On Some Cograph Extensions and Their Algorithms

This paper reviews some classes of graphs related to cographs. The characterization of cographs by which there is no induced path on three edges (P4) is relaxed in various ways. Considered are classes of graphs that are uniquely representable by trees. Three such classes studied here are: P4-reducible, P4-sparse and P4-extendible graphs. In addition, we show that the class of P4-lite graphs has also a unique tree representation. This class is further studied with respect to belonging to perfectly orderable graphs. It is shown that P4-lite graphs are a proper subset of perfectly orderable graph class known as P4-simplicial graphs. Linear time sequential recognition algorithms are discussed, together with NC class parallel algorithms. We show that the recognition of these classes belongs to NC class of algorithms and hint to possible extensions of existing algorithms. 1.0 Introduction This paper deals with several classes of perfect and near-perfect graphs related to complement reducible graphs cographs. Especially interesting is the existence of graphs that have a unique tree representation, which allows us to devise polynomial time algorithms for various problems. This report focuses primarily on this type of cograph extensions and generalizations. The text is organized as follows. An overview of cographs is given first. Section 3 contains the study of four classes of cograph extensions. Section 4 discusses further one of the families and points to a link between cograph extensions and graph decompositions. The report closes with the discussion of recognition algorithms for given classes of graphs.