Partial drawing extensibility (PDE) complexity

Establish whether the partial drawing extensibility problem—deciding if a given partial straight-line drawing of a graph can be extended to a full straight-line planar drawing—is ER-complete.

Background

PDE asks whether a partial placement of some vertices (and possibly edges) can be extended to a full straight-line planar drawing. It is NP-hard, and ER-hardness is known under a polygonal domain constraint.

The authors highlight PDE as a central unresolved problem in geometric graph drawing regarding ER-completeness.

References

A tantalizing and fundamental open question in this area is the complexity of the partial drawing extensibility problem~\ourref[Graph(s)!Partial Drawing Extensibility (Open)]{p:partialdrawingext}: Is it -complete to test whether a partial straight-line embedding of a graph can be extended to a straight-line embedding of the full grap? The problem is only known to be -hard if the realization has to lie within a polygon~\ourref[Graph(s)!Graph in a Polygonal Domain]{p:GraphinPD}.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Section 'Graph Drawing' (overview)