Well-quasi-ordered classes of bounded clique-width
Abstract
We study classes of graphs with bounded clique-width that are well-quasi-ordered by the induced subgraph relation, in the presence of labels on the vertices. We prove that, given a finite presentation of a class of graphs, one can decide whether the class is labelled-well-quasi-ordered. This answers positively to two conjectures of Pouzet in the restricted case of bounded clique-width classes. Namely, we prove that being labelled-well-quasi-ordered by a set of size 2 or by a well-quasi-ordered infinite set are equivalent conditions, and that in such cases, one can freely assume that the graphs are equipped with a total ordering on their vertices. Finally, we provide a structural characterization of those classes as those that are of bounded clique-width and do not existentially transduce the class of all finite paths.
Cite
@article{arxiv.2601.18571,
title = {Well-quasi-ordered classes of bounded clique-width},
author = {Maël Dumas and Aliaume Lopez},
journal= {arXiv preprint arXiv:2601.18571},
year = {2026}
}
Comments
well-quasi-ordering, clique-width, automata theory, monoids, factorization forests, gap embedding