Hereditary 2-WQO Graph Classes Have Bounded Clique-Width

2026-07-12

AI Summary

This study addresses a central problem at the intersection of graph theory and model theory: whether hereditary 2-well-quasi-ordered (2-WQO) graph classes necessarily have bounded clique-width, thereby verifying Pouzet’s long-standing conjecture on the equivalence of various WQO notions. By characterizing such graph classes via forbidden induced subgraphs, the authors show they belong to the class of monadically dependent structures. Combining this with Ramsey-theoretic arguments, they rule out the existence of large well-connected sets, which implies bounded clique-width. This work establishes, for the first time, a direct link between hereditary 2-WQO graph classes and bounded clique-width, fully confirming Pouzet’s conjecture that a hereditary graph class is 2-WQO if and only if it is ∀-WQO, thus significantly advancing the integration of WQO theory and structural graph theory.

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.

Problem

• hereditary graph class
• 2-WQO
• clique-width
• well-quasi-ordering
• monadic dependence

Innovation

• clique-width
• well-quasi-ordering
• monadic dependence
• hereditary graph classes
• Ramsey theory