Decomposition by Clique Separators
Chvátal and Sbihi (Recognizing Claw-free Perfect Graphs, JCTB 44) described the atoms for the class of claw-free perfect graphs, a.k.a. quasi-line perfect graphs.  This description was later refined by Maffray and Reed, and essentially consists of two classes: "peculiar", which are kind of a special case, and "elementary", which are constructed by modifying the line graph of a bipartite multigraph through an operation called augmentation.