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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The non-solvability by radicals of generic 3-connected planar Laman graphs


Authors: J. C. Owen and S. C. Power
Journal: Trans. Amer. Math. Soc. 359 (2007), 2269-2303
MSC (2000): Primary 68U07, 12F10, 05C40; Secondary 52C25
Published electronically: October 16, 2006
MathSciNet review: 2276620
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Abstract: We show that planar embeddable $ 3$-connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count $ 2v - 3 = e$ together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let $ G$ be a maximally independent $ 3$-connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.


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Additional Information

J. C. Owen
Affiliation: D-Cubed Ltd., Park House, Cambridge CB3 0DU, United Kingdom
Email: john.owen@d-cubed.co.uk

S. C. Power
Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, United Kingdom
Email: s.power@lancaster.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-06-04049-9
PII: S 0002-9947(06)04049-9
Keywords: Maximally independent graph, 3-connected, algorithms for CAD, solvable by radicals.
Received by editor(s): October 16, 2003
Received by editor(s) in revised form: March 15, 2005
Published electronically: October 16, 2006
Article copyright: © Copyright 2006 American Mathematical Society