Globally Rigid Research Articles

Uniquely Realisable Graphs in Analytic Normed Planes

Abstract A framework $(G,p)$ in Euclidean space $\mathbb{E}^{d}$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in $\mathbb{E}^{2}$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $\mathbb{R}^{2} \setminus \{0\}$. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $e\in E$. We also prove that the analogous necessary conditions hold in $d$-dimensional normed spaces.

Four-regular graphs with extremal rigidity properties

A graph G=(V,E) is called k-edge rigid (k-edge globally rigid, resp.), if it stays rigid (globally rigid, resp.) after the deletion of at most k−1 edges. We can define k-vertex rigidity and k-vertex global rigidity in a similar manner. It is known that if G is 3-edge rigid (2-edge globally rigid, 2-vertex globally rigid) with |V|≥5 then |E|≥2|V| holds. Furthermore, the graphs that satisfy the edge count with equality are all 4-regular.In this paper we show that for a 4-regular graph G the properties of 3-edge rigidity, 2-edge global rigidity, and essential 6-edge connectivity are equivalent. By sharpening a result of H. Fleischner, F. Genest, and B. Jackson we give a new inductive construction for the family of 4-regular and essentially 6-edge connected graphs (and hence also for the 4-regular graphs with these rigidity properties). We prove that G is 2-vertex globally rigid if and only if it is 4-vertex connected and essentially 6-edge connected.We also consider 2-vertex rigid graphs G=(V,E) with minimum size |E|=2|V|−1 as well as with |E|=2|V|. In the former case we use our results on essentially 6-edge connected graphs to develop a new inductive construction, complementing an earlier, different construction of B. Servatius. In the latter case we characterize the edge pairs of G whose deletion preserves rigidity, and use this result to verify the correctness of a construction of 3-vertex rigid graphs on |V|≥6 vertices and with |E|=2|V|+2 edges, proposed by S.A. Motevallian, C. Yu, and B.D.O. Anderson.

Coincident-point rigidity in normed planes

A bar-joint framework (G,p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

Redundantly globally rigid braced triangulations

By mapping the vertices of a graph G to points in ℝ3, and its edges to the corresponding line segments, we obtain a three-dimensional realization of G. A realization of G is said to be globally rigid if its edge lengths uniquely determine the realization, up to congruence. The graph G is called globally rigid if every generic three-dimensional realization of G is globally rigid. We consider global rigidity properties of braced triangulations, which are graphs obtained from maximal planar graphs by adding extra edges, called bracing edges. We show that for every even integer n ≥ 8 there exist braced triangulations with 3n − 4 edges which remain globally rigid if an arbitrary edge is deleted from the graph. The bound is best possible. This result gives an affirmative answer to a recent conjecture. We also discuss the connections between our results and a related more general conjecture, due to S. Tanigawa and the third author.

Global rigidity of 2-dimensional direction-length frameworks with connected rigidity matroids

A 2-dimensional direction-length framework ( G , p ) consists of a multigraph G = ( V ; D , L ) with realisation p : V → R 2 . The edges of G represent geometric constraints: edges in L have fixed length, and edges in D have fixed gradient. A direction-length framework ( G , p ) is globally rigid if every ( G , q ) which satisfies the same geometric constraints as ( G , p ) can be obtained from ( G , p ) by an isometry of the plane. We characterise global rigidity for the class of 2-dimensional direction-length frameworks ( G , p ) for which p is generic and the rigidity matroid of G is connected. Specifically, we show that such frameworks are globally rigid if and only if both D and L are non-empty, and every 2-separation of G is direction-balanced. This extends previous work by Jackson and Jordán (2010).

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 \({\mathbb {R}}^d\).

Minimally globally rigid graphs

A graph G=(V,E) is globally rigid in Rd if for any generic placement p:V→Rd of the vertices, the edge lengths p(u)−p(v),uv∈E uniquely determine p, up to congruence. In this paper we consider minimally globally rigid graphs, in which the deletion of an arbitrary edge destroys global rigidity. We prove that if G=(V,E) is minimally globally rigid in Rd on at least d+2 vertices, then |E|≤(d+1)|V|−d+22. This implies that the minimum degree of G is at most 2d+1. We also show that the only graph in which the upper bound on the number of edges is attained is the complete graph Kd+2. It follows that every minimally globally rigid graph in Rd on at least d+3 vertices is flexible in Rd+1.As a counterpart to our main result on the sparsity of minimally globally rigid graphs, we show that in two dimensions, dense graphs always contain nontrivial globally rigid subgraphs. More precisely, if some graph G=(V,E) satisfies |E|≥5|V|, then G contains a subgraph on at least seven vertices that is globally rigid in R2. If the well-known “sufficient connectivity conjecture” is true, then our methods also extend to higher dimensions.Finally, we discuss a conjectured strengthening of our main result, which states that if a pair of vertices {u,v} is linked in G in Rd+1, then {u,v} is globally linked in G in Rd. We prove this conjecture in the d=1,2 cases, along with a variety of related results.

Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

A bar-joint framework (G,p) in a (non-Euclidean) real normed plane X is the combination of a finite, simple graph G and a placement p of the vertices in X. A framework (G,p) is globally rigid in X if every other framework (G,q) in X with the same edge lengths as (G,p) arises from an isometry of X. The weaker property of local rigidity in normed planes (where only (G,q) within a neighbourhood of (G,p) are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for ℓp-norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in X, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in X.

Globally rigid powers of graphs

The characterization of rigid graphs in Rd for d≥3 is a major open problem in rigidity theory. The same holds for globally rigid graphs. In this paper our goal is to give necessary and/or sufficient conditions for the (global) rigidity of the square G2 (and more generally, the power Gk) of a graph G in Rd, for some values of k,d. Our work is motivated by some results and conjectures of M. Cheung and W. Whiteley from 2008, the Molecular Theorem of N. Katoh and S. Tanigawa from 2011, which settled the case of rigidity for k=2,d=3, and the potential applications in molecular conformation and sensor network localization.We first consider the case when k=d and characterize those graphs G for which Gd is globally rigid in Rd, for all d≥1, and then focus on the case when k=d−1. We provide a new, direct proof for the 3-dimensional bar-and-joint version of the Molecular Theorem (d=3) and a necessary condition for the rigidity of Gd−1 in Rd, for all d≥3. We conjecture that this condition is also sufficient.The global rigidity of square graphs in R3 is still an open problem. We formulate a Molecular Global Rigidity Conjecture, which proposes a combinatorial characterization of globally rigid square graphs in terms of vertex partitions and edge count conditions. We prove that the condition is necessary. For the general case we give a best possible connectivity based sufficient condition by showing that if G is 3-edge-connected then Gd−1 is globally rigid in Rd, for all d≥3.Our results imply affirmative answers to the conjectures of M. Cheung and W. Whiteley in two special cases.

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension d>2. We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in {mathbb {R}}^d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in {mathbb {R}}^{2d} to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

Global rigidity of (quasi-)injective frameworks on the line

A realization of a graph G is a pair (G,p) where p maps the vertices of G into Euclidean space Rd. The realization is injective if p is injective and quasi-injective if for each edge of G, p maps the endpoints of the edge to different points in space. The realization is globally rigid if any realization (G,q) in Rd with the same edge lengths is congruent to (G,p).In this paper we characterize graphs that have an injective (quasi-injective, respectively) non-globally rigid realization in R1, and we show that the problem of recognizing these graphs is NP-complete in both the injective and the quasi-injective cases. Our characterizations are based on the notion of NAC-colorings, which have been used previously to investigate similar problems in the plane. We also give an overview of related results and open problems in rigidity theory.

Spectral conditions for graph rigidity in the Euclidean plane

Rigidity is the property of a structure that does not flex. It is well studied in discrete geometry and mechanics, and has applications in material science, engineering and biological sciences. A bar-and-joint framework is a pair (G,p) of graph G together with a map p of the vertices of G into a Euclidean space. We view the edges of (G,p) as bars and the vertices as universal joints. The vertices can move continuously as long as the distances between pairs of adjacent vertices are preserved. The framework is rigid if any such motion preserves the distances between all pairs of vertices. In 1970, Laman obtained a combinatorial characterization of rigid graphs in the Euclidean plane. In 1982, Lovász and Yemini discovered a new characterization and proved that every 6-connected graph is rigid. Combined with a combinatorial characterization of global rigidity given by Jackson and Jordán in 2009, it is actually proved that every 6-connected graph is globally rigid. Consequently, if the algebraic connectivity of a graph is greater than 5, then it is globally rigid. In this paper, we improve this bound and show that for a graph G with minimum degree δ≥6, if its algebraic connectivity is greater than 2+1δ−1, then G is rigid and if its algebraic connectivity is greater than 2+2δ−1, then G is globally rigid. Our results imply that every connected regular Ramanujan graph with degree at least 8 is globally rigid. We also prove a more general result giving a sufficient spectral condition for the existence of k edge-disjoint spanning rigid subgraphs. The same condition implies that a graph contains k edge-disjoint spanning 2-connected subgraphs. This result extends previous spectral conditions for packing edge-disjoint spanning trees.

On the global rigidity of tensegrity graphs

A tensegrity graph is a graph with edges labeled as bars, cables and struts. A realization of a tensegrity graph T is a pair (T,p), where p maps the vertices of T into Rd for some d≥1. The realization is globally rigid if any realization (T,q) in Rd in which the bars have the same length and the cables and struts are not longer and not shorter, respectively, is an isometric image of (T,p). A tensegrity graph is weakly globally rigid in Rd if it has a generic globally rigid realization in Rd, and strongly globally rigid in Rd if every generic realization in Rd is globally rigid.In this paper we give a necessary condition for weak global rigidity in Rd and prove that in the d=1 case the same condition is also sufficient. In particular, our results imply that a tensegrity graph has a generic globally rigid realization in R1 if and only if it has a generic universally rigid realization in R1. We also show that recognizing strongly globally rigid tensegrity graphs in Rd is co-NP-hard for all d≥1.

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.

Global Rigidity of 2D Linearly Constrained Frameworks

AbstractA linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints that fixes the distances between some pairs of points and additionally restricts some of the points to lie in given affine subspaces. It is globally rigid if the configuration is uniquely defined by the constraint system. We show that a generic linearly constrained framework in $\mathbb{R}^2$ is globally rigid if and only if it is redundantly rigid and “balanced”. For unbalanced generic frameworks, we determine the precise number of solutions to the constraint system whenever the rigidity matroid of the framework is connected. We obtain a stress matrix sufficient condition and a Hendrickson type necessary condition for a generic linearly constrained framework to be globally rigid in $\mathbb{R}^d$.

Global rigidity of periodic graphs under fixed-lattice representations

In [9] Hendrickson proved that (d+1)-connectivity and redundant rigidity are necessary conditions for a generic (non-complete) bar-joint framework to be globally rigid in Rd. Jackson and Jordán [10] confirmed that these conditions are also sufficient in R2, giving a combinatorial characterization of graphs whose generic realizations in R2 are globally rigid. In this paper, we establish analogues of these results for infinite periodic frameworks under fixed lattice representations. Our combinatorial characterization of globally rigid generic periodic frameworks in R2 in particular implies toroidal and cylindrical counterparts of the theorem by Jackson and Jordán.

Global rigidity of direction-length frameworks

A 2-dimensional direction-length framework is a collection of points in the plane which are linked by pairwise constraints that fix the direction or length of the line segments joining certain pairs of points. We represent it as a pair (G,p), where G=(V;D,L) is a ‘mixed’ graph and p:V→R2 is a point configuration for V. It is globally rigid if every direction-length framework (G,q) which satisfies the same constraints can be obtained from (G,p) by a translation or a rotation by 180°. We characterise the mixed graphs G with the property that every generic framework (G,p) is globally 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.

Global Rigidity of Unit Ball Graphs

A $d$-dimensional bar-and-joint framework $(G,p)$, where $G$ is a graph and $p$ maps the vertices of $G$ to points in $\mathbb{R}^d$, is said to be globally rigid if every $d$-dimensional framework...

GROLO: Realistic Range-Based Localization for Mobile IoTs Through Global Rigidity

We study the realistic problem of range-based localization for mobile Internet of Things (IoT) without using GPS. This problem arises from the real-world applications and is characterized by the following three challenges: 1) inaccurate devices, such as the low cost motion sensors and moving parts in the mobile IoT nodes, make it infeasible to localize the nodes using their speed and direction; 2) when a team of IoT nodes keep moving, some nodes may get lost and be disconnected from the network due to the large accumulated errors in the distances; and 3) although the theory of global rigidity ensures the position uniqueness of nodes, there are still nonlocalizable nodes in a globally rigid graph. To address these challenges, we propose a distributed localization protocol, called GROLO. GROLO is able to perform efficient distributed localization through an adaptive global rigidity formation maintenance mechanism especially designed for the resource limited IoT nodes. GROLO is able to localize all of the nodes periodically in a mobile IoT by short distance adjustment. Furthermore, GROLO requires only necessary additional neighbor distance measurements. We evaluated GROLO and other localization protocols using both simulated nodes and ten real mobile nodes in an IoT. The results show that GROLO is promising for the localization and formation control with inaccurate mobile IoT in realistic environments.