Radically solvable graphs

TL;DR

This paper characterizes which planar graphs produce radically solvable generic frameworks in the Euclidean plane, linking graph structure to algebraic solvability of vertex coordinates.

Contribution

It provides a graph-theoretic characterization of radically solvable frameworks and conjectures this extends beyond planar graphs.

Findings

Radical solvability depends only on the underlying graph.

Characterization of planar graphs with radically solvable frameworks.

Conjecture that the characterization applies to all graphs.

Abstract

A 2-dimensional framework is a straight line realisation of a graph in the Euclidean plane. It is radically solvable if the set of vertex coordinates is contained in a radical extension of the field of rationals extended by the squared edge lengths. We show that the radical solvability of a generic framework depends only on its underlying graph and characterise which planar graphs give rise to radically solvable generic frameworks. We conjecture that our characterisation extends to all graphs.