Reference for (odd-hole,antihole)-free graphs? 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.
In fact, it is relatively easy. Instead for studying independent set problem in (odd-hole,antihole)-free graphs, we take complement of the graphs and try to find a maximum clique in it. Thus it becomes maximum clique problem in (hole, anti-odd-hole)-free graphs.

In section 2 of the paper "Triangulated Neighborhoods in Even-hole-free Graphs" by da Silva and Vuskovic, they stated that Farber first shows 


  There are $O(n^2)$ maximal cliques in any 4-hole free graphs.


Then their main theorem stated that


  There are $O(n + m)$ maximal cliques in even-hole-free graphs, and all the maximal cliques can be found in time $O(n^2m)$.


Since we are dealing with (hole, anti-odd-hole)-free graphs which is clearly even-hole-free, finding a maximum clique takes at most $O(n^2m)$ time.

4-hole-free are critical to these kind of results like a poly-time algorithm for $\overline{K_{2,m}}$-free graphs, so the real challenge may be studying independent set problem in (hole, anti-odd-hole)-free graphs instead, which becomes the maximum clique problem in (odd-hole, anti-hole)-free graphs.



Edit:

Oh, another thought came out. (hole, anti-odd-hole)-free graphs are almost weakly chordal in the following sense: since 4-hole-free implies there are only anti-holes with size 4~7 remains (any k-anti-hole with size > 7 contains a 4-hole), and it is also anti-odd-hole-free which restrict the size of anti-holes down to 4 and 6, it is almost no holes/antiholes in the graph! Thus a poly-time algorithm seems plausible for such graphs.