Forbidden induced subgraphs for perfectness of claw-free graphs of independence number at least 4

For every graph X, we consider the class of all connected {K(1,3), X}-free graphs which are distinct from an odd cycle and have independence number at least 4, and we show that all graphs in the class are perfect if and only if X is an induced subgraph of some of P6, K1 ∪ P5, 2P3, Z2 or K1 ∪ Z1. Furthermore, for X chosen as 2K1 ∪ K3, we list all eight imperfect graphs belonging to the class; and for every other choice of X, we show that there are infinitely many such graphs. In addition, for X chosen as B(1,2), we describe the structure of all imperfect graphs in the class.