The Nonsolvability by Radicals of Generic 3-connected Planar Graphs

Abstract

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.

Continuous curves from infinite Kempe linkages

“…, |V |} is algebraically independent over Q. (For Laman graphs standard algebraic geometry can be used to show that the notions essentially coincide; see [17].) For infinite frameworks vertex genericity is a less definitive term because it is possible for infinite graphs to have two vertex generic frameworks, one of which is rigid while the other is not.…”

Characterizing Graphs with Convex and Connected Cayley Configuration Spaces

“…To the best of our knowledge, the only results of a similar flavor are: the result of [31] that shows the equivalence of Tree-or Triangle-decomposability [30] and Quadratic or Radical realizability for planar graphs; and the result characterizing the sampling complexity of 1-dof Henneberg-1 graphs [13].…”

Radically solvable graphs

“…cease to be rigid when any edge is removed. A conjectured characterisation of quadratically/radically solvable minimally rigid generic frameworks was given in [9] and this conjecture was verified for the special case when the underlying graph is 3-connected and planar in [10].…”

“…This property has been extensively studied and we refer the reader to [15] for an excellent survey of the area. Previous work on quadratic/radical solvability [9,10] considered generic frameworks which are minimally rigid i.e. cease to be rigid when any edge is removed.…”

“…We develop a reduction scheme in Section 7 which shows how the radical or quadratic solvability of a rigid graph is related to the corresponding property for a derived graph which may be chosen to be minimally rigid. We use this and the main result of [10] to show in Theorem 7.5 that a rigid 3-connected planar graph is radically solvable if and only if it is globally rigid. This leads us to consider rigid graphs which are not 3-connected i.e.…”