Rigid Graph Research Articles

Generic Symmetry-Forced Infinitesimal Rigidity: Translations and Rotations

We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the underlying symmetry group consists of rotations and translations. Along the way, we use tropical geometry to show how a construction of Edmonds and Rota that associates a matroid to a submodular function can be used to give a description of the algebraic matroid underlying a Hadamard product of two linear spaces in terms of the matroids underlying each linear space. This leads to new, short, proofs of Laman's theorem, and a theorem of Jord{a}n, Kaszanitzky, and Tanigawa characterizing the minimally generically symmetry-forced rigid graphs in the plane when the symmetry group contains only rotations.

The Rigidity of Infinite Graphs II

The Rigidity of Infinite Graphs II

New Upper Bounds for the Number of Embeddings of Minimally Rigid Graphs

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 embeddings in $$\mathbb {C}^d$$ and $$\mathbb {R}^d$$ coincide. For decades, only the trivial bound of $$O(2^{d|V|})$$ was known on the number of embeddings. Recently, matrix permanent bounds have led to a small improvement for $$d\ge 5$$ . This work improves upon the existing upper bounds for the number of embeddings in $$\mathbb {R}^d$$ and $$S^d$$ , by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively eliminates vertices or vertex paths. For the most important cases of $$d=2$$ and $$d=3$$ , the new bounds are $$O(3.7764^{|V|})$$ and $$O(6.8399^{|V|})$$ , respectively. In general, we improve the exponent basis in the asymptotic behavior with respect to the number of vertices of the recent bound mentioned above by the factor of $$\sqrt{2}$$ . Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic root counts.

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.

Rainbow and monochromatic circuits and cocircuits in binary matroids

Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if an n-element rank r binary matroid M is colored with exactly r colors, then M either contains a rainbow colored circuit or a monochromatic cocircuit. As the class of binary matroids is closed under taking duals, this immediately implies that if M is colored with exactly n−r colors, then M either contains a rainbow colored cocircuit or a monochromatic circuit. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids.Motivated by a conjecture of Bérczi, Schwarcz and Yamaguchi, we also analyze the relation between the covering number of a binary matroid and the maximum number of colors or the maximum size of a color class in any of its rainbow circuit-free colorings. For simple graphic matroids, we show that there exists a rainbow circuit-free coloring that uses each color at most twice only if the graph is (2,3)-sparse, that is, it is independent in the 2-dimensional rigidity matroid. Furthermore, we give a complete characterization of minimally rigid graphs admitting such a coloring.

Neural Network-Based Formation Control With Target Tracking for Second-Order Nonlinear Multiagent Systems

This article proposes a distance-based formation control and target tracking for multiagent systems, where agents are modeled using second-order nonlinear systems in the presence of disturbance. By applying a rigid graph theory, we developed a neural network (NN)-based backstepping controller to address the distance-based formation control problem of nonlinear multiagent systems. To compensate for the unknown nonlinearity in the system dynamics, the radial basis function NN was used where the NN tuning law was derived based on Lyapunov stability theory. We rigorously proved the uniform ultimate boundedness of the formation distance error and NN weights’ norm estimation error. Finally, using simulation results, we demonstrated the proposed method’s performance on the second-order, nonlinear multiagent systems. To provide further evaluation, we compared the proposed distance-based method and existing displacement-based methods.

Aligning Spatially Constrained Graphs

We focus on the problem of aligning graphs that have a spatial basis. In such graphs, which we refer to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rigid graphs</i> , nodes have preferred positions relative to their graph neighbors. Rigid graphs can be used to abstract objects in diverse applications such as large biomolecules, where edges corresponding to chemical bonds have preferred lengths, functional connectomes of the human brain, where edges corresponding to co-firing regions of the brain have preferred anatomical distances, and mobile device/ sensor communication logs, where edges corresponding to point-to-point communications across devices have distance constraints. Effective analysis of such graphs must account for edge lengths in addition to topological features. For instance, when identifying conserved patterns through graph alignment, it is important for matched edges to have correlated lengths, in addition to topological similarity. In this paper, we formulate the problem of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rigid graph alignment</i> and present a method for solving it. Our formulation of rigid graph alignment simultaneously aligns the topology of the input graphs, as well as the geometric structure represented by the edge lengths, which is solved using a block coordinate descent technique. Using detailed experiments on real and synthetic datasets, we demonstrate a number of important desirable features of our method: (i) it significantly outperforms topological and structural aligners on a wide range of problems; (ii) it scales to problems in important real-world applications; and (iii) it has excellent stability properties, in view of noise and missing data in typical applications.

Globally rigid graphs are fully reconstructible

AbstractAd-dimensional framework is a pair$(G,p)$, where$G=(V,E)$is a graph andpis a map fromVto$\mathbb {R}^d$. The length of an edge$uv\in E$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 the graphGand its edge lengths uniquely determine$(G,p)$, up to congruence. A graphGis called globally rigid in$\mathbb {R}^d$if everyd-dimensional generic framework$(G,p)$is globally rigid.In this paper, we consider the problem of reconstructing a graph from the set of edge lengths arising from a generic framework. Roughly speaking, a graphGis strongly reconstructible in$\mathbb {C}^d$if the set of (unlabeled) edge lengths of any generic framework$(G,p)$ind-space, along with the number of vertices ofG, uniquely determine bothGand the association between the edges ofGand the set of edge lengths. It is known that ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then it is strongly reconstructible in$\mathbb {C}^d$. We strengthen this result and show that, under the same conditions,Gis in fact fully reconstructible in$\mathbb {C}^d$, which means that the set of edge lengths alone is sufficient to uniquely reconstructG, without any constraint on the number of vertices (although still under the assumption that the edge lengths come from a generic realization).As a key step in our proof, we also prove that ifGis globally rigid in$\mathbb {R}^d$on at least$d+2$vertices, then thed-dimensional generic rigidity matroid ofGis connected. Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.

Realizations of Rigid Graphs

A minimally rigid graph, also called Laman graph, models a planar framework which is rigid for a general choice of distances between its vertices. In other words, there are finitely many ways, up to isometries, to realize such a graph in the plane. Using ideas from algebraic and tropical geometry, we derive a recursive formula for the number of such realizations. Combining computational results with the construction of new rigid graphs via gluing techniques, we can give a new lower bound on the maximal possible number of realizations for graphs with a given number of vertices.

Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

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.

Computational Design of Planar Multistable Compliant Structures

This article presents a method for designing planar multistable compliant structures. Given a sequence of desired stable states and the corresponding poses of the structure, we identify the topology and geometric realization of a mechanism—consisting of bars and joints—that is able to physically reproduce the desired multistable behavior. In order to solve this problem efficiently, we build on insights from minimally rigid graph theory to identify simple but effective topologies for the mechanism. We then optimize its geometric parameters, such as joint positions and bar lengths, to obtain correct transitions between the given poses. Simultaneously, we ensure adequate stability of each pose based on an effective approximate error metric related to the elastic energy Hessian of the bars in the mechanism. As demonstrated by our results, we obtain functional multistable mechanisms of manageable complexity that can be fabricated using 3D printing. Further, we evaluated the effectiveness of our method on a large number of examples in the simulation and fabricated several physical prototypes.

On Spectra and Spectral Measures of Schreier and Cayley Graphs

Abstract We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the rooted $d$-regular tree, which provide examples of the following interesting phenomena. For $d=2$ and any $m\geq 2$, we get an uncountable family of non-quasi-isometric Cayley graphs with the same Laplacian spectrum, a union of two intervals, which we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another. For each $d\geq 3, m\geq 1$, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct, moreover, a complete system of eigenfunctions that are strongly localized.

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.

Minimum Size Highly Redundantly Rigid Graphs in the Plane

A graph G is said to be k-vertex rigid in mathbb {R}^d if G-X is rigid in mathbb {R}^d for all subsets X of the vertex set of G with cardinality less than k. We determine the smallest number of edges in a k-vertex rigid graph on n vertices in mathbb {R}^2, for all kge 4. We also consider k-edge-rigid graphs, defined by removing edges, as well as k-vertex globally rigid and k-edge globally rigid graphs in mathbb {R}^d. For d=2 we determine the corresponding tight bounds for each of these versions, for all kge 3. Our results complete the solutions of these extremal problems in the plane. The result on k-vertex rigidity verifies a conjecture of Kaszanitzky and Király (Graphs Combin, 32:225–240, 2016). We also determine the degree of vertex redundancy of powers of cycles, with respect to rigidity in the plane, answering a question of Yu and Anderson (Int J Robust Nonlinear Control, 19(13):1427–1446, 2009).

Distance‐based formation control for multi‐lane autonomous vehicle platoons

This paper investigates the formation control of connected autonomous vehicle (CAV) platoons moving in multi lanes using distance-based formation control techniques based on rigid graphs and V2V communication. A hierarchical architecture is proposed to decompose the cooperative control into velocity planning and vehicle dynamic control. A new velocity planning method is first developed via a distributed distance-based formation controller so that each vehicle can keep platoon and change lane. Then, for the vehicle dynamics with nonlinearities and bounded disturbances, an adaptive controller is designed for regulating driving/braking torque to achieve the longitudinal velocity output of the velocity planner. The steering controller is designed to adjust the yaw angle of each vehicle to track and change lanes. Furthermore, stability analysis is conducted based on the Lyapunov theory. Finally, the applications of the proposed control designs to various of automated highway system (AHS) scenarios including lane-change, curve lane and platoon overtaking, are simulated and numerically analysed to validate the effectiveness of theoretical results.

A note on generic rigidity of graphs in higher dimension

The characterization of rigid graphs in Rd is known only in the low dimensional cases (d=1,2) and is a major open problem in higher dimensions. In this note we consider the other extreme case when d is close to n, the number of vertices of the graph. It turns out that there is a fairly simple characterization as long as n−d is at most four. We also characterize globally rigid graphs in this range.

Ubiquitous Tracking for Autonomous Underwater Vehicle With IoUT: A Rigid-Graph-Based Solution

Tracking an autonomous underwater vehicle (AUV) has been regarded as one of the most key applications for Internet of Underwater Things (IoUT). However, the strong mobility of AUV as well as asynchronous clock, stratification effect, and high energy consumption of acoustic communication make it challenging to achieve such a task. To handle the above issues, this article develops a ubiquitous tracking scheme for AUV. The tracking scheme is divided into two stages, i.e.: 1) motion prediction and 2) persistent tracking. In the first stage, an unscented transform-based localization estimator is utilized by sensor nodes to acquire the initial position of AUV, through which a terminal sliding-mode velocity observer is designed to predict the mobility trajectory of AUV. With the predicted mobility trajectory, a minimum rigid-graph-based tracking strategy is developed in the second stage to enable ubiquitously tracking. For the designed tracking strategy, the posterior Cramer-Rao lower bound is selected as the benchmark to optimize the network topology, such that a minimum rigid graph can be generated to balance the tradeoff between tracking accuracy and energy consumption. Particularly, the duty-cycle mechanism and the unscented Kalman filtering are jointly adopted to prolong the network lifetime and improve the tracking accuracy. Finally, simulation and experimental results are presented to show the effectiveness of our approach.

Rigid realizations of graphs with few locations in the plane

Adiprasito and Nevo (2018) proved that there exists a set of 76 points in R3 such that every triangulated planar graph has an infinitesimally rigid realization in which each vertex is mapped to a point in this set.In this paper we show that there exists a set of 26 points in the plane such that every planar graph which is generically rigid in R2 has an infinitesimally rigid realization in which each vertex is mapped to a point in this set.It is known that a similar result, with a set of constant size, does not hold for the family of all generically rigid graphs in Rd, d≥2. We show that there exists a constant c such that for every positive integer n there is a set of c(n) points in the plane such that every generically rigid graph in R2 on n vertices has an infinitesimally rigid realization on this set. This bound is tight up to a constant factor.