X-free graphs are those that contain no graph from X as an induced subgraph. A hole is a cycle with at least 4 vertices. An odd-hole is a hole with an odd number of vertices. An antihole is the complement of a hole.
The (odd-hole,odd-antihole)-free graphs are precisely the perfect graphs; this is the Strong Perfect Graph Theorem. It is possible to find the largest independent set (and largest clique) in a perfect graph in polynomial time, but the only known method of doing so requires building a semi-definite program to compute the Lovász theta number.
The (hole,antihole)-free graphs are called weakly chordal, and constitute a rather easy class for many problems (including INDEPENDENT SET and CLIQUE).
Does anyone know if (odd-hole,antihole)-free graphs have been studied or written about?
These graphs occur quite naturally in constraint satisfaction problems where the graph of related variables forms a tree. Such problems are rather easy, so it would be nice if there were a way to find a largest
independent set clique for graphs in this family without having to compute the Lovász theta.
Equivalently, one wants to find a largest independent set for (hole, odd-antihole)-free graphs. Hsien-Chih Chang points out below why this is a more interesting class for INDEPENDENT SET than (odd-hole, antihole)-free graphs.