Optimal path length in the planar graph product structure theorem

Determine whether every n-vertex planar graph G can be contained in a product H ⊠ P ⊠ K_c, with H a planar graph of treewidth at most 3, such that the path P has |V(P)| = O(√n), or, more strongly, |V(P)| = O(tw(G)).

Background

The Planar Graph Product Structure Theorem represents every planar graph as a subgraph of H ⊠ P ⊠ K_c, where H has bounded treewidth and P is a path. The paper establishes sublinear upper bounds on the size of P, including bounds of order n{(1+ε)/2} and bounds depending on the treewidth of G.

Planar graphs of treewidth Θ(√n) show that a path factor of order Ω(√n) is sometimes necessary. The stated problem asks whether this lower-bound order can always be achieved, or whether the path length can instead be bounded linearly in tw(G).

References

Can a version of the Planar Graph Product Structure Theorem be proved with $|V(P)| O(\sqrt{n})$ or with $|V(P)| O(\tw(G))$.

— Short Paths in the Planar Graph Product Structure Theorem  (2502.01927 - Hendrey et al., 4 Feb 2025) in Section 6, item 1 (Open problems)

We conclude with a number of open problems that arise from this work: Can a version of the Planar Graph Product Structure Theorem be proved with $|V(P)| O(\sqrt{n})$ or with $|V(P)| O(\tw(G))$.

— Short Paths in the Planar Graph Product Structure Theorem  (2502.01927 - Hendrey et al., 4 Feb 2025) in Section 6, item 1