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Characterize planar graphs whose edges can all be touched or crossed by a convex curve

Determine which planar graphs admit planar straight-line drawings in which a given convex curve touches all edges or crosses all edges, and provide a precise characterization of this class.

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Background

Beyond placing vertices or crossing faces, the paper raises the edge-centric question: for which planar graphs can a convex curve touch or cross all edges in a planar straight-line drawing? The authors provide examples demonstrating that not all planar graphs have such drawings, including the octahedral graph K_{2,2,2} (not all edges can be crossed) and a 30-vertex graph constructed from nine octahedra (not all edges can be touched).

Despite these examples, a full characterization remains unresolved; the authors explicitly state they do not settle the question, marking it as an open direction for further research.

References

This naturally raises the question of which planar graphs have planar drawings where all edges are touched or crossed by a given convex curve. We do not settle this question, but in this section we provide an example showing that it is nontrivial: not every planar graph has such a drawing.

— Stabbing Faces By a Convex Curve (2508.17549 - Eppstein, 24 Aug 2025) in Appendix, Section “A graph whose edges cannot be touched by a convex curve”