Equivalent definition of cograph A cograph is simple graph defined by the criteria


*

*$K_1$ is a cograph,

*If X is a cograph, then so is its graph complement, and

*If X and Y are cographs, then so is their graph union X union Y
and also it is said that a graph is called cograph if G does not contain the path graph $P_4$ as an induced subgraph. How these definition are equivalent or atleast how first one is implying second? I am totally struck in which direction to proceed. Any reference or hint?
 A: To see that the first definition implies the second, you can use induction on $n$, the number of vertices.
The base case is trivial.  For the induction step, the cograph $G$ can be obtained either by the union of two smaller cographs $G_1$ and $G_2$, or by complementing such a graph.
In the first case, apply induction on both $G_1$ and $G_2$ to deduce that $G$ has no $P_4$.  In the second case, $G$ is obtained by complementing a cograph $G'$ obtained from the union of two cographs.  We just argued that $G'$ is $P_4$-free, so it suffices to observe that the complement of a $P_4$ is a $P_4$, and that complementing a graph without a $P_4$ cannot create one.
In the other direction (defintion 2 implies definition 1), the proof I know of relies on the fact that if $G$ is a $P_4$-free graph, then for any $X \subseteq V(G)$, one of $G[X]$ or the complement of $G[X]$ is disconnected ($G[X]$ denotes the subgraph induced by $X$).  It's provable by induction.  Then, this fact can be used to show that $G$ was obtained by taking the union of two cographs, or by complementing it.
EDIT : 
The full proof can be find on page 6, Theorem 19 of this document : 
https://homepages.warwick.ac.uk/~masgax/Graph-Theory-notes.pdf
