Tight grid-minor bounds for apex-minor-free graphs of bounded radius

Determine tight bounds, as functions of the radius r and the number t of vertices of an apex graph A, on the size of grid minors in A-minor-free graphs of radius r.

Background

The paper establishes an O(rt2) upper bound on the side length of a grid minor excluded from an A-minor-free graph of radius r, where A is an apex graph with t vertices. It also gives an Ω(r√t) lower bound and notes an additional Ω(t) lower bound arising from nested-triangle graphs. The authors explicitly identify closing the gap between these bounds as an open problem.

References

It is an open problem to determine tight bounds on the size of grid-minors in $A$-minor-free graphs with radius $r$, where $A$ is an apex graph with $t$ vertices. \cref{PolyApexMinorTheorem} provides an $O(rt2)$ upper bound, and \cref{LowerBoundIntro} provides an $\Omega(r\sqrt{t})$ lower bound.

— Polynomial Bounds in the Apex Minor Theorem  (2503.04228 - Hendrey et al., 6 Mar 2025) in Section Open Problems

It is an open problem to determine tight bounds on the size of grid-minors in $A$-minor-free graphs with radius $r$, where $A$ is an apex graph with $t$ vertices. \cref{PolyApexMinorTheorem} provides an $O(rt2)$ upper bound, and \cref{LowerBoundIntro} provides an $\Omega(r\sqrt{t})$ lower bound. There is also an $\Omega(t)$ lower bound because it is known that if $H$ is the nested triangles graph with $t$ vertices and $H$ is a minor of the $n\times n$ grid, then $n\Omega(t)$ (that is, \cref{PlanarGrid} is tight up to a constant factor).

— Polynomial Bounds in the Apex Minor Theorem  (2503.04228 - Hendrey et al., 6 Mar 2025) in Section "Open Problems"

It is an open problem to determine tight bounds on the size of grid-minors in $A$-minor-free graphs with radius $r$, where $A$ is an apex graph with $t$ vertices.

— Polynomial Bounds in the Apex Minor Theorem  (2503.04228 - Hendrey et al., 6 Mar 2025) in Section 5, “Open Problems”