Recursive DFS reads top-to-bottom: visit the node, mark it, recurse into each neighbor. The closure pattern (visit defined inside dfs_recursive) keeps visited and order out of the public signature. The function is post-order if you append to order AFTER the recursive call instead of before. The only risk is Python's default recursion limit of 1000 frames: deep chains (linked-list-shaped graphs, long paths) crash with RecursionError and you should switch to the iterative version below.

Iterative DFS replaces the call stack with an explicit Python list. list.pop() is O(1) from the end, which gives LIFO semantics, the same as recursion. To match the recursive visit order exactly, push neighbors in reverse so the first neighbor lands on top of the stack and is visited next. The iterative version handles arbitrarily deep graphs without bumping sys.setrecursionlimit, which is the right move for production code that might process untrusted input. The check if node in visited: continue after the pop matters: a node can be pushed twice before either copy is processed.

Cycle detection in a directed graph needs three colors, not two. White = unseen, gray = currently on the recursion stack, black = fully processed. A back-edge (the only kind of edge that creates a cycle in a DFS tree) is exactly an edge from gray to gray. A two-color visited set is wrong because it cannot tell the difference between 'I've seen this node before in another branch' (cross-edge, not a cycle) and 'I'm currently inside this node' (back-edge, cycle). The same coloring drives topological sort: emit a node when it turns black, then reverse the list.