Finding pairs of vertices which would disconnect a graph
Assume that $G=(V,E)$ does contain any articulation point (otherwise the answer is easy).

One simple approach that requires $O(|V| \cdot |E|)$ time is that of computing the articulation points of $G-v$ for each $v \in V$.

If you find any articulation point $u$ then removing the two of vertices $\{u,v\}$ disconnects the graph.

In practice this amounts to running the depth first search needed to compute the articulation points of $G-v$ on $G$, while ignoring the edges to $v$.