arXiv subjects

Jan Legerský

Publications and source records attributed to Jan Legerský.

At least 19 recordsLinked to original sources

arXiv2026

Learning Minimally Rigid Graphs with High Realization Counts

For minimally rigid graphs, the same edge-length data can admit multiple realizations (up to translations and rotations). Finding graphs with exceptionally many realizations is an extremal problem in rigidity theory, but exhaustive search quickly becomes infeasible due to the super-exponential growth of the number of candidate graphs and the high cost of realization-count evaluation. We propose a reinforcement-learning approach that constructs minimally rigid graphs via 0- and 1-extensions, also known as Henneberg moves. We optimize realization-count invariants using the Deep Cross-Entropy Method with a policy parameterized by a Graph Isomorphism Network encoder and a permutation-equivariant extension-level action head. Empirically, our method matches the known optima for planar realization counts and improves the best known bounds for spherical realization counts, yielding new record graphs.

cs.LG↗

arXiv2026

Stable cuts, NAC-colourings and flexible realisations of graphs

A (2-dimensional) realisation of a graph $G$ is a pair $(G,p)$, where $p$ maps the vertices of $G$ to $\mathbb{R}^2$. A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. We say that $G$ is rigid if every generic realisation of $G$ is rigid; otherwise, $G$ is flexible. In this paper, we investigate the relationship between stable cuts and graphs which are either flexible, or admit a flexible (not necessarily generic) realisation with positive edge lengths. We strengthen a result of Chen and Yu, who proved that every $n$-vertex graph with at most $2n-4$ edges has a stable cut, by showing that every flexible graph has a stable cut. The existence of a stable cut is a sufficient, but not necessary, condition for a flexible realisation to exist. Using a result of Le and Pfender on stable cuts, we prove a conjecture of Grasegger, Legerský and Schicho that characterises the minimally rigid graphs which admit a flexible realisation. Additionally, we investigate the number of NAC-colourings in various graphs. A NAC-colouring is a type of edge colouring introduced by Grasegger, Legerský and Schicho, who showed that the existence of such a colouring characterises the existence of a flexible realisation with positive edge lengths. We provide an upper bound on the number of NAC-colourings for arbitrary graphs, and construct families of graphs, including rigid and minimally rigid ones, for which this number is exponential in the number of vertices.

math.CO↗

arXiv2025

Flexible realizations existence: NP-completeness on sparse graphs and algorithms

One of the questions in Rigidity Theory is whether a realization of the vertices of a graph in the plane is flexible, namely, if it allows a continuous deformation preserving the edge lengths. A flexible realization of a connected graph in the plane exists if and only if the graph has a NAC-coloring, which is a surjective edge coloring by two colors such that for each cycle, either all the edges have the same color, or there are at least two edges of each color. The question whether a graph has a NAC-coloring, and hence also the existence of a flexible realization, has been proven to be NP-complete. We show that this question is also NP-complete on graphs with maximum degree five and on graphs with the average degree at most $4+\varepsilon$ for every fixed $\varepsilon >0$. We also show that NAC-colorings can be counted in linear time for graphs with bounded treewidth. Since the only existing implementation of checking the existence of a NAC-coloring is rather naive, we propose new algorithms along with their implementation, which is significantly faster. We also focus on searching all NAC-colorings of a graph, since they provide useful information about its possible flexible realizations.

cs.CG↗

arXiv2025

PyRigi -- a general-purpose Python package for the rigidity and flexibility of bar-and-joint frameworks

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 admits only finitely many ways, up to isometries, of being drawn in the plane once the edge lengths are fixed, whether it has a unique embedding, or whether it satisfied such properties even after the removal of any of its edges. By implementing algorithms from the scientific literature, PyRigi enables the exploration of rigidity properties of structures that would be out of reach for computations by hand. With reliable and robust algorithms, as well as clear, well-documented methods that are closely connected to the underlying mathematical definitions and results, PyRigi aims to be a practical and powerful general-purpose tool for the working mathematician interested in rigidity theory. PyRigi is open source and easy to use, and awaits researchers to benefit from its computational potential.

math.MG↗

arXiv2024

Pentagonal bipyramids lead to the smallest flexible embedded polyhedron

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 with one face subdivided into three is the only polyhedron with fewer vertices for which the existence of a self-intersection-free flex was open. Since subdividing a face into three does not change the mobility, we focus on flexible pentagonal bipyramids. When a bipyramid flexes, the distance between the two opposite vertices of the two pyramids changes; associating the position of the bipyramid to this distance yields an algebraic map that determines a nontrivial extension of rational function fields. We classify flexible pentagonal bipyramids with respect to the Galois group of this field extension and provide examples for each class, building on a construction proposed by Nelson. Surprisingly, one of our constructions yields a flexible pentagonal bipyramid that can be extended to an embedded flexible polyhedron with 8 vertices. The latter hence solves the open question.

math.MG↗

arXiv2024

Constructing reflection-symmetric flexible realisations of graphs

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 necessary combinatorial condition on graphs with reflection-symmetric flexible realisations. This condition is based on a specific type of edge colouring, where edges are assigned one of three colours in a symmetric way. From some of these colourings we also construct concrete reflection-symmetric realisations with their corresponding symmetry preserving motion. We study also a specific class of reflection-symmetric realisations consisting of triangles and parallelograms.

math.CO↗

arXiv2023

Flexibility and rigidity of frameworks consisting of triangles and parallelograms

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 lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal flexibility, the existence of at least two classes of an equivalence relation based on 3- and 4-cycles and being a non-trivial subgraph of the Cartesian product of graphs. We study the algorithmic aspects and the rotationally symmetric version of the problem. The results are illustrated on frameworks obtained from tessellations by regular polygons.

math.CO↗

arXiv2022

Flexing infinite frameworks with applications to braced Penrose tilings

A planar framework -- a graph together with a map of its vertices to the plane -- is flexible if it allows a continuous deformation preserving the distances between adjacent vertices. Extending a recent previous result, we prove that a connected graph with a countable vertex set can be realized as a flexible framework if and only if it has a so-called NAC-coloring. The tools developed to prove this result are then applied to frameworks where every 4-cycle is a parallelogram, and countably infinite graphs with $n$-fold rotational symmetry. With this, we determine a simple combinatorial characterization that determines whether the 1-skeleton of a Penrose rhombus tiling with a given set of braced rhombi will have a flexible motion, and also whether the motion will preserve 5-fold rotational symmetry.

math.CO↗

arXiv2022

On the existence of paradoxical motions of generically rigid graphs on the sphere

We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our interpretation of realizations allows us to provide a combinatorial characterization of these graphs in terms of the existence of particular colorings of the edges. Moreover, we determine necessary relations for flexibility between the spherical lengths of the edges. We conclude by classifying all possible motions on the sphere of the complete bipartite graph with $3+3$ vertices where no two vertices coincide or are antipodal.

math.CO↗

arXiv2022

Zero-sum cycles in flexible polyhedra

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 there is a cycle of edges whose lengths sum up to zero once suitably weighted by 1 and -1. We do this via elementary combinatorial considerations, made possible by a well-known compactification of the three-dimensional affine space as a quadric in the four-dimensional projective space. The compactification is related to the Euclidean metric, and allows us to use a simple degeneration technique that reduces the problem to its one-dimensional analogue, which is trivial to solve.

math.MG↗

arXiv2021

Zero-sum cycles in flexible non-triangular polyhedra

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 faces, we showed in a previous work the existence of cycles with a sign assignment for their edges, such that the signed sum of the edge lengths along the cycle is zero. In this work, we extend this result to flexible non-triangular polyhedra.

math.MG↗

arXiv2020

Bracing frameworks consisting of parallelograms

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 combinatorially which choices of braces make a grid of squares infinitesimally rigid using a bracing graph: a bipartite graph whose vertices are the columns and rows of the grid, and a row and column are adjacent if and only if they meet at a braced square. Duarte and Francis generalized the notion of the bracing graph to rhombic carpets, proved that the connectivity of the bracing graph implies rigidity and stated the other implication without proof. Nagy Kem gives the equivalence in the infinitesimal setting. We consider continuous deformations of braced frameworks consisting of a graph from a more general class and its placement in the plane such that every 4-cycle forms a parallelogram. We show that rigidity of such a braced framework is equivalent to the non-existence of a special edge coloring, which is in turn equivalent to the corresponding bracing graph being connected.

math.CO↗

arXiv2020

Combinatorics of Bricard's octahedra

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. The explanations of these rules rely on the use of a well-known creation of modern algebraic geometry, the moduli space of stable rational curves with marked points, for the description of configurations of graphs on the sphere. Once one accepts the objects and the rules, the classification becomes elementary (though not trivial) and can be enjoyed without the need of a very deep background on the topic.

math.MG↗

arXiv2020

FlexRiLoG -- A SageMath Package for Motions of Graphs

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 special edge colorings. The package computes and illustrates the colorings and the motions. We present the structure and usage of the package.

cs.MS↗

arXiv2020

On the Classification of Motions of Paradoxically Movable Graphs

Edge lengths of a graph are called flexible if there exist infinitely many non-congruent realizations of the graph in the plane satisfying these edge lengths. It has been shown recently that a graph has flexible edge lengths if and only if the graph has a special type of edge coloring called NAC-coloring. We address the question how to determine all possible proper flexible edge lengths from the set of all NAC-colorings of a graph. We do so using restrictions to 4-cycle subgraphs.

math.CO↗

arXiv2020

Flexible placements of graphs with rotational symmetry

We study the existence of an $n$-fold rotationally symmetric placement of a symmetric graph in the plane allowing a continuous deformation that preserves the symmetry and the distances between adjacent vertices. We show that such a flexible placement exists if and only if the graph has a NAC-colouring satisfying an additional property on the symmetry; a NAC-colouring is a surjective edge colouring by two colours such that every cycle is either monochromatic, or there are at least two edges of each colour.

math.CO↗

arXiv2019

Graphs with Flexible Labelings allowing Injective Realizations

We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinatorial condition on existence of such a labeling based also on NAC-colorings of the graph. By introducing new tools for the construction of such labelings, we show that the necessary condition is also sufficient up to 8 vertices, but this is not true in general for more vertices.

math.CO↗

arXiv2018

On the maximal number of real embeddings of minimally rigid graphs in $\mathbb{R}^2$, $\mathbb{R}^3$ and $S^2$

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 is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings. In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in $\mathbb{R}^3$. One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration. Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in $\mathbb{R}^2$ and $\mathbb{R}^3$, while in the case of $S^2$ we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 (Bartzos et al, 2018).

math.AG↗