12 papers
On rigidity properties of unit-distance graphs
Sean Dewar, Georg Grasegger, Alison La Porta +2
A unit-distance graph is a graph which admits a realisation in Euclidean space in which every edge has unit length. Imposing further geometric conditions on the non-edges gives a f…
Rigidity on compact surfaces through hyperbolic symmetries
Sean Dewar, Alison La Porta, Rebecca Monks +3
Generically the rigidity of bar-joint structures admits combinatorial characterisations in the Euclidean plane and, more generally, for frameworks on the sphere and the torus. The…
The -dimensional realisation number of a rigid graph
Sean Dewar, Anthony Nixon, Ben Smith
Determining the number of (complex) realisations of a rigid graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we provide tw…
Generalised Erdős distance theory on graphs
Sean Dewar, Nora Frankl, Samuel Mansfield +3
The famous Erdős distinct distances problem asks the following: how many distinct distances must exist between a set of points in the plane? There are many generalisations of t…
A tropical approach to rigidity: counting realisations of frameworks
Oliver Clarke, Sean Dewar, Daniel Green Tripp +4
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 f…
Stable cuts, NAC-colourings and flexible realisations of graphs
Katie Clinch, Dániel Garamvölgyi, John Haslegrave +3
A (2-dimensional) realisation of a graph is a pair , where maps the vertices of to . A realisation is flexible if it can be continuously deformed w…
The -fold circuit property for matroids
Bill Jackson, Anthony Nixon, Ben Smith
Double circuits were introduced by Lovász in 1980 as a fundamental tool in his derivation of a min-max formula for the size of a maximum matching in linear matroids. This formula w…
Triangulated spheres with holes in triangulated surfaces
Katie Clinch, Sean Dewar, Niloufar Fuladi +6
Let denote a sphere with holes. Given a triangulation of a surface , we consider the question of when contains a spanning subgraph such t…
Single-cell 3D genome reconstruction in the haploid setting using rigidity theory
Sean Dewar, Georg Grasegger, Kaie Kubjas +2
This article considers the problem of 3-dimensional genome reconstruction for single-cell data, and the uniqueness of such reconstructions in the setting of haploid organisms. We c…
Angular constraints on planar frameworks
Sean Dewar, Georg Grasegger, Anthony Nixon +4
Consider a collection of points in the plane and the sets of slopes or directions of the lines between pairs of points. It is known that the algebraic matroid on the set of directi…
Rigidity of nearly planar classes of graphs
Sean Dewar, Georg Grasegger, Eleftherios Kastis +2
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 vari…
Rigid frameworks with dilation constraints
Sean Dewar, Anthony Nixon, Andrew Sainsbury
We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean -space under additional dilation constraints in specified coordinate directions. In this settin…