Product structure with a bounded-size planar factor

Prove or disprove the conjecture that there exists an integer c ≥ 1 such that every n-vertex planar graph G of treewidth at most k and path-partition-width at most b, where n ≥ kb, is contained in H ⊠ P ⊠ K_c for some planar graph H of treewidth at most 3 with |V(H)| ≤ b and some path P with |V(P)| ≤ k.

Background

The paper observes that any representation G ⊆ H ⊠ P ⊠ K_c induces a path-partition of G, yielding the lower bound |V(H)| ≥ ppw(G)/c. This can be linear in the number of vertices, for example when G has a vertex of linear degree.

The conjecture proposes that, up to a constant factor in the complete-graph component, path-partition-width is also an upper bound on the size of H, while treewidth controls the number of path layers.

References

There is an integer $c 1$ such that for any integers $n,k,b 1$ with $n kb$, every $n$-vertex planar graph $G$ with treewidth at most $k$ and path-partition-width at most $b$ is contained in $H \boxtimes P \boxtimes K_c$ for some planar graph $H$ with treewidth at most $3$ and $|V(H)| b$ and some path $P$ with $|V(P)| k$.

— Short Paths in the Planar Graph Product Structure Theorem  (2502.01927 - Hendrey et al., 4 Feb 2025) in Section 6, item 2, Conjecture