Pandichotomy of planar graph classes in the plane

Determine whether every planar graph, every planar 3-tree, and every planar bipartite graph is pandichotomous in the Euclidean plane.

Background

The paper establishes several sufficient conditions for planar or bipartite graphs to admit geometric realizations in the Euclidean plane, including outerplanarity of the short subgraph and caterpillar structure of the long subgraph. It also proves that some graphs of bounded degeneracy are not pandichotomous in the plane. The authors leave unresolved whether the broader classes of all planar graphs, planar 3-trees, or planar bipartite graphs nevertheless have the stronger property of being pandichotomous in two dimensions.

References

Some interesting open questions remain, such as: \item Is every planar graph, planar $3$-tree, or planar bipartite graph pandichotomous in $2$?

— Geometric realizations of dichotomous ordinal graphs  (2503.07361 - Angelini et al., 10 Mar 2025) in Section Conclusion

Is every planar graph, planar $3$-tree, or planar bipartite graph pandichotomous in~$2$?

— Geometric realizations of dichotomous ordinal graphs  (2503.07361 - Angelini et al., 10 Mar 2025) in Section Conclusion