Subset-strengthened bounds for triconnected cubic planar graphs

Establish subset-robust linear free-set bounds for triconnected cubic planar graphs by proving that, for any such graph G and any subset X ⊆ V(G), there exists S ⊆ X of size Ω(|X|) that is a free set of G.

Background

Linear free-set bounds are known globally for triconnected cubic planar graphs, but current methods do not guarantee extracting large free subsets from an arbitrary prescribed vertex set X.

Achieving a subset-robust linear bound would strengthen the applicability of free-set techniques to tasks that require fixing chosen vertex subsets across multiple embeddings.

References

We conclude with a list of open problems: Is the following strengthening of \cref{fs-cubic} true: For any triconnected cubic planar graph $G$ and any subset $X$ of vertices of $G$, there exists a set $S\subseteq X$ of size $\Omega(|X|)$ that is a free set of $G$?

— Free Sets in Planar Graphs: History and Applications  (2403.17090 - Dujmović et al., 2024) in Section Open Problems (enumerated item 6b)