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Algebraic methods for counting Euclidean embeddings of rigid graphs
[article]
2009
arXiv
pre-print
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A major open problem concerns the number of embeddings of such graphs, up to rigid motions, in Euclidean space. ...
The study of (minimally) rigid graphs is motivated by numerous applications, mostly in robotics and bioinformatics. ...
Walter for insightful discussions on the number of embeddings of the K 3,3 . ...
arXiv:0906.1437v2
fatcat:n5p43kegove7domhjmfmeikrgq
Algebraic Methods for Counting Euclidean Embeddings of Rigid Graphs
[chapter]
2010
Lecture Notes in Computer Science
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In this paper, we introduce a novel approach to generate an interactive layout of such a metabolic network taking its hierarchical structure into account and present methods for navigation and exploration ...
Our approach supports bundled edge routes heuristically minimizing a given cost function based on the number of bends, the number of edge crossings and the density of edges within a bundle. ...
For large graphs, the brute-force method testing all grid positions naturally takes longer compared to the simulated annealing method. ...
doi:10.1007/978-3-642-11805-0_19
fatcat:cdtf6cwv3vdmna24yln6dtzfoq
Counting Euclidean embeddings of rigid graphs
[article]
2017
arXiv
pre-print
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Consequently, counting the number of Euclidean embeddings of a given rigid graph, reduces to the problem of counting roots of the corresponding polynomial system. ...
A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. ...
We can obtain an upper bound of 52 and 48 Euclidean embeddings for the two of these graphs. ...
arXiv:1402.1484v2
fatcat:2ykaup7wmzhyzdpka5bhwsc2h4
The m-Bézout Bound and Distance Geometry
[chapter]
2021
Lecture Notes in Computer Science
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Our work is inspired by the application of the m-Bézout bound to counting Euclidean embeddings of distance graphs. ...
Counting embeddings is an algebraic question once one constructs a system whose solutions correspond to the different embeddings. ...
Rigidity theory studies the properties of graphs that have rigid embeddings in Euclidean space for fixed edge weights that represent length between points. ...
doi:10.1007/978-3-030-85165-1_2
fatcat:usaavbo3qzbcddmuwistt434ie
On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2
2019
Journal of symbolic computation
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Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R d or on a sphere and other manifolds which in addition satisfy certain edge length constraints. ...
this classification for all 6-vertex graphs. ...
An embedding of a simple graph G = (V, E) in a euclidean space R d is a map from the set V to R d . ...
doi:10.1016/j.jsc.2019.10.015
fatcat:uwzuztg3dja2fpapyvqnnqfruq
New upper bounds for the number of embeddings of minimally rigid graphs
[article]
2021
arXiv
pre-print
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By definition, a rigid graph in 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. Naturally, the complex solutions of such systems extend the notion of rigidity to ℂ^d. ...
EB and IZE are members of team AROMATH, joint between INRIA Sophia-Antipolis, France, and NKUA. ...
arXiv:2010.10578v2
fatcat:tebye4r44rekpkjwu3v5eebd5y
On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs
[article]
2020
arXiv
pre-print
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We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. ...
Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d≥ 5, both in the Euclidean and spherical case. ...
We thank Georg Grasegger for providing us with runtimes of the combinatorial algorithm that calculates c d (G) and for explaining to EB the counting of embeddings for triangle-free Geiringer graphs. ...
arXiv:2005.14485v2
fatcat:hfrwv4en3zcatorbb3pcgb35yy
On the maximal number of real embeddings of minimally rigid graphs in R^2, R^3 and S^2
[article]
2018
arXiv
pre-print
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Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space R^d or on a sphere and which in addition satisfy certain edge length constraints. ...
this classification for all 6-vertex graphs. ...
Acknowledgments This work is part of the project ARCADES that has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement ...
arXiv:1811.12800v1
fatcat:ak3ahn6qszhvxczshtxkvgji3i
Rigidity for sticky discs
2019
Proceedings of the Royal Society A
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Our approach is to study the space of packings with a fixed contact graph. ...
We study the combinatorial and rigidity properties of disc packings with generic radii. ...
We thank Meera Sitharam and Wai Yeung Lam for helpful feedback on an earlier version of this paper. The anonymous referees made a number of suggestions that improved the exposition. ...
doi:10.1098/rspa.2018.0773
pmid:30853847
pmcid:PMC6405455
fatcat:bsk2wub44bhcjo2imehth3rdgm
Rigidity for sticky disks
[article]
2019
arXiv
pre-print
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Our approach is to study the space of packings with a fixed contact graph. ...
We study the combinatorial and rigidity properties of disk packings with generic radii. ...
Acknowledgements We thank Meera Sitharam and Wai Yeung Lam for helpful feedback on an earlier version of this paper. The anonymous referees made a number of suggestions that improved the exposition. ...
arXiv:1809.02006v2
fatcat:gxlpifa6d5ellfufeiihk2ywx4
Characterizing the Universal Rigidity of Generic Frameworks
2014
Discrete & Computational Geometry
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A framework is universally rigid if any framework in any dimension with the same graph and edge lengths is a Euclidean image of it. ...
Connelly showed that the existence of such a positive semi-definite stress matrix is sufficient for universal rigidity, so this provides a characterization of universal rigidity for generic frameworks. ...
We do not know a characterization of generically universally rigid graphs. See Fig. 1 for examples of embedded graphs with increasing types of rigidity. ...
doi:10.1007/s00454-014-9590-9
fatcat:6yriunvwzfg7jmwqgr6kqe3siy
The assembly modes of rigid 11-bar linkages
[article]
2017
arXiv
pre-print
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Rigid 11-bar linkages, where n=7, are the simplest planar linkages for which these questions were still open. ...
A related question concerns the number of assembly modes of rigid mechanisms as a function of their nodes n, which is uniquely defined given m. ...
Both authors were inspired to work on this problem at the Workshop on Discrete and Algebraic Geometry in September 2010, at Val d'Ajol, France. ...
arXiv:1010.6214v2
fatcat:6x7cuvagqzbcfoblhapyxfhlpi
Lower Bounds on the Number of Realizations of Rigid Graphs
2018
Experimental Mathematics
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Toward this goal, for graphs that are minimally rigid in the plane, we take advantage of a recently published algorithm, which is the fastest available method, although its complexity is still exponential ...
Combining computational results with the theory of constructing new rigid graphs by gluing, we give a new lower bound on the maximal possible number of (complex) realizations for graphs with a given number ...
We use the state-ofthe-art computer algebra tools to count the number of embeddings as the maximum number of complex solutions of polynomial systems. ...
doi:10.1080/10586458.2018.1437851
pmid:32655833
pmcid:PMC7324120
fatcat:w57dsqachzd75huxjcvjo6budq
Euclidean distance geometry and applications
[article]
2012
arXiv
pre-print
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Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. ...
We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics. ...
John, Therese Malliavin, Benoît Masson, Michael Nilges and Maxim Sviridenko for co-authoring some of the papers we wrote on different facets of this topic. ...
arXiv:1205.0349v1
fatcat:jpr2dxrcj5a6degsanug3bbhcu
Geometric algorithms for sensor networks
2011
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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This paper surveys the use of geometric methods for wireless sensor networks. ...
The close relationship of sensor nodes with their embedded physical space imposes a unique geometric character on such systems. ...
A graph is globally rigid if it admits a unique embedding in the plane, subject to global rotations and translations. The theory of graph rigidity in two dimensions is relatively well understood. ...
doi:10.1098/rsta.2011.0215
pmid:22124080
fatcat:2m3bwwdjnramzcrtkjip6om3t4
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