Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity theory by replacing G with an arbitrary representable matroid M. The ideas of rigidity independence and parallel independence, as well as Laman's and Recski's combinatorial characterizations of 2-dimensional rigidity for graphs, can naturally be extended to this winder setting. As we explain, many of these fundamental concepts really depend only on the matroid associated with G (or its Tutte polynomial) and have little to do with the special nature of graphic matroids or the field R.
Our main result is a "nesting theorem" relating the various kinds of independence. Immediate corollaries include generalizations of Laman's Theorem, as well as the equality of 2-rigidity and 2-parallel independence. A key tool in our study is the space of photos of M, a natural algebraic variety whose irreducibility is closely related to the notions of rigidity independence and parallel independence.
The number of points on this variety, when working over a finite field, turns out to be an interesting Tutte polynomial evaluation.
We define and study slider-pinning rigidity, giving a complete combinatorial characterization. This is done via direction-slider networks, which are a generalization of Whiteley's direction networks.
The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks. In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces. Using this new matrix, we derive necessary Maxwell-Laman-type counting conditions for a well-positioned bar-joint framework in a real normed vector space to be infinitesimally rigid. Moreover, we derive symmetry-extended counting conditions for a bar-joint framework with a non-trivial symmetry group to be isostatic (…
We give Henneberg-type constructions for three families of sparse colored graphs arising in the rigidity theory of periodic and other forced symmetric frameworks. The proof method, which works with Laman-sparse finite covers of colored graphs highlights the connection between these sparse colored families and the well-studied matroidal (k, l)-sparse families.
A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in d-dimensional Euclidean space is generalised to the non-Euclidean l^p norms and to countably infinite graphs. For all dimensions and norms it is shown t…
Laman Graphs are Generically Bearing Rigid in Arbitrary Dimensions
University of Sheffield（谢菲尔德大学）；Australian National University（澳大利亚国立大学）；Technion–Israel Institute of Technology（以色列理工学院）；Gwangju Institute of Science and Technology（光州科学技术院）
In this note we prove a lower bound for the rank of 2-dimensional generic rigidity matroid for regular graphs of degree four and five. Also, we give examples to show the order of the bound we give is sharp.
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…
A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in $\bR^2$. A more general theory is developed for frameworks in $\bR^3$ whose vertices are constrained to move on a two-dimensional smooth submanifold $\M$. Furthermore, when $\M$ is a union of concentric spheres, or a union of parallel planes or a union of concentric cylinders, necessary and sufficient combinatorial conditions are obtained for the minimal rigidity of generic frameworks.
We showed in the first paper of this series that the generic $C_2^1$-cofactor matroid is the unique maximal abstract $3$-rigidity matroid. In this paper we obtain a combinatorial characterization of independence in this matroid. This solves the cofactor counterpart of the combinatorial characterization problem for the rigidity of generic 3-dimensional bar-joint frameworks. We use our characterization to verify that the counterparts of conjectures of Dress (on the rank function) and Lovász and Yemini (which suggested a sufficient connectivity condition for rigidity) hold for this matroid.
The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a {\it maximum matroid for a graph} $G$ is introduced, and the existence and uniqueness of the maximum matroid for any graph $G$ is proved. The maximum matroid for $K_3$ is shown to be the cycle (or graphic) matroid. This result is pursued in two directions - to determine the maximum matroid for the $m$-cycle $C_m$ and to determine the maximum matroid for the complete graph $K_m$. The maximum matroid for $K_4$ is the matroid whose bases are the Laman graphs, related to structural rigidity o…
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture.
We prove that the linear matroid that defines generic rigidity of $d$-dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linked by bars) can be obtained from the union of ${d+1 \choose 2}$ graphic matroids by applying variants of Dilworth truncation $n_r$ times, where $n_r$ denotes the number of rods. This leads to an alternative proof of Tay's combinatorial characterizations of generic rigidity of rod-bar frameworks and that of identified body-hinge frameworks.
We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the self-stress; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects…
Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, …
A foundational theorem of Laman provides a counting characterisation of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterisation was obtained for frameworks in three dimensions whose vertices are constrained to concentric spheres or to concentric cylinders. Noting that the plane and the sphere have 3 independent locally tangential infinitesimal motions while the cylinder has 2, we obtain here a Laman-Henneberg theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions. Such…
A bar-joint framework $(G,p)$ in $\mathbb{R}^d$ is rigid if the only edge-length preserving continuous motions of the vertices arise from isometries of $\mathbb{R}^d$. It is known that, when $(G,p)$ is generic, its rigidity depends only on the underlying graph $G$, and is determined by the rank of the edge set of $G$ in the generic $d$-dimensional rigidity matroid $\mathcal{R}_d$. Complete combinatorial descriptions of the rank function of this matroid are known when $d=1,2$, and imply that all circuits in $\mathcal{R}_d$ are generically rigid in $\mathbb{R}^d$ when $d=1,2$. Determining the ra…
A bar-and-joint framework is a finite set of points together with specified distances between selected pairs. In rigidity theory we seek to understand when the remaining pairwise distances are also fixed. If there exists a pair of points which move relative to one another while maintaining the given distance constraints, the framework is flexible; otherwise, it is rigid. Counting conditions due to Maxwell give a necessary combinatorial criterion for generic minimal bar-and-joint rigidity in all dimensions. Laman showed that these conditions are also sufficient for frameworks in R^2. However, t…
We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorithms. To prove our rigidity theorem we introduce and develop periodic direction networks and Z2-graded-sparse colored graphs.
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…
A conjecture of Graver from 1991 states that the generic $3$-dimensional rigidity matroid is the unique maximal abstract $3$-rigidity matroid with respect to the weak order on matroids. Based on a close similarity between the generic $d$-dimensional rigidity matroid and the generic $C_{d-2}^{d-1}$-cofactor matroid from approximation theory, Whiteley made an analogous conjecture in 1996 that the generic $C_{d-2}^{d-1}$-cofactor matroid is the unique maximal abstract $d$-rigidity matroid for all $d\geq 2$. We verify the case $d=3$ of Whiteley's conjecture in this paper. A key step in our proof i…
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph $G=(V,E)$, a multiset $R=\{r1,..., r_t\}$ of vertices in $V$, and a matroid ${\cal M}$ on $R$. We prove a necessary and sufficient condition for $G$ to be decomposed into $t$ edge-disjoint subgraphs $G_1=(V_1,T_1),..., G_t=(V_t,T_t)$ such that (i) for each $i$, $G_i$ is a tree with $r_i\in V_i$, and (ii) for each $v\in V$, the multiset $\{r_i\in R\mid v\in V_i\}$ is a base of ${\cal M}$. I…
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…
We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial matroids and associated linear representation results that may be interesting in their own right. The same techniques immediately yield a Maxwell-Laman-type combinatorial characterization for frameworks embedded in 2-dimensional cones that arise as quotients of the plane by a finite order rotation.
We develop a rigidity theory for bar-joint frameworks in Euclidean $d$-space in which specified classes of edges are allowed to change length in a coordinated fashion that requires differences of lengths to be preserved within each class. Rigidity for these coordinated frameworks is a generic property, and we characterize the rigid graphs in terms of redundant rigidity in the standard $d$-dimensional rigidity matroid. We also interpret our main results in terms of matroid unions.
Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from $K_4$ graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses cla…
We show that, for points along the moment curve, the bar-and-joint rigidity matroid and the hyperconnectivity matroid coincide, and that both coincide with the $C^{d-2}_{d-1}$-cofactor rigidity of points along any (non-degenerate) conic in the plane. For hyperconnectivity in dimension two, having the points in the moment curve is no loss of generality. We also show that, restricted to bipartite graphs, the bar-and-joint rigidity matroid is freer than the hyperconnectivity matroid.
A two-dimensional direction-length framework $(G,p)$ consists of a multigraph $G=(V;D,L)$ whose edge set is formed of "direction" edges $D$ and "length" edges $L$, and a realisation $p$ of this graph in the plane. The edges of the framework represent geometric constraints: length edges fix the distance between their endvertices, whereas direction edges specify the gradient of the line through both endvertices. A direction-length framework $(G,p)$ is globally rigid if every framework $(G,q)$ which satisfies the same direction and length constraints as $(G,p)$ can be obtained by translating $(G,…
We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of $d$-dimensional normed spaces (including all $\ell^p$ spaces with $p\not=2$). Complete combinatorial characterisations are obtained for half-turn rotation in the $\ell^1$ and $\ell^\infty$-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of $(2,2,0)$-gain-tight graphs.
We show that a generic framework $(G,p)$ on the cylinder is globally rigid if and only if $G$ is a complete graph on at most four vertices or $G$ is both redundantly rigid and $2$-connected. To prove the theorem we also derive a new recursive construction of circuits in the simple $(2,2)$-sparse matroid, and a characterisation of rigidity for generic frameworks on the cylinder when a single designated vertex is allowed to move off the cylinder.
Rigidity and reconstruction in matroids of highly connected graphs
HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization and the MTA-ELTE Momentum Matroid Optimization Research Group（匈牙利研究与教育网络-厄特沃什·罗兰大学埃格瓦里组合优化研究组和匈牙利科学院-厄特沃什·罗兰大学动量拟阵优化研究组）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究与教育网络阿尔弗雷德·雷尼数学研究所）
Sufficient conditions for bipartite rigidity, symmetric completability and hyperconnectivity of graphs
HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization（匈牙利研究与创新网络-罗兰大学埃格瓦里组合优化研究组）；Queen Mary University of London（伦敦玛丽女王大学）；Department of Operations Research, ELTE Eötvös Loránd University（罗兰大学运营系）
A G-gain graph is a graph whose oriented edges are labeled invertibly from a group G. Zaslavsky proposed two matroids of G-gain graphs, called frame matroids and lift matroids, and investigated linear representations of them. Each matroid has a canonical representation over a field F if G is isomorphic to a subgroup of F^{\times} in the case of frame matroids or G is isomorphic to an additive subgroup of F in the case of lift matroids. The canonical representation of the frame matroid of a complete graph is also known as a Dowling geometry, as it was first introduced by Dowling for finite grou…
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.
Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically pinned (completely immobilized) bar-and-joint frameworks and prove that they also form matroids. We address five problems on graded sparse graphs: {\bf Decision}, {\bf Extraction}, {\bf Components}, {\bf Optimization}, and {\bf Extension}. We extend our {\bf pebble game algorithms} to solve them.
Rank Contributions of Vertices in Rigidity Matroids of Clique Covered Graphs
Queen Mary University of London（伦敦大学玛丽女王学院）；HUN-REN-ELTE Egerváry Research Group on Combinatorial Optimization（匈牙利研究与教育网络-罗兰大学组合优化Egerváry研究组）；Department of Operations Research, ELTE Eötvös Loránd University（罗兰大学运筹学系）
In recent years, various notions of algebraic independence have emerged as a central and unifying theme in a number of areas of applied mathematics, including algebraic statistics and the rigidity theory of bar-and-joint frameworks. In each of these settings the fundamental problem is to determine the extent to which certain unknowns depend algebraically on given data. This has, in turn, led to a resurgence of interest in algebraic matroids, which are the combinatorial formalism for algebraic (in)dependence. We give a self-contained introduction to algebraic matroids together with examples hig…
This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a) algebraic matroids, we expose cryptomorphisms making them accessible to techniques from commutative algebra. This allows us to introduce for each circuit in an algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly- nomial in classical Galois theory, and studying the matroid structure with multivariate methods. For (b) matroid…
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 the Euclidean plane. 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 ver…
