The family of $(\alpha \omega)$-graphs are of interest for several reasons. For example, any minimal counter-example to Berge's Strong Perfect Graph Conjecture belongs to this family. This paper accounts for all (4,3)-graphs. One of these is not attainable by existing techniques for generating $(\alpha +1,\omega)$-graphs from $(\alpha, \omega)$-graphs. Keywords: perfect graphs, minimal imperfect graphs, $(\alpha,\omega)$-graphs.