What is a formal way to define a complement of a directed graph? On the wikipedia link -> Here it is not quite clear for me if we take the edges of the opposite direction. In any case, could anyone please clearly give a definition of the complement of the directed graph?
Thanks in advance
Say we have a directed graph $D=(V,A)$. Let $A_{all}$ denote the set of all possible directed edges on the vertex set $V$. In other words, if $u$ and $v$ are vertices of $V$, then both the directed edges $u\rightarrow v$ and $v \rightarrow u$ belong to $A_{all}$.
The complement of $D$ will be the directed graph $\overline{D} = (V, A_{all}\setminus A)$. In other words, the complement has every possible directed edge not in $D$.
The directed graph formed by reversing the direction of all the edges in $D$ is the transpose or reverse of $D$, and is in general not the same as the complement.
See this link: Wikipedia - Transpose of Digraph