We obtain a complete classification of graphs H for which the class of $(\text{triangle},H)$-free graphs is well-quasi-ordered by the induced subgraph relation and an almost complete classification of graphs H for which the class of $(\text{triangle},H)$-free graphs has bounded clique-width. In particular, we show that for these graph classes, well-quasi-orderability implies boundedness of clique-width. To obtain our results, we further refine a known method based on canonical decomposition. This leads to a new decomposition technique that is applicable to both notions, well-quasi-orderability and clique-width.

中文翻译：

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无三角形图类的类宽度和良好拟序

我们获得图H的完全分类，其中$（\text{三角形}，H）$诱导子图关系和图H的几乎完整分类对无图是很好的拟序，对于$（\text{三角形}，H）$无图具有界线宽度。特别地，我们表明对于这些图类，良好的可排序性暗示了集团宽度的有界性。为了获得我们的结果，我们进一步完善了基于规范分解的已知方法。这导致了一种新的分解技术，该技术适用于两种概念：准拟序性和集团宽度。