It was proved by Huynh, Mohar, Šámal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for fixed $t$, if a graph $G$ is $K_t$-minor-free and contains every $n$-vertex planar graph as a subgraph, then $G$ has $2^{Ω(n)}$ vertices. On the other hand, we construct a polynomial size $K_4$-minor-free graph containing every $n$-vertex tree as an induced subgraph, and a polynomial size $K_7$-minor-free graph containing every $n$-vertex $K_4$-minor-free graph as induced subgraph. This answers several problems raised recently by Bergold, Iršič, Lauff, Orthaber, Scheucher and Wesolek. We study more generally the order of universal graphs for various classes (of graphs of bounded degree, treedepth, pathwidth, or treewidth), if the universal graphs retain some of the structure of the original class.

翻译：Huynh、Mohar、Šámal、Thomassen与Wood于2021年证明：任何包含所有可数平面图作为子图的可数图必然具有无限团子式。本文证明了该结果的有限量化版本：对固定参数$t$，若图$G$不含$K_t$子式且包含所有$n$顶点平面图作为子图，则$G$的顶点数至少为$2^{Ω(n)}$。另一方面，我们构造了一个多项式大小的不含$K_4$子式的图，它包含所有$n$顶点树作为导出子图，以及一个多项式大小的不含$K_7$子式的图，它包含所有$n$顶点无$K_4$子式图作为导出子图。这解决了Bergold、Iršič、Lauff、Orthaber、Scheucher与Wesolek近期提出的若干问题。我们进一步研究了各类图（有界度图、树深图、路径宽图或树宽图）的泛图阶数，当这些泛图保留原图类的部分结构时，问题更具一般性。