ER-Completeness of Graph Realizability for Embedded Graphs

Determine whether the graph realizability problem (straight-line embedding with prescribed edge lengths) remains ER-complete when the drawing is required to be crossing-free (embedded).

Background

Graph realizability with unit edge lengths is ER-complete in general but allows degenerate overlaps; requiring a crossing-free embedding is a stricter variant.

The survey explicitly notes this embedded variant’s complexity is unresolved.