Well-quasi-ordered classes of bounded clique-width

TL;DR

This paper proves that for classes of graphs with bounded clique-width, it is decidable whether they are well-quasi-ordered with labels, solving an open problem and confirming conjectures in this area.

Contribution

It establishes the decidability of labelled-well-quasi-ordering for bounded clique-width graph classes and provides a structural characterization of such classes.

Findings

Decidability of labelled-well-quasi-ordering for bounded clique-width classes.

Equivalence of conditions for labelled-well-quasi-ordering with small or infinite label sets.

Structural characterization involving bounded clique-width and transduction properties.

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 solves an open problem raised by Daligault, Rao and Thomass\'e 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.