Dice Question Streamline Icon: https://streamlinehq.com

Subset-strengthened bounds for bounded-degree planar graphs

Develop subset-robust free-set bounds for bounded-degree planar graphs by proving that, for any planar graph G of bounded maximum degree and any subset X ⊆ V(G), there exists S ⊆ X of size Ω(|X|^{0.8}) that is a free set of G.

Information Square Streamline Icon: https://streamlinehq.com

Background

Current strongest bounds in bounded-degree planar graphs leverage dual circumference to obtain global free-set lower bounds of Ω(n{0.8}/Δ4), but they do not guarantee subset selection from an arbitrary X ⊆ V(G).

A subset-robust strengthening would make the powerful dual-circumference method flexible for applications requiring prescribed vertex subsets (e.g., partial simultaneous embeddings).

References

We conclude with a list of open problems: Is the following strengthening of \cref{dual_planar_concrete} true: For any planar graph $G$ of bounded degree and any subset $X$ of vertices of $G$, there exists a set $S\subseteq X$ of size $\Omega(|X|{0.8})$ that is a free set of $G$?

— Free Sets in Planar Graphs: History and Applications (2403.17090 - Dujmović et al., 25 Mar 2024) in Section Open Problems (enumerated item 6a)