We provide an algorithm to decide whether a class of finite graphs that has bounded linear clique width is well-quasi-ordered by the induced subgraph relation in the presence of a labelling of the vertices, where the class is given by an $\mathsf{MSO}$-transduction from finite words. This study leverages tools from automata theory, and the proof scheme allows to derive a weak version of the Pouzet conjecture for classes of bounded linear clique-width. We also provide an automata based characterization of which classes of $\mathsf{NLC}$ graphs are labelled-well-quasi-ordered by the induced subgraph relation, where we recover the results of Daligault Rao and Thomass\'e by encoding the models into trees with the gap embedding relation of Dershowitz and Tzameret.

翻译：本文提出一种算法，用于判定一类具有有界线性团宽的有限图类，在顶点标记存在的情况下，是否在导出子图关系下构成良拟序；该类图通过从有限词上的$\mathsf{MSO}$可定义转换得到。本研究利用自动机理论工具，其证明方案可推导出有界线性团宽图类上的Pouzet猜想的弱形式。我们还给出了基于自动机的刻画，说明哪些$\mathsf{NLC}$图类在导出子图关系下具有标记良拟序性质，通过将模型编码为带有Dershowitz与Tzameret的间隙嵌入关系的树，恢复了Daligault、Rao与Thomassé的结论。