Generic Global Rigidity Research Articles

Vertex Splitting, Coincident Realisations, and Global Rigidity of Braced Triangulations

We give a relatively short graph theoretic proof of a result of Jordán and Tanigawa that a 4-connected graph which has a spanning plane triangulation as a proper subgraph is generically globally rigid in \({\mathbb {R}}^3\). Our proof is based on a new sufficient condition for the so called vertex splitting operation to preserve generic global rigidity in \({\mathbb {R}}^d\).

Graph Reconstruction from Unlabeled Edge Lengths

A d-dimensional framework is a pair (G, p), where G=(V,E) is a graph and p is a map from V to mathbb {R}^d. The length of an edge uvin E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in mathbb {R}^d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in mathbb {R}^d is globally rigid for some graph G on nge d+2 vertices (where dge 2), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For d=1,2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid.

Generically Globally Rigid Graphs Have Generic Universally Rigid Frameworks

We show that any graph that is generically globally rigid in ℝd has a realization in ℝd that is both generic and universally rigid. This also implies that the graph also must have a realization in ℝd that is both infinitesimally rigid and universally rigid; such a realization serves as a certificate of generic global rigidity. Our approach involves an algorithm by Lovász, Saks and Schrijver that, for a sufficiently connected graph, constructs a general position orthogonal representation of the vertices, and a result of Alfakih that shows how this representation leads to a stress matrix and a universally rigid framework of the graph.

Stress Matrices and Global Rigidity of Frameworks on Surfaces

In 2005, Bob Connelly showed that a generic framework in $${\mathbb {R}}^d$$Rd is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in $${\mathbb {R}}^3$$R3. For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum-rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jordan, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $${{\mathbb {R}}}^2$$ R 2 . The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in $${\mathbb {R}}^3$$ R 3 when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.

Generic global rigidity of body–bar frameworks

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid inRd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.When the configuration p is generic, redundant rigidity and (d+1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d=2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d⩾3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient.However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in Rd for any d⩾1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in Rd, for any d⩾1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.

New Classes of Counterexamples to Hendrickson’s Global Rigidity Conjecture

We examine the generic local and global rigidity of various graphs in ℝd . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for d>3, and we define a construction that generates infinitely many graphs in ℝ5. Finally, we state some conjectures for further exploration.

Formal Theory of Noisy Sensor Network Localization

Graph theory has been used to characterize the solvability of the sensor network localization problem. If sensors correspond to vertices and edges correspond to sensor pairs between which the distance is known, a significant result in the theory of range-based sensor network localization is that if the graph underlying the sensor network is generically globally rigid and there is a suitable set of anchors at known positions, then the network can be localized, i.e., a unique set of sensor positions can be determined that is consistent with the data. In particular, for planar problems, provided the sensor network has three or more noncollinear anchors at known points, all sensors are located at generic points, and the intersensor distances corresponding to the graph edges are precisely known rather than being subject to measurement noise, generic global rigidity of the graph is necessary and sufficient for the network to be localizable (in the absence of any further information). In practice, however, distance measurements will never be exact, and the equations whose solutions deliver sensor positions in the noiseless case in general no longer have a solution. This paper then argues that if the distance measurement errors are not too great and otherwise the associated graph is generically globally rigid and there are three or more noncollinear anchors, the network will be approximately localizable, in the sense that estimates can be found for the sensor positions which are near the correct values; in particular, a bound on the position errors can be found in terms of a bound on the distance errors. The sensor positions in this case can be found by minimizing a cost function which, although nonconvex, does have a global minimum.

Rigidity, global rigidity, and graph decomposition

The recent combinatorial characterization of generic global rigidity in the plane by Jackson and Jordán (2005) [10] recalls the vital relationship between connectivity and rigidity that was first pointed out by Lovász and Yemini (1982) [13]. The Lovász–Yemini result states that every 6-connected graph is generically rigid in the plane, while the Jackson–Jordán result states that a graph is generically globally rigid in the plane if and only if it is 3-connected and edge-2-rigid. We examine the interplay between the connectivity properties of the connectivity matroid and the rigidity matroid of a graph and derive a number of structure theorems in this setting, some well known, some new. As a by-product we show that the class of generic rigidity matroids is not closed under 2-sum decomposition. Finally we define the configuration index of the graph and show how the structure theorems can be used to compute it.

Global Rigidity: The Effect of Coning

Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al. in Characterizing generic global rigidity, arXiv:0710.0907v1, 2007) running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity. There is a technique which transfers rigidity to one dimension higher: coning. Here we confirm that the cone on a graph is generically globally rigid in ℝd+1 if and only if the graph is generically globally rigid in ℝd . As a corollary, we see that a graph is generically globally rigid in the d-dimensional sphere $\mathbb{S}^{d}$if and only if it is generically globally rigid in ℝd .