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From mostly-free to free: existence of a constant γ

Investigate whether there exists a constant γ>0 such that every mostly free vertex subset S—i.e., one that can be mapped to any set P of |S| points in general position while admitting a straight-line crossing-free drawing—contains a free subset of size at least γ|S|.

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Background

The notion of mostly free relaxes free sets by requiring universality only for point sets in general position. Outerplanar graphs demonstrate that entire vertex sets can be mostly free, yet the largest free subset need not exceed 2n/3 in some examples.

Establishing a universal γ would relate partial universality under general position constraints to full universality for arbitrary point sets, refining the structure of free subsets.

References

We conclude with a list of open problems: Say that a subset $S$ of vertices in a planar graph $G$ is {mostly free} if, for every set $P$ of $|S|$ points in general position, there exists a straight-line crossing-free drawing\ of $G$ in which each vertex in $S$ is drawn at a distinct point in $P$. Is there a constant $\gamma >0$ such that every mostly free set $S$ contains a free subset of size at least $\gamma|S|$?

— Free Sets in Planar Graphs: History and Applications (2403.17090 - Dujmović et al., 25 Mar 2024) in Section Open Problems (enumerated item 5)