Snakes and Ladders: a Treewidth Story
Steven Chaplick, Steven Kelk, Ruben Meuwese, Matus Mihalak, Georgios, Stamoulis

TL;DR
This paper proves that reducing long ladder structures in a graph to length 4 does not change its treewidth, simplifying computations and resolving an open problem in algorithmic phylogenetics.
Contribution
It establishes that long ladders can be safely reduced to length 4 without affecting treewidth, impacting graph minor theory and phylogenetics.
Findings
Long ladders can be reduced to length 4 without changing treewidth
Bound is tight; longer ladders do affect treewidth
The chain reduction rule preserves treewidth in phylogenetic display graphs
Abstract
Let be an undirected graph. We say that contains a ladder of length if the grid graph is an induced subgraph of that is only connected to the rest of via its four cornerpoints. We prove that if all the ladders contained in 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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Taxonomy
TopicsAlgorithms and Data Compression · Genomics and Phylogenetic Studies · Evolutionary Algorithms and Applications
