Articulation vertex in complementary graph
Hint: An equivalent characteristic of "an articulation vertex $v$" is as follows:


  $v$ is an articulation vertex in $G$ $\iff$ there exists two vertices $x,y$ such that $v$ is on all the paths between $x,y$.


So it suffices to check all the pairs $x,y$ in $G'$ to see whether there exists some path between them which does not pass $v$.



More hints:

Consider two cases:


In $G$, $v$ is on all the paths of $x,y$. Then is $e = (x,y) \in G'$?
In $G$, $v$ is not on all the paths of $x,y$. Where are $x$ and $y$ if $v$ is removed in $G$? Then,

2.1 What if $(x,y) \notin G$?
2.2 What if $(x,y) \in G$? Can you find a path $x \leadsto y$ in $G'$ which does not pass $v$?