Detect Cycle in Directed Graph
A medium Graph problem included in Love Babbar 450, Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 2
- Core for
- 11 roles
- Platform
- GeeksforGeeks
The problem
Given the number of vertices and a list of edges in a directed graph, determine whether the graph contains a cycle.
Example 1
- Input
- n = 4, edges = [[0,1],[1,2],[2,0],[2,3]]
- Output
- true
- Why
- There is a cycle 0->1->2->0.
Example 2
- Input
- n = 3, edges = [[0,1],[1,2]]
- Output
- false
- Why
- The directed edges form a chain with no cycle.
Constraints
- 1 <= n <= 100
- 0 <= edges.length <= 1000
How to think about it
Updated 2026-09-09In directed graphs, reaching a visited node is not enough to declare a cycle: two independent branches can converge on the same target (a cross-edge) without forming a loop. A cycle requires a back-edge pointing to an ancestor that is still active on the current call path.
Approaches, worst first
DFS with recursion stack tracking
time O(V + E) · space O(V)
Maintain two boolean arrays: `visited` and `onPath`. When entering a node, mark both true. If an adjacent neighbor is already `onPath`, a back-edge exists. When backtracking, unmark `onPath`.
Kahn algorithm with in-degree arrayWrite this one
time O(V + E) · space O(V + E)
Compute in-degrees for all vertices. Enqueue all vertices with in-degree 0. Repeatedly dequeue vertices, decrement neighbor in-degrees, and count visited nodes. If the count of peeled nodes is strictly less than n, a cycle prevented resolution.
Where people lose marks · 3
- Forgetting to clear `onPath[u]` during DFS post-order backtracking, which causes diamond-shaped DAG structures to be falsely flagged as cycles.
- Disjoint subgraphs require checking every vertex index from 0 to n - 1; starting only at node 0 misses cycles residing in other components.
- Using simple DSU on directed graphs; standard union-find ignores edge direction and falsely identifies directed DAGs as cyclic.
The theory behind it
Graph — the ground this problem stands on. All Graph problems
What Graph is
A graph is a network of individual points, called vertices or nodes, connected by lines called edges. Think of a subway transit map, an electrical circuit, or a web of social friends. Unlike a tree, a graph has no designated top node and no parent-child hierarchy. Connections can run one-way or both ways, and paths can loop back on themselves to form closed cycles.
When to reach for it
Reach for graph algorithms when inputs describe relationships, networks, flights between cities, course prerequisites, or clone networks. Signals include finding the shortest route across unweighted connections, ordering tasks that depend on earlier tasks, counting isolated clusters, or checking whether a path contains an infinite loop. Whenever problems present pairs of related entities and ask for reachability, distances, or dependencies, graph representations apply.
How the pattern works
First convert edge lists into an adjacency list, mapping each node to an array of its neighbors. Choose your exploration strategy based on the goal: use a queue and breadth-first search to find the shortest path in unweighted networks, or use recursion and depth-first search to explore full paths and detect cycles. Because graphs can have loops, always track visited nodes in a set or boolean array. Add nodes to the visited set at the moment they enter the queue so they are never visited twice.
What each operation costs
| Operation | Time |
|---|---|
| visit all nodes and edges via search | O(v + e) |
| topological sort using in-degree counts | O(v + e) |
| shortest path using dijkstra with a min-heap | O((v + e) log v) |
What usually goes wrong with Graph
- Adding a node to the visited set when popping from the queue instead of when pushing, which lets neighboring nodes enqueue duplicate entries and wastes memory.
- Failing to check for cycles in directed graphs when finding prerequisite orders, causing topological sort routines to hang or return incomplete lists.
- Assuming an input graph is fully connected and scanning from only a single starting node, missing disconnected islands and isolated components.
Which roles need this problem
Graph is a core topic for these 11 roles — if you're targeting one of them, this problem is early in your path, not optional.
Secondary for 6 more roles, including Performance Engineer, Search Engineer, Information Retrieval Engineer.
Companies that have asked it
Tags taken from the problem's own GeeksforGeeks page — not a copied list.
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