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Linear-size free sets in planar graphs of treewidth at most 4

Determine whether every n-vertex planar graph of treewidth at most 4 contains a free set—i.e., a vertex subset S that can be mapped to any set P of |S| points while admitting a straight-line crossing-free drawing—of size Ω(n).

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Background

Linear free-set bounds are known for treewidth ≤2 and ≤3, but cannot be generalized to treewidth ≤5 due to constructions linked to shortness exponents and dual circumference.

The case of treewidth 4 is a natural frontier: resolving it would clarify how structural graph width constraints interact with geometric flexibility of straight-line planar embeddings.

References

As noted in the introduction to this section, \cref{fs-tw3} cannot be generalized to all $n$-vertex planar graphs of treewidth at most $5$. This leaves open the question of whether a linear bound is possible for planar graphs of treewidth at most $4$.

— Free Sets in Planar Graphs: History and Applications (2403.17090 - Dujmović et al., 25 Mar 2024) in Subsection “Subclasses of planar graphs with linear-sized free sets”