Lower Bounds on the Number of Realizations of Rigid Graphs

Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. 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 exponen...

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Bibliographic Details
Published in:	Experimental mathematics Vol. 29; no. 2; pp. 125 - 136
Main Authors:	Grasegger, Georg, Koutschan, Christoph, Tsigaridas, Elias
Format:	Journal Article
Language:	English
Published:	Taylor & Francis 01-06-2020
Subjects:	graph realization Laman graph minimally rigid graph Original
Online Access:	Get full text
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Summary:	Computing the number of realizations of a minimally rigid graph is a notoriously difficult problem. 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 of vertices. We extend these ideas to rigid graphs in three dimensions and we derive similar lower bounds, by exploiting data from extensive Gröbner basis computations.
Bibliography:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/uexm.
ISSN:	1058-6458 1944-950X
DOI:	10.1080/10586458.2018.1437851