Snakes and Ladders: a Treewidth Story
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Let G be an undirected graph. We say that G contains a ladder of length k if the 2 × (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.
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