On forbidden induced subgraphs for K1, 3-free perfect graphs

Abstract Considering connected K 1 , 3 -free graphs with independence number at least 3, Chudnovsky and Seymour (2010) showed that every such graph, say G , is 2 ω -colourable where ω denotes the clique number of G . We study ( K 1 , 3 , Y ) -free graphs, and show that the following three statements are equivalent. • [(1)] Every connected ( K 1 , 3 , Y ) -free graph which is distinct from an odd cycle and which has independence number at least 3 is perfect. • [(2)] Every connected ( K 1 , 3 , Y ) -free graph which is distinct from an odd cycle and which has independence number at least 3 is ω -colourable. • [(3)] Y is isomorphic to an induced subgraph of P 5 or Z 2 (where Z 2 is also known as hammer). Furthermore, for connected ( K 1 , 3 , Y ) -free graphs (without an assumption on the independence number), we show a similar characterisation featuring the graphs P 4 and Z 1 (where Z 1 is also known as paw).