  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 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 $\omega^2$ where $\omega$ denotes the number of vertices in a
largest clique of $G$. We prove that (banner, odd hole)-free graphs are
perfect-divisible. %
