Related papers: Rigidity of Graphs and Frameworks: A Matroid Theor…
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…
A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a…
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…
We define the notion of affine rigidity of a hypergraph and prove a variety of fundamental results for this notion. First, we show that affine rigidity can be determined by the rank of a specific matrix which implies that affine rigidity is…
A picture P of a graph G = (V,E) consists of a point P(v) for each vertex v in V and a line P(e) for each edge e in E, all lying in the projective plane over a field k and subject to containment conditions corresponding to incidence in G. A…
We define periodic frameworks as graphs on the torus, using the language of gain graphs. We present some fundamental definitions and results about the infinitesimal rigidity of graphs on a torus of fixed size and shape, and find necessary…
We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only…
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.…
A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain…
For a finite point set $E\subset \mathbb{R}^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k + 1$ points in E such that the distance between a pair of points is specified if the…
The combinatorial characterization of generic rigidity for bar-joint frameworks in dimensions $d \ge 3$ has been a long-standing open problem in discrete geometry. While the two-dimensional case was resolved in 1927 by Pollaczek-Geiringer…
We show that any graph that is generically globally rigid in $\mathbb{R}^d$ has a realization in $\mathbb{R}^d$ that is both generic and universally rigid. This also implies that the graph also must have a realization in $\mathbb{R}^d$ that…
We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean $d$-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity.…
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 $xy\in E$ in $(G,p)$ is the distance between $p(x)$ and $p(y)$. A vertex pair $\{u,v\}$ of $G$ is…
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…
A bar framework determined by a finite graph $G$ and configuration $\bf p$ in $d$ space is universally rigid if it is rigid in any ${\mathbb R}^D \supset {\mathbb R}^d$. We provide a characterization of universally rigidity for any graph…
Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic graphs under fixed lattice representations. A periodic graph is vertex-redundantly rigid…
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…
Let (G,P) be a bar framework of n vertices in general position in R^d, d <= n-1, where G is a (d+1)-lateration graph. In this paper, we present a constructive proof that (G,P) admits a positive semi-definite stress matrix with rank n-d-1.…
A graph is said to be rigid if, given a generic realisation of the graph as a bar-and-joint framework in the plane, there exist only finitely many other realisations of the graph with the same edge lengths modulo rotations, reflections and…