We prove that several natural graph classes have tree-decompositions with minimum width such that each bag has bounded treewidth. For example, every planar graph has a tree-decomposition with minimum width such that each bag has treewidth at most 3. This treewidth bound is best possible. More generally, every graph of Euler genus $g$ has a tree-decomposition with minimum width such that each bag has treewidth in $O(g)$. This treewidth bound is best possible. Most generally, every $K_p$-minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function $f(p)$. In such results, the assumption of an excluded minor is justified, since we show that analogous results do not hold for the class of 1-planar graphs, which is one of the simplest non-minor-closed monotone classes. In fact, we show that 1-planar graphs do not have tree-decompositions with width within an additive constant of optimal, and with bags of bounded treewidth. On the other hand, we show that 1-planar $n$-vertex graphs have tree-decompositions with width $O(\sqrt{n})$ (which is the asymptotically tight bound) and with bounded treewidth bags. Moreover, this result holds in the more general setting of bounded layered treewidth, where the union of a bounded number of bags has bounded treewidth.

翻译：我们证明了几类自然图具有最小宽度的树分解，其中每个包具有有界树宽。例如，每个平面图都存在一个最小宽度的树分解，其中每个包的树宽至多为3。该树宽界是最优的。更一般地，每个欧拉亏格为$g$的图都存在一个最小宽度的树分解，其中每个包的树宽为$O(g)$。该树宽界是最优的。最一般地，每个不含$K_p$子式的图都存在一个最小宽度的树分解，其中每个包的树宽至多为某个多项式函数$f(p)$。在这些结果中，排除子式的假设是合理的，因为我们证明类似结论不适用于1-平面图类，这是最简单的非子式封闭单调类之一。事实上，我们证明1-平面图不存在宽度在最优值加性常数范围内且包具有有界树宽的树分解。另一方面，我们证明具有$n$个顶点的1-平面图存在宽度为$O(\sqrt{n})$（这是渐近紧界）且包具有有界树宽的树分解。此外，该结果在更一般的有界分层树宽设置中成立，其中有界数量包的并集具有有界树宽。