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：设 $G$ 是一个无向图。若 $2 \times (k+1)$ 网格图是 $G$ 的诱导子图，且仅通过其四个角点与 $G$ 的其余部分相连，则称 $G$ 包含长度为 $k$ 的梯子。我们证明：若 $G$ 中包含的所有梯子均缩减至长度 4，则树宽保持不变（并且该界是紧的）。这一结果表明，在计算图的树宽时，长梯子可直接缩减，且有界树宽图的极小禁止子图不能包含长梯子。我们的结果还解决了算法系统发育学中的一个开放问题：用于简化两棵进化树比较的公共链归约规则，在展示图中保持树宽不变。