2003/11/04 by J. C. Owen, John C. Owen, S. C. Power +3
Computer Science · Engineering · Mathematics · #05C40 (primary) #12F10 #13P99 (secondary) #52C25 #68U07 #Advanced Materials and Mechanics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Computational Geometry and Mesh Generation #FOS: Mathematics #Structural Analysis and Optimization #math.AC #math.CO #msc:05C40 #msc:12F10 #msc:13P99 #msc:52C25 #msc:68U07
paper · pdf · doi:10.48550/arxiv.math/0311037
45 pages, 11 figures
arxiv created 2003/11/04 · openalex publication_date 2003/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD 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 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 dimensions for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these dimensions 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 dimension field.