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arXiv · 2607.10939

Hereditary 2-WQO Graph Classes Have Bounded Clique-Width

Julien Duron·Nikolas Mählmann·Szymon Toruńczyk
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Abstract

A graph class is $k$-WQO if its $k$-labeled graphs are well-quasi-ordered under label-preserving induced subgraph embeddings. We show that every hereditary graph class that is $2$-WQO has bounded clique-width. Combined with the recent result of Dumas and Lopez, this confirms a long-standing conjecture of Pouzet: A hereditary graph class is $2$-WQO if and only if it is $k$-WQO for all $k\geq 2$, if and only if it is $\forall$-WQO, that is, its labeled graphs are well-quasi-ordered for every possible well-quasi-ordered label set. Our proof builds on a recent structure/non-structure dichotomy for the model theoretic notion of monadic dependence by Dreier, Mählmann, and Toruńczyk. Through the non-structure characterization by forbidden induced subgraphs, we show that every hereditary $2$-WQO graph class is monadically dependent. Leveraging the Ramsey-theoretic structural properties provided by monadic dependence, we then establish bounded clique-width by ruling out the existence of large well-linked sets, which are the canonical obstructions for clique-width.

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BibTeXRIS

Julien Duron, Nikolas Mählmann, Szymon Toruńczyk. 2026-07-26. Hereditary 2-WQO Graph Classes Have Bounded Clique-Width. https://arxiv.org/abs/2607.10939

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