On forbidden induced subgraphs for [formula omitted]-free perfect graphs

Abstract

Considering connected K1,3-free graphs with independence number at least 3, Chudnovsky and Seymour (2010) showed that every such graph, say G, is 2ω-colourable where ω denotes the clique number of G. We study (K1,3,Y)-free graphs, and show that the following three statements are equivalent. •[(1)] Every connected (K1,3,Y)-free graph which is distinct from an odd cycle and which has independence number at least 3 is perfect.•[(2)] Every connected (K1,3,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 P5 or Z2 (where Z2 is also known as hammer). Furthermore, for connected (K1,3,Y)-free graphs (without an assumption on the independence number), we show a similar characterisation featuring the graphs P4 and Z1 (where Z1 is also known as paw).