Detect cycles in a genealogy graph during a Depth-first search

I'm loading horse genealogical data recursively. For some wrong sets of data my recursion never stops... and that is because there are cycles in the data.

How can I detect those cycles to stop recurring?

I thought of while recurring maintain a hashTable with all "visited" horses. But that will find some false positives, because a horse CAN be twice in a tree.

What cannot happen is that a horse appear as a father or grandfather or great grandfather of ITSELF.

Pseudo code:

void ProcessTree(GenTreeNode currentNode, Stack<GenTreeNode> seen)
{
   if(seen.Contains(currentNode)) return;
   // Or, do whatever needs to be done when a cycle is detected

   ProcessHorse(currentNode.Horse); // Or whatever processing you need

   seen.Push(currentNode);

   foreach(GenTreeNode childNode in currentNode.Nodes)
   {
      ProcessTree(childNode, seen);
   }

   seen.Pop();
}

The basic idea is to keep a list of all of the nodes that we've already seen on our way down to the current node; if was get back to a node that we already went through, then you know that we've formed a cycle (and we should skip the value, or do whatever needs to be done)