On the structure of (banner, odd hole)-free graphs

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A banner is a graph which consists of a hole on four vertices and a single vertex with precisely one neighbor on the hole. We prove that a (banner, odd hole)-free graph is either perfect, or does not contain a stable set on three vertices, or contains a homogeneous set. Using this structure result, we design a polynomial-time algorithm for recognizing (banner, odd hole)-free graphs. We also design polynomialtime algorithms to find, for such a graph, a minimum coloring and largest stable set.