Topological Sort (DFS)
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 2
- Core for
- 12 roles
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 = 5, edges = [[0,1],[1,2],[2,3],[3,4]]
- Output
- [0,1,2,3,4]
- Why
- Linear chain 0->1->2->3->4 has unique topological order.
Example 2
- Input
- numVertices = 4, edges = [[0,1],[1,2],[2,3]]
- Output
- [0,1,2,3]
- Why
- Linear chain 0->1->2->3 has unique topological order.
Constraints
- 1 <= numVertices <= 100
- 0 <= edges.length <= numVertices * (numVertices - 1) / 2
How to think about it
Updated 2026-09-29In 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.
More Graph problems
Target Roles
Core requirement for 12 roles:
Track in your role's order
Pick your target role and all 370 problems resequence to what that interview actually asks.
Start freeProblem set and role mapping as of .