A cograph is a simple graph that does not contain an induced path on 4 vertices. A graph G is if the vertices of G can be colored in k colors such that, for each color, the subgraph induced by the vertices assigned the color is a cograph. A graph that is and is not , but becomes whenever a vertex is removed, is called graph. Two general constructions are provided that produce critical graphs from color critical graphs and hypergraphs. A characterization is also given for when a general composition of graphs (path-joins) is critical. The characterization is used to provide an upper bound for the fewest number of vertices of a graph.
