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 that 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 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 polynomial‐time algorithms to find, for such a graph, a minimum coloring, and largest stable set. A graph G is perfectly divisible if every induced subgraph H of G contains a set X of vertices such that X meets all largest cliques of H, and X induces a perfect graph. The chromatic number of a perfectly divisible graph G is bounded by ω2 where ω denotes the number of vertices in a largest clique of G. We prove that (banner, odd hole)‐free graphs are perfectly divisible.