Planarity characterization for subcubic induced subdivision containment

Prove or refute the conjecture that, assuming P ≠ NP, for every subcubic graph H, H-Induced Subdivision Containment is solvable in polynomial time if and only if H is planar.

Background

The paper proves NP-completeness of H-Induced Subdivision Containment for one fixed subcubic nonplanar graph H. It also reviews polynomial-time results for several planar patterns and uses these facts to formulate a proposed complexity dichotomy.

The conjecture asserts that planarity exactly characterizes the polynomial-time cases among subcubic pattern graphs, under the usual assumption that P and NP are distinct.

References

A~natural conjecture stems from our paper and existing algorithms, under P $\neq$ NP. For any subcubic graph $H$, $H$-ISC is in P if and only if $H$ is planar.

— Induced Disjoint Paths Without an Induced Minor  (2502.05289 - Aboulker et al., 7 Feb 2025) in Section “Open questions”, Conjecture 1