Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded pathwidth and bounded maximum degree has a tree-partition of bounded width, with the extra property that the underlying tree has bounded pathwidth. Moreover, we prove a lower bound showing that the bound on the pathwidth of the underlying tree is within a constant factor of optimal.

翻译：具有有界树宽和有界最大度的图已知存在有界宽的树划分。如果将有界树宽的假设加强为有界路径宽，我们能得到什么结论？我们证明，每个具有有界路径宽和有界最大度的图都存在有界宽的树划分，且额外满足底层树具有有界路径宽的性质。此外，我们证明了一个下界，表明底层树路径宽的界在常数因子内是最优的。