How do I check if a directed graph is acyclic?

I would try to sort the graph topologically, and if you can't, then it has cycles.

Doing a simple depth-first-search is not good enough to find a cycle. It is possible to visit a node multiple times in a DFS without a cycle existing. Depending on where you start, you also might not visit the entire graph.

You can check for cycles in a connected component of a graph as follows. Find a node which has only outgoing edges. If there is no such node, then there is a cycle. Start a DFS at that node. When traversing each edge, check whether the edge points back to a node already on your stack. This indicates the existence of a cycle. If you find no such edge, there are no cycles in that connected component.

As Rutger Prins points out, if your graph is not connected, you need to repeat the search on each connected component.

As a reference, Tarjan's strongly connected component algorithm is closely related. It will also help you find the cycles, not just report whether they exist.

Lemma 22.11 on the Book Introduction to Algorithms (Second Edition) states that:

A directed graph G is acyclic if and only if a depth-first search of G yields no back edges

Solution1： Kahn algorithm to check cycle. Main idea: Maintain a queue where node with zero in-degree will be added into queue. Then peel off node one by one until queue is empty. Check if any node's in-edges are existed.

Solution2: Tarjan algorithm to check Strong connected component.

Solution3: DFS. Use integer array to tag current status of node: i.e. 0 --means this node hasn't been visited before. -1 -- means this node has been visited, and its children nodes are being visited. 1 -- means this node has been visited, and it's done. So if a node's status is -1 while doing DFS, it means there must be a cycle existed.