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

Bartzos, Evangelos; Legerský, Jan


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    <subfield code="u">Department of Informatics and Telecommunications, National Kapodistrian University of Athens, and ATHENA Research Center</subfield>
    <subfield code="a">Bartzos, Evangelos</subfield>
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    <subfield code="a">&lt;p&gt;This is supplementary material for the paper&amp;nbsp;&lt;em&gt;On the Maximal Number of Real Embeddings of&amp;nbsp;Minimally Rigid Graphs in&amp;nbsp;&lt;span class="math-tex"&gt;\(\mathbb{R}^2\)&lt;/span&gt;,&amp;nbsp;&lt;span class="math-tex"&gt;\(\mathbb{R}^3\)&lt;/span&gt;&amp;nbsp;and&amp;nbsp;&lt;span class="math-tex"&gt;\(S^2\)&lt;/span&gt;&lt;/em&gt;. It includes source code and results. See also&amp;nbsp;the&amp;nbsp;&lt;a href="http://jan.legersky.cz/project/spatialgraphembeddings/"&gt;project website&lt;/a&gt;. The material extends the results of our &lt;a href="https://doi.org/10.1145/3208976.3208994"&gt;conference paper&lt;/a&gt; (ISSAC 2018).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Construction of Laman and Geiringer graphs&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;MATLAB and SageMath code for construction of Laman and&amp;nbsp;Geiringer graphs using Henneberg steps.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Computation of mixed volume and complex solutions&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Functions that&amp;nbsp;compute the mixed volume and the (complex) embeddings in the plane, space or sphere of a given graph using different algebraic formulations.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Coupler curve visualization and sampling method&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Implementation of&amp;nbsp;our&amp;nbsp;method for obtaining edge lengths of a Geiringer&amp;nbsp;graph with many real spatial embeddings.&lt;br&gt;
Visualization of coupler curves of the 7-vertex Geiringer graph with the maximal number of&amp;nbsp;real embeddings (G48).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Maple parametric search&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Maple worksheet illustrating a method to improve the number of real embeddings of G48 using RootFinding[Parametric] subpackage.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Results&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Mixed volumes and numbers of real and complex embeddings in the plane, space and sphere.&lt;br&gt;
Edge lengths giving high&amp;nbsp;numbers of real embeddings, in the plane, space and sphere.&amp;nbsp;&lt;br&gt;
Maple script verifying the results.&lt;/p&gt;</subfield>
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