28 citations · 110 across the 30 of their papers we have counts for
33 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…
The minimum degree question for the Maker Breaker Domination Game
Jakob Führer, Georg Grasegger, Paul Hametner +1
The Maker Breaker Domination Game is a two player game played on a graph in which the players take turns to claim a vertex from the graph. The aim of the Dominator is to claim…
Counting fibres of the Hadamard product using Bergman fans
Oliver Clarke, Sean Dewar, Matteo Gallet +3
We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associa…
Computing the number of realisations of a rigid graph
Sean Dewar, Georg Grasegger, Josef Schicho +2
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 w…
PyRigi -- a general-purpose Python package for the rigidity and flexibility of bar-and-joint frameworks
Matteo Gallet, Georg Grasegger, Matthias Himmelmann +1
We present PyRigi, a novel Python package designed to study the rigidity properties of graphs and frameworks. Among many other capabilities, PyRigi can determine whether a graph ad…
Explorations on the number of realizations of minimally rigid graphs
Georg Grasegger
Rigid graphs have only finitely many realizations. In the recent years significant progress was made in computing the number of such realizations. With this progress it was also po…
Pentagonal bipyramids lead to the smallest flexible embedded polyhedron
Matteo Gallet, Georg Grasegger, Jan Legerský +1
Steffen's polyhedron was believed to have the least number of vertices among polyhedra that can flex without self-intersections. Maksimov clarified that the pentagonal bipyramid wi…
Constructing reflection-symmetric flexible realisations of graphs
Sean Dewar, Georg Grasegger, Jan Legerský
We study reflection-symmetric realisations of symmetric graphs in the plane that allow a continuous symmetry and edge-length preserving deformation. To do so, we identify a necessa…
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…
The number of realisations of a rigid graph in Euclidean and spherical geometries
Sean Dewar, Georg Grasegger
A graph is -rigid if for any generic realisation of the graph in (equivalently, the -dimensional sphere ), there are only finitely many non-congr…
Calligraphs and sphere realizations
Matteo Gallet, Georg Grasegger, Niels Lubbes +1
We introduce a recursive procedure for computing the number of realizations of a minimally rigid graph on the sphere up to rotations. We accomplish this by combining two ingredient…
On the uniqueness of collections of pennies and marbles
Sean Dewar, Georg Grasegger, Kaie Kubjas +2
In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit -spheres that may touch but not overlap. Given…
Flexibility and rigidity of frameworks consisting of triangles and parallelograms
Georg Grasegger, Jan Legerský
A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge l…
Minimal counterexamples to Hendrickson's conjecture on globally rigid graphs
Georg Grasegger
In this paper we consider the class of graphs which are redundantly -rigid and -connected but not globally -rigid, where is the dimension. This class arises from c…
Graph rigidity properties of Ramanujan graphs
Sebastian M. Cioabă, Sean Dewar, Georg Grasegger +1
A recent result of Cioabă, Dewar and Gu implies that any -regular Ramanujan graph with is globally rigid in . In this paper, we extend these results and…
Coupler curves of moving graphs and counting realizations of rigid graphs
Georg Grasegger, Boulos El Hilany, Niels Lubbes
A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. T…
Quotient graphs of symmetrically rigid frameworks
Sean Dewar, Georg Grasegger, Eleftherios Kastis +1
A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in that admit some non-trivial s…
Zero-sum cycles in flexible non-triangular polyhedra
Matteo Gallet, Georg Grasegger, Jan Legerský +1
Finding necessary conditions for the geometry of flexible polyhedra is a classical problem that has received attention also in recent times. For flexible polyhedra with triangular…
Zero-sum cycles in flexible polyhedra
Matteo Gallet, Georg Grasegger, Jan Legerský +1
We show that if a polyhedron in the three-dimensional affine space with triangular faces is flexible, i.e., can be continuously deformed preserving the shape of its faces, then the…
Bracing frameworks consisting of parallelograms
Georg Grasegger, Jan Legerský
A rectangle in the plane can be continuously deformed preserving its edge lengths, but adding a diagonal brace prevents such a deformation. Bolker and Crapo characterized combinato…
Combinatorics of Bricard's octahedra
Matteo Gallet, Georg Grasegger, Jan Legerský +1
We re-prove the classification of flexible octahedra, obtained by Bricard at the beginning of the XX century, by means of combinatorial objects satisfying some elementary rules. Th…
FlexRiLoG -- A SageMath Package for Motions of Graphs
Georg Grasegger, Jan Legerský
In this paper we present the SageMath package FlexRiLoG (short for flexible and rigid labelings of graphs). Based on recent results the software generates motions of graphs using s…