A polynomial time membership test and solutions to the minimum coloring and maximum weight clique and stable set problems are given for certain families of graphs. In particular, it is shown that for any arbitrary hereditary family of graphs, these problems can be solved quickly for the entire family whenever they can be solved quickly for the clique cutset free members of the family. If the graphs in the family are perfect, then a similar statement holds for the mimimum weight clique and stable set problems. These results are obtained by applying a polynomial time algorithm for determining whether a graph has a clique cutset. Several questions of Gavril concerning "clique separable" perfect graphs are answered, and a polynomial time recognition algorithm for Gallai's "i-triangulated" perfect graphs is given. Keywords: graph algorithms, cliques, stable sets, perfect graphs, clique separable graphs, i-triangulated graphs.