Paul, Patrick and I have decided to let Lovász graphs denote the class of graph where alpha=theta, named of course after the great László Lovász.
Earlier we found that this class of graphs includes the KE graphs. Patrick points out that the Lovász graphs are a proper superclass of these graphs. So the Lovász graphs are a generalization of both the class of König-Egerváry graphs as well as the class of perfect graphs. Interestingly, both of these classes contain the bipartite graphs as a subclass.
The Lovász graphs–like the KE graphs–are not a hereditary class. The 5-cycle with a single pendant adjoining one of its vertices is a Lovasz graph (with alpha=theta=3) whereas the 5-cycle (with alpha=2, and theta=sqrt(5)), an induced subgraph, is not a Lovász graph.
We would like to know a characterization of these graphs.