We prove blow-up structure theorems for graphs excluding a tree or an apex-tree as a minor. First, we show that for every $t$-vertex tree $T$ with $t\geq 3$ and radius $h$, and every graph $G$ excluding $T$ as a minor, there exists a graph $H$ with pathwidth at most $2h-1$ such that $G$ is contained in $H\boxtimes K_{t-2}$ as a subgraph. This improves on a recent theorem of Dujmović, Hickingbotham, Joret, Micek, Morin, and Wood (2024), who proved the same result but with a larger bound on the order of the complete graph in the product. Second, we show that for every $t$-vertex tree $T$ with $t\geq 2$, radius $h$ and maximum degree $d$, and every graph $G$ excluding the apex-tree $T^+$ as a minor, where $T^+$ is the tree obtained by adding a universal vertex to $T$, there exists a graph $H$ with treewidth at most $4h-1$ such that $G$ is contained in $H\boxtimes K_{2(t-1)d}$. The bound on the treewidth of $H$ is best possible up to a factor $2$, and improves on a $2^{h+2}-4$ bound that follows from a recent result of Dujmović, Hickingbotham, Hodor, Joret, La, Micek, Morin, Rambaud, and Wood (2024).

翻译：我们证明了不含树或顶树作为子式的图的爆炸结构定理。首先，对于任意具有$t$个顶点（$t\geq 3$）、半径$h$的树$T$，以及任意不含$T$作为子式的图$G$，存在一个路径宽度不超过$2h-1$的图$H$，使得$G$可作为子图包含于$H\boxtimes K_{t-2}$中。这改进了Dujmović、Hickingbotham、Joret、Micek、Morin和Wood（2024）的近期定理——他们证明了相同结果，但乘积中完全图的阶数上界更大。其次，对于任意具有$t$个顶点（$t\geq 2$）、半径$h$、最大度$d$的树$T$，以及任意不含顶树$T^+$（即向$T$添加一个万能顶点所得之树）作为子式的图$G$，存在一个树宽不超过$4h-1$的图$H$，使得$G$可包含于$H\boxtimes K_{2(t-1)d}$中。此$H$树宽的上界在因子$2$意义下是最优的，且改进了由Dujmović、Hickingbotham、Hodor、Joret、La、Micek、Morin、Rambaud和Wood（2024）近期结果导出的$2^{h+2}-4$上界。