The nonsolvability by radicals of generic 3connected planar Laman graphs
Authors:
J. C. Owen and S. C. Power
Journal:
Trans. Amer. Math. Soc. 359 (2007), 22692303
MSC (2000):
Primary 68U07, 12F10, 05C40; Secondary 52C25
Published electronically:
October 16, 2006
MathSciNet review:
2276620
Fulltext PDF Free Access
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Additional Information
Abstract: We show that planar embeddable connected Laman graphs are generically nonsoluble. 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 vertexedge count 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 be a maximally independent 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:
DCubed Ltd., Park House, Cambridge CB3 0DU, United Kingdom
Email:
john.owen@dcubed.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/S0002994706040499
PII:
S 00029947(06)040499
Keywords:
Maximally independent graph,
3connected,
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
