So, let $tin[v]$ denote the the depth first search time of node $v$. Now, we introduce the array $fup[v]$, which is the minimum of $tin[v]$, the DFS time of all nodes $p$ that are connected to node $v$ via back-edge $(v, p)$ and all the values of $fup[to]$ for each vertex to which is a direct child of $v$ in the DFS tree.

Now, there is a back edge from node $v$ or it's descendants if there is a son $to$ of node $v$ such that $fup[to] \leq tin[v]$.(If $fup[to] = tin[v]$, it means back edge comes directly to $v$, otherwise it comes to some ancestor).