遗传 2-WQO 图类具有有界团宽度
Hereditary 2-WQO Graph Classes Have Bounded Clique-Width
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中文总结 AI 辅助
研究遗传图类，利用一元依赖的结构/非结构二分法及相关性质，通过排除良连接集证明遗传 2-WQO 图类有有界团宽度，证实 Pouzet 猜想。
中文摘要 AI 辅助
如果一个图类的 k 标记图在保持标签的诱导子图嵌入下是良拟序的，那么它就是 k-WQO。我们证明了每个遗传的 2-WQO 图类都有有界团宽度。结合 Dumas 和 Lopez 的最新结果，这证实了 Pouzet 的一个长期猜想：一个遗传图类是 2-WQO 当且仅当对于所有 k≥2 它是 k-WQO，当且仅当它是 ∀-WQO，即其标记图对于每个可能的良拟序标签集都是良拟序的。我们的证明基于 Dreier、Mählmann 和 Toruńczyk 最近关于一元依赖模型理论概念的结构/非结构二分法。通过禁止诱导子图的非结构特征，我们表明每个遗传的 2-WQO 图类都是一元依赖的。利用一元依赖提供的拉姆齐理论结构性质，我们通过排除大的良连接集的存在来建立有界团宽度，而良连接集是团宽度的典型障碍。
英文摘要
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.
