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Graph embeddings in the plane, space and sphere - source code and results

Bartzos, Evangelos; Legerský, Jan


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{
  "inLanguage": {
    "alternateName": "eng", 
    "@type": "Language", 
    "name": "English"
  }, 
  "description": "<p>This is supplementary material for the paper&nbsp;<em>On the Maximal Number of Real Embeddings of&nbsp;Minimally Rigid Graphs in&nbsp;<span class=\"math-tex\">\\(\\mathbb{R}^2\\)</span>,&nbsp;<span class=\"math-tex\">\\(\\mathbb{R}^3\\)</span>&nbsp;and&nbsp;<span class=\"math-tex\">\\(S^2\\)</span></em>. It includes source code and results. See also&nbsp;the&nbsp;<a href=\"http://jan.legersky.cz/project/spatialgraphembeddings/\">project website</a>. The material extends the results of our <a href=\"https://doi.org/10.1145/3208976.3208994\">conference paper</a> (ISSAC 2018).</p>\n\n<p><strong>Construction of Laman and Geiringer graphs</strong></p>\n\n<p>MATLAB and SageMath code for construction of Laman and&nbsp;Geiringer graphs using Henneberg steps.</p>\n\n<p><strong>Computation of mixed volume and complex solutions</strong></p>\n\n<p>Functions that&nbsp;compute the mixed volume and the (complex) embeddings in the plane, space or sphere of a given graph using different algebraic formulations.</p>\n\n<p><strong>Coupler curve visualization and sampling method</strong></p>\n\n<p>Implementation of&nbsp;our&nbsp;method for obtaining edge lengths of a Geiringer&nbsp;graph with many real spatial embeddings.<br>\nVisualization of coupler curves of the 7-vertex Geiringer graph with the maximal number of&nbsp;real embeddings (G48).</p>\n\n<p><strong>Maple parametric search</strong></p>\n\n<p>Maple worksheet illustrating a method to improve the number of real embeddings of G48 using RootFinding[Parametric] subpackage.</p>\n\n<p><strong>Results</strong></p>\n\n<p>Mixed volumes and numbers of real and complex embeddings in the plane, space and sphere.<br>\nEdge lengths giving high&nbsp;numbers of real embeddings, in the plane, space and sphere.&nbsp;<br>\nMaple script verifying the results.</p>", 
  "license": "https://opensource.org/licenses/GPL-3.0", 
  "creator": [
    {
      "affiliation": "Department of Informatics and Telecommunications, National Kapodistrian University of Athens, and ATHENA Research Center", 
      "@type": "Person", 
      "name": "Bartzos, Evangelos"
    }, 
    {
      "affiliation": "Research Institute for Symbolic Computation, Johannes Kepler University Linz, Austria", 
      "@id": "https://orcid.org/0000-0002-8122-668X", 
      "@type": "Person", 
      "name": "Legersk\u00fd, Jan"
    }
  ], 
  "url": "https://zenodo.org/record/1495153", 
  "datePublished": "2018-11-24", 
  "version": "special_issue", 
  "keywords": [
    "rigid graph", 
    "real embedding", 
    "coupler curve", 
    "mixed volume"
  ], 
  "@context": "https://schema.org/", 
  "identifier": "https://doi.org/10.5281/zenodo.1495153", 
  "@id": "https://doi.org/10.5281/zenodo.1495153", 
  "@type": "SoftwareSourceCode", 
  "name": "Graph embeddings in the plane, space and sphere - source code and results"
}
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