Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the ladders contained in $G$ are reduced to length 4, the treewidth remains unchanged (and that this bound is tight). Our result indicates that, when computing the treewidth of a graph, long ladders can simply be reduced, and that minimal forbidden minors for bounded treewidth graphs cannot contain long ladders. Our result also settles an open problem from algorithmic phylogenetics: the common chain reduction rule, used to simplify the comparison of two evolutionary trees, is treewidth-preserving in the display graph of the two trees.

翻译：令 $G$ 为一个无向图。我们称 $G$ 包含长度为 $k$ 的梯子，若 $2 \times (k+1)$ 网格图是 $G$ 的一个诱导子图，且仅通过其四个角点与 $G$ 的其余部分连接。我们证明，若 $G$ 中包含的所有梯子均被缩减至长度 4，则树宽保持不变（且该界是紧的）。我们的结果表明，在计算图的树宽时，长梯子可以直接简化，且有界树宽图的最小禁子结构不能包含长梯子。该结果还解决了算法系统发育学中的一个未解决问题：用于简化两棵进化树比较的公共链归约规则，在两棵树的显示图中保持树宽不变。