Clique-width and well-quasi-ordering of triangle-free graph classes.

We obtain a complete classification of graphs H for which the class of (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 (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. [ABSTRACT FROM AUTHOR]