Cograph

In graph theory, a cograph, or complement-reducible graph, or P₄-free graph, is a graph that can be generated from the single-vertex graph K₁ by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K₁ and is closed under complementation and disjoint union.

Cographs have been discovered independently by several authors since the 1970s; early references include Jung (1978), Lerchs (1971), Seinsche (1974), and Sumner (1974). They have also been called D*-graphs,hereditary Dacey graphs (after the related work of James C. Dacey Jr. on orthomodular lattices), and 2-parity graphs. They have a simple structural decomposition involving disjoint union and complement graph operations that can be represented concisely by a labeled tree, and used algorithmically to efficiently solve many problems such as finding the maximum clique that are hard on more general graph classes.

Special cases of the cographs include the complete graphs, complete bipartite graphs, cluster graphs, and threshold graphs. The cographs are, in turn, special cases of the distance-hereditary graphs, comparability graphs, and perfect graphs.

Any cograph may be constructed using the following rules:

The cographs may be defined as the graphs that can be constructed using these operations, starting from the single-vertex graphs. Alternatively, instead of using the complement operation, one can use the join operation, which consists of forming the disjoint union ${\displaystyle G\cup H}$ and then adding an edge between every pair of a vertex from ${\displaystyle G}$ and a vertex from ${\displaystyle H}$.

...
Wikipedia