• simplification in expression. reduced 3 different checks to 2.
  current expression:

low[node] = min(low[node], tin[ancestor], low[to])

proposed expression:

low[node] = min(low[node], low[to])    // whether "to" is descendants or already visited ancestor (i.e. back-edge)

• complete explaination

We can further improve the intuition behind the low[] array. We know that low[node] stores the lowest entry time of any node that is reachable from node, including via back edges. The tin[node] represents the time when node was first discovered during DFS.

When we encounter a back-edge from node to one of its ancestors (say ancestor), the earlier logic suggests we consider tin[ancestor] to update low[node]. However, we can simplify and strengthen this by observing that if ancestor can reach further up in the DFS tree (through its own back-edges), we can directly use low[ancestor] instead.

So instead of:

low[node] = min(low[node], tin[ancestor])

We use:

low[node] = min(low[node], low[ancestor])

This works because low[ancestor] already reflects the lowest reachable ancestor for ancestor, and since node can reach ancestor, it can also reach everything that ancestor can.

This simplification helps propagate the true lowest entry time more effectively through the DFS tree and back-edges.

Thus, the final update rule becomes:

low[cur] = min(low[cur], low[to])

for all adjacent to, whether they are descendants in the DFS tree or already visited ancestors (i.e., back-edges).