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Algebraic methods for counting Euclidean embeddings of rigid graphs [article]

Ioannis Z. Emiris, Elias P. Tsigaridas, Antonios Varvitsiotis
2009 arXiv   pre-print
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]

Ioannis Z. Emiris, Elias P. Tsigaridas, Antonios E. Varvitsiotis
2010 Lecture Notes in Computer Science  
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]

Ioannis Z. Emiris, Ioannis Psarros
2017 arXiv   pre-print
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]

Evangelos Bartzos, Ioannis Z. Emiris, Charalambos Tzamos
2021 Lecture Notes in Computer Science  
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

Evangelos Bartzos, Ioannis Z. Emiris, Jan Legerský, Elias Tsigaridas
2019 Journal of symbolic computation  
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]

Evangelos Bartzos and Ioannis Z. Emiris and Raimundas Vidunas
2021 arXiv   pre-print
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]

Evangelos Bartzos and Ioannis Z. Emiris and Josef Schicho
2020 arXiv   pre-print
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]

Evangelos Bartzos, Ioannis Z. Emiris, Jan Legerský, Elias Tsigaridas
2018 arXiv   pre-print
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

Robert Connelly, Steven J. Gortler, Louis Theran
2019 Proceedings of the Royal Society A  
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]

Robert Connelly, Steven J. Gortler, Louis Theran
2019 arXiv   pre-print
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

Steven J. Gortler, Dylan P. Thurston
2014 Discrete & Computational Geometry  
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]

Ioannis Z. Emiris
2017 arXiv   pre-print
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

Georg Grasegger, Christoph Koutschan, Elias Tsigaridas
2018 Experimental Mathematics  
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]

Leo Liberti, Carlile Lavor, Nelson Maculan, Antonio Mucherino
2012 arXiv   pre-print
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

J. Gao, L. Guibas
2011 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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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