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.

翻译：我们研究了在顶点带有标签的情况下，关于导出子图关系具有良拟序性的有界团宽度图类。我们证明，给定一个图类的有限表示，可以判定该类是否具有标签良拟序性。这解决了Daligault、Rao和Thomassé在2010年提出的一个开放问题，并在有界团宽度类的受限情形下对Pouzet的两个猜想给出了肯定回答。具体而言，我们证明了被大小为2的集合良拟序与被无限良拟序集合良拟序是等价条件，且在此类情形下，可以自由假设这些图的顶点上配备了全序关系。最后，我们通过结构特征刻画了这些图类：它们既是具有有界团宽度的图类，同时也不存在将全体有限路径类进行存在性转换的可能。