A graph $A$ is "apex" if $A-z$ is planar for some vertex $z\in V(A)$. Eppstein [Algorithmica, 2000] showed that for a minor-closed class $\mathcal{G}$, the graphs in $\mathcal{G}$ with bounded radius have bounded treewidth if and only if some apex graph is not in $\mathcal{G}$. In particular, for every apex graph $A$ and integer $r$, there is a minimum integer $g(A,r)$ such that every $A$-minor-free graph with radius $r$ has treewidth at most $g(A,r)$. We show that if $t=|V(A)|$ then $g(A,r)\in O^\ast(r^9t^{18})$ which is the first upper bound on $g(A,r)$ with polynomial dependence on both $r$ and $t$. More precisely, we show that every $A$-minor-free graph with radius $r$ has no $16rt^2 \times 16rt^2$ grid minor, which implies the first result via the Polynomial Grid Minor Theorem. A key example of an apex graph is the complete bipartite graph $K_{3,t}$, since $K_{3,t}$-minor-free graphs include and generalise graphs embeddable in any fixed surface. In this case, we prove that every $K_{3,t}$-minor-free graph with radius $r$ has no $4r(1+\sqrt{t})\times 4r(1+\sqrt{t})$ grid minor, which is tight up to a constant factor.

翻译：若存在顶点$z\in V(A)$使得$A-z$为平面图，则称图$A$为"顶点图"。Eppstein [Algorithmica, 2000]证明了：对于次封闭图类$\mathcal{G}$，当且仅当某个顶点图不属于$\mathcal{G}$时，$\mathcal{G}$中半径有界的图才具有有界的树宽。特别地，对于任意顶点图$A$和整数$r$，存在最小整数$g(A,r)$，使得每个半径为$r$且不含$A$子式的图的树宽至多为$g(A,r)$。我们证明当$t=|V(A)|$时，$g(A,r)\in O^\ast(r^9t^{18})$，这是首次给出的$g(A,r)$上界在$r$和$t$上均具有多项式依赖性。更精确地说，我们证明每个半径为$r$且不含$A$子式的图都不包含$16rt^2 \times 16rt^2$网格子式，这一结论通过多项式网格次定理可推导出上述结果。完全二分图$K_{3,t}$是顶点图的关键示例，因为不含$K_{3,t}$子式的图包含并可推广至任何固定曲面可嵌入的图类。对此情形，我们证明每个半径为$r$且不含$K_{3,t}$子式的图都不包含$4r(1+\sqrt{t})\times 4r(1+\sqrt{t})$网格子式，该结果在常数因子范围内是紧的。