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SGE-nomap for pairs of planar graphs

Establish whether, for every positive integer n, every pair of n-vertex planar graphs admits a simultaneous geometric embedding without mapping (SGE-nomap) on some point set of size n.

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Background

Simultaneous geometric embedding without mapping asks for a single point set supporting straight-line planar embeddings of multiple graphs without a fixed vertex mapping. While negative results are known for larger collections of graphs (currently up to 30), the two-graph case remains unresolved.

Resolving the two-graph case would settle a central question in simultaneous planar embeddings and influence related problems in matched and partial embeddings.

References

The following well known and still wildly open problem was asked by \citet{DBLP:journals/comgeo/BrassCDEEIKLM07} in 2003, and is also listed among selected list of graph drawing problems in (see also Problem 12): Does every pair of $n$-vertex planar graphs has SGE-nomap (for every positive integer $n$)?

— Free Sets in Planar Graphs: History and Applications (2403.17090 - Dujmović et al., 25 Mar 2024) in Section Applications, Subsection Simultaneous Geometric Embeddings – Without Mapping