Recognizing cographs and threshold graphs through a classification of their edges

In this work, we attempt to establish recognition properties and characterization for two classes of perfect graphs, namely cographs and threshold graphs, leading to constant-time parallel recognition algorithms. We classify the edges of an undirected graph as either free, semi-free or actual, and define the class of A-free graphs as the class containing all the graphs with no actual edges. Then, we define the actual subgraph Ga of a non-A-free graph G as the subgraph containing all the actual edges of G. We show properties and characterizations for the class of A-free graphs and the actual subgraph Ga of a cograph G, and use them to derive structural and recognition properties for cographs and threshold graphs. These properties imply parallel recognition algorithms which run in O(1) time using O(nm) processors.  2000 Elsevier Science B.V. All rights reserved.