  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.
