Topological Sort (DFS)
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 a directed acyclic graph with numVertices vertices and a list of directed edges, return a topological ordering of the vertices using depth-first search.
Example 1
- Input
- numVertices = 4, edges = [[1,0],[2,0],[3,1],[3,2]]
- Output
- [3,1,2,0]
- Why
- Vertex 3 has no dependencies, then 1 and 2, then 0 depends on both 1 and 2.
Example 2
- Input
- numVertices = 2, edges = [[1,0]]
- Output
- [1,0]
- Why
- Vertex 1 must come before vertex 0 in topological order.
Constraints
- 1 <= numVertices <= 100
- 0 <= edges.length <= numVertices * (numVertices - 1) / 2
How to think about it
Updated 2026-09-09In a DAG, a vertex cannot finish processing until all nodes it points to have completely finished theirs. This post-order traversal naturally completes vertices in reverse topological order: the deepest sink finishes first, and the ultimate source finishes last. Pushing to a stack on exit and reading top-to-bottom yields the dependency order.
Approaches, worst first
Repeated in-degree zero source scanning
time O(V^2 + E) · space O(V + E)
Compute in-degrees for all vertices. Repeatedly scan the entire vertex list to find any vertex with zero in-degree, append it to the order, mark it removed, and subtract incoming edges from its neighbors. Scanning all vertices on every step wastes quadratic time compared to stack tracking.
DFS post-order stack orderingWrite this one
time O(V + E) · space O(V + E)
Maintain a boolean visited array and a recursion stack or list. Run a recursive DFS from each unvisited vertex. After visiting all outgoing edges of vertex u, append u to the list. Reversing the list at the end gives the topological ordering.
Where people lose marks · 3
- Pushing the vertex onto the output stack upon first entering DFS instead of during backtracking post-order produces an invalid ordering.
- Disconnected graph components will be skipped if the loop over all vertices 0 to numVertices - 1 is omitted.
- Interpreting edge pairs [u, v] backwards; verify whether an edge represents `u -> v` or `v -> u` before constructing adjacency lists.
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.
Track this in your role's order
Pick your target role and all 370 problems — including this one — resequence to what that interview actually asks. Free.
Start freeMore Graph problems
Problem set and role mapping as of .