Short-path product structure beyond planar graphs

Establish whether the product structure theorems for graphs embeddable on any fixed surface and for graphs excluding a fixed apex graph as a minor can be strengthened so that the path factor P satisfies |V(P)| = O(n^{1−ε}) for some fixed ε > 0.

Background

Product structure theorems analogous to the planar theorem are known for graphs embeddable on any fixed surface and for graphs excluding a fixed apex graph as a minor.

The paper obtains sublinear path bounds for planar graphs and asks whether a comparable polynomially sublinear bound extends to these broader graph classes, with a fixed positive exponent improvement over the linear bound.