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arXiv · 2302.10662

Snakes and Ladders: a Treewidth Story

Steven Chaplick·Steven Kelk·Ruben Meuwese·Matus Mihalak·Georgios Stamoulis
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Abstract

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.

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Steven Chaplick, Steven Kelk, Ruben Meuwese, Matus Mihalak, Georgios Stamoulis. 2024-01-30. Snakes and Ladders: a Treewidth Story. https://arxiv.org/abs/2302.10662

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