Related papers: Characterizing Generic Global Rigidity
A bar-joint framework $(G,p)$ in the Euclidean space $\mathbb{E}^d$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ in $\mathbb{E}^d$ with the edge lengths of $(G,p)$. Building on key results of…
We prove that if $G$ is the graph of a connected triangulated $(d-1)$-manifold, for $d\geq 3$, then $G$ is generically globally rigid in $\mathbb R^d$ if and only if it is $(d+1)$-connected and, if $d=3$, $G$ is not planar. The special case…
A graph $G = (V,E)$ is globally rigid in $\mathbb{R}^d$ if for any generic placement $p : V \rightarrow \mathbb{R}^d$ of the vertices, the edge lengths $||p(u) - p(v)||, uv \in E$ uniquely determine $p$, up to congruence. In this paper we…
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…
We examine the generic local and global rigidity of various graphs in R^d. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity and hypothesized that they…
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…
A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge…
A bar-joint framework $(G,p)$ in Euclidean $d$-space is rigid if the only edge-length-preserving continuous motions arise from isometries of $\mathbb{R}^d$. In the generic case, rigidity is determined by the generic $d$-dimensional rigidity…
Following a review of related results in rigidity theory, we provide a construction to obtain generically universally rigid frameworks with the minimum number of edges, for any given set of n nodes in two or three dimensions. When a set of…
A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known…
A $d$-dimensional (bar-and-joint) framework $(G,p)$ consists of a graph $G=(V,E)$ and a realisation $p:V\to \mathbb{R}^d$. It is rigid if every continuous motion of the vertices which preserves the lengths of the edges is induced by an…
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…
A tensegrity is a structure made from cables, struts and stiff bars. A $d$-dimensional tensegirty is universally rigid if it is rigid in any dimension $d'$ with $d'\geq d$. The celebrated super stability condition due to Connelly gives a…
We explore the rigidity of generic frameworks in 3-dimensions whose underlying graph is close to being planar. Specifically we consider apex graphs, edge-apex graphs and their variants and prove independence results in the generic…
A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive.…
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…
A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius…
A $d$-dimensional tensegrity framework $(T,p)$ is an edge-labeled geometric graph in ${\mathbb R}^d$, which consists of a graph $T=(V,B\cup C\cup S)$ and a map $p:V\to {\mathbb R}^d$. The labels determine whether an edge $uv$ of $T$…
The celebrated result of Gortler-Healy-Thurston (independently, Jackson-Jord\'an for $d=2$) shows that the global rigidity of graphs realised in the $d$-dimensional Euclidean space is a generic property. Extending this result to the global…
We consider a generalization of the concept of $d$-flattenability of graphs - introduced for the $l_2$ norm by Belk and Connelly - to general $l_p$ norms, with integer $P$, $1 \le p < \infty$, though many of our results work for $l_\infty$…