Narrowing the gap in the clique-width dichotomy for $(H_1, H_2)$-free graphs

We continue a recent systematic study into the clique-width of (H1, H2)-free graphs and present five new classes of (H1, H2)-free graphs of bounded clique-width. As a consequence, we have reduced the number of open cases from 13 to 8. Four of the five new graph classes have in common that one of their two forbidden induced subgraphs is the triangle. As such, we generalize corresponding results for H-free bipartite graphs. Our new results are based on the exploitation of a common technique, which aims to reduce the graph under consideration to a k-partite graph that can be decomposed in a certain way which, as we prove, ensures boundedness of their clique-width. We also discuss the (immediate) consequences of our work for the study into determining the complexity of the Vertex Colouring problem for (H1, H2)-free graphs.