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arXiv · 2212.02737

Induced subgraphs and tree-decompositions VII. Basic obstructions in $H$-free graphs

Tara Abrishami·Bogdan Alecu·Maria Chudnovsky·Sepehr Hajebi·Sophie Spirkl
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Abstract

We say a class $\mathcal{C}$ of graphs is clean if for every positive integer $t$ there exists a positive integer $w(t)$ such that every graph in $\mathcal{C}$ with treewidth more than $w(t)$ contains an induced subgraph isomorphic to one of the following: the complete graph $K_t$, the complete bipartite graph $K_{t,t}$, a subdivision of the $(t\times t)$-wall or the line graph of a subdivision of the $(t \times t)$-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Wei{\ss}auer) to prove that the class of all $H$-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph $H$) is clean if and only if $H$ is a forest whose components are subdivided stars. Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest $H$ as above, we show that forbidding certain connected graphs containing $H$ as an induced subgraph (rather than $H$ itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer $\eta$, a complete description of unavoidable connected induced subgraphs of a connected graph $G$ containing $\eta$ vertices from a suitably large given set of vertices in $G$. This is of independent interest, and will be used in subsequent papers in this series.

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Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl. 2022-12-06. Induced subgraphs and tree-decompositions VII. Basic obstructions in $H$-free graphs. https://doi.org/10.1016/j.jctb.2023.10.008

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