A Description of Claw-Free Perfect Graphs

It is known that all claw-free perfect graphs can be decomposed via clique-cutsets into two types of indecomposable graphs respectively called elementary and peculiar (1988, V. Chvátal and N. Sbihi,J. Combin. Theory Ser. B44, 154–176). We show here that every elementary graph is made up in a well-defined way of a line-graph of bipartite graph and some local augments consisting of complements of bipartite graphs. This yields a complete description of the structure of claw-free Berge graphs and a new proof of their perfectness.