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.

References

It is open whether the problem remains -complete for embedded graphs.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “Graph Realizability” entry