράγμα βελτιώθηκε για κάθε διάσταση. Στην ειδική περίπτωση του $d=2$, το ασυμπτωτικό άνω φράγμα βελτιώθηκε σε $\mathcal(3.7764^)$. Είναι γνωστό ότι ο αριθμός των λύσεων ενός τετράγωνου αλγεβρικού συστήματος σχετίζεται με τον αριθμό των εμβυθίσεων. Συγκεκριμένα, ο αριθμός των μιγαδικών λύσεων ενός τέτοιου αλγεβρικού συστήματος επεκτείνει την έννοια των πραγματικών εμβυθίσεων στον μιγαδικό χώρο, επιτρέποντάς μας να φράξουμε τις μιγαδικές λύσεις χρησιμοποιώντας εργαλεία από τη μιγαδική αλγεβρική γεωμετρία. Σε αυτή την διπλωματική, μετρώντας τους outdegree-περιορισμένους προσανατολισμούς ενός γραφήματος που σχετίζονται με τα αλγεβρικά φράγματα [Bartzos et al., 2020], βελτιώνουμε τα υπάρχοντα άνω φράγματα, για την κλάση των ελαχιστικώς άκαμπτων γραφημάτων, στον αριθμό των εμβυθίσεων.In graph theory, a rigid graph is a graph that has a finite number of embeddings in $\mathbb{R}^d$ up to rigid motions, with respect to a set of edge length constraints. An embedding of graph in $\mathbb{R}^d$ is an assignment of vertices to points in $\mathbb{R}^d$, which also induces a set of edge lengths that correspond to the distances between the connected vertices. An important class of rigid graphs is the class of minimally rigid graphs. A minimally rigid graph, is a graph that is rigid and has the property that the removal of any edge yields a graph that is not rigid. It is a major open problem to find tight upper bounds on the number of the embeddings in $\mathbb{R}^d$. For a long period, only the trivial bound of $\mathcal{O}(2^{d \cdot |V|})$ was known on the number of embeddings, that is derived from the direct application of B\'ezout's Theorem. In [Bartzos et al., 2020], the bound was improved for $d\geq5$, using matrix permanents. Recently in [Bartzos et al., 2021], the asymptotic bound was improved in all dimension. In the special case of $d=2$, the asymptotic upper bound was improved to $\mathcal(3.7764^)$. It is known that the number of solutions of a well-constrained algebraic system is related to the number of embeddings. In particular, the number of the complex solutions of such an algebraic system extends the notion of real embeddings in the complex space, allowing us to bound the complex solutions, using tools from the complex algebraic geometry. In this thesis, by counting outdegree-constrained orientations of a graph that are related to the algebraic bounds [Bartzos et al., 2020], we improve the existing upper bounds, for the class of minimally rigid graphs, on the number of embeddings

On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs

International audienceRigid graph theory is an active area with many open problems, especially regarding embeddings in R^d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S^2 and C36. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall

Lower bounds on the number of realizations of rigid graphs

Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. Towards this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential. Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gr\"obner basis computations