In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints (also called constrained non-crossing Laman frameworks) on a given generic set of $n$ points. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in $O(n^4)$ time and O(n) space, or, slightly different implementation, in $O(n^3)$ time and $O(n^2)$ space. In particular, we obtain that the set of all the constrained non-crossing Laman frameworks on a given point set is connected by flips which restore the Laman property.
In 1983 Kalai proved an incredible generalisation of Cayley's formula for the number of trees on a labelled vertex set to a formula for a class of $r$-dimensional simplicial complexes. These simplicial complexes generalise trees by means of being homologically $\Bbb Q$-acyclic. In this text we consider a different generalisation of trees to the class of pure dimensional simplicial complexes that \textit{minimally connect a vertex set} (in the sense that the removal of any top dimensional face disconnects the complex). Our main result provides an upper and lower bound for the number of these mi…
Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than 180 degrees. In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. We provide two proofs, which both yield efficient embedding algorithms. One based on Henneberg inductive constructions from combinatorial rigidity theory, the other on a generalization of Tutte's barycentric embeddings to directed grap…
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 determi…
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 embeddin…
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-Me…
Rigid graph theory is an active area with many open problems, especially regarding embeddings in $\mathbb{R}^d$ 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 B{é}zout (m-B{é}zout) bounds of algebraic systems since they are fast…
A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the matroidal class of (k,l)-sparse graphs for suitable k and l. A characterisation of infinitesimal rigidity is obtained for product norms and it is shown that K_6 - e (respectively, K_7) is the smallest minimally rigid graph for the class of 2 x 2 symmetric (respectively, hermitian) matrices with the trace norm.
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 si…
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 …
We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the symmetry group consists of rotations and translations. Along the way, we use tropical geometry to show how a construction of Edmonds that associates a matroid to a submodular function can be used to give a description of the algebraic matroid of a Hadamard product of two linear spaces in terms of the matroids of each linear space. This leads to new, short, proofs of Laman's theorem, and a theorem of Jord{á}n, Kaszanitzky, and Tanigawa, and M…
The planar rigidity problem asks, given a set of m pairwise distances among a set P of n unknown points, whether it is possible to reconstruct P, up to a finite set of possibilities (modulo rigid motions of the plane). The celebrated Maxwell-Laman Theorem from Rigidity Theory says that, generically, the rigidity problem has a combinatorial answer: the underlying combinatorial structure must contain a spanning minimally-rigid graph (Laman graph). In the case where the system is not rigid, its inclusion-wise maximal rigid substructures (rigid components) are also combinatorially characterized vi…
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results we develop a new technique for deriving linear representations of sparsity matroids on colored graphs and extend the direction network method of proving rigidity characterizations to handle reflections.
When all non-edge distances of a graph realized in $\mathbb{R}^{d}$ as a {\em bar-and-joint framework} are generically {\em implied} by the bar (edge) lengths, the graph is said to be {\em rigid} in $\mathbb{R}^{d}$. For $d=3$, characterizing rigid graphs, determining implied non-edges and {\em dependent} edge sets remains an elusive, long-standing open problem. One obstacle is to determine when implied non-edges can exist without non-trivial rigid induced subgraphs, i.e., {\em nucleations}, and how to deal with them. In this paper, we give general inductive construction schemes and proof tech…
Laman graphs are fundamental to rigidity theory. A graph G with n vertices and m edges is a generic minimally rigid graph (Laman graph), if m=2n-3 and every induced subset of k vertices spans at most 2k-3 edges. We consider the verification problem: Given a graph G with n vertices, decide if it is Laman. We present an algorithm that takes O(T(n)+n log n) time, where T(n) is the best time to extract two edge disjoint spanning trees from G or decide no such trees exist. Our algorithm exploits a known construction called red-black hierarchy (RBH), that is a certificate for Laman graphs. First, we…
