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arXiv · 2411.11842

Induced subgraphs and tree decompositions XVII. Anticomplete sets of large treewidth

Maria Chudnovsky·Sepehr Hajebi·Sophie Spirkl
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Abstract

Two sets $X, Y$ of vertices in a graph $G$ are "anticomplete" if $X\cap Y=\varnothing$ and there is no edge in $G$ with an end in $X$ and an end in $Y$. We prove that every graph $G$ of sufficiently large treewidth contains two anticomplete sets of vertices each inducing a subgraph of large treewidth unless $G$ contains, as an induced subgraph, a highly structured graph of large treewidth that is an obvious counterexample to this statement. These are: complete graphs, complete bipartite graphs and "interrupted $s$-constellations." The latter is a slightly adjusted version of a well-known construction by Bonamy et al.

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Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl. 2024-11-18. Induced subgraphs and tree decompositions XVII. Anticomplete sets of large treewidth. https://arxiv.org/abs/2411.11842

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