Well-quasi-ordered classes of bounded clique-width
This solves a decade-old open problem in graph theory and structural combinatorics, providing decision procedures and characterizations for well-quasi-ordering in bounded clique-width classes.
The paper solves an open problem by proving that, given a finite presentation of a graph class with bounded clique-width, one can decide whether the class is labelled-well-quasi-ordered. It also confirms two conjectures of Pouzet for such classes and provides a structural characterization.
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 solves an open problem raised by Daligault, Rao and Thomassé in 2010, and 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.