Pattern this teaches:Topological Sort
Find Eventual Safe States
A medium Graph interview guide. Task statement, worked examples, intuition, and step-by-step solutions.
- Topic
- Graph
- Sheets
- 1
- Core for
- 12 roles
The problem
Given a directed graph with n nodes labeled from 0 to n-1 and a list of edges, find all nodes that are eventually safe. A node is eventually safe if every possible path starting from it leads to a terminal node (a node with no outgoing edges).
Example 1
- Input
- n = 7, edges = [[0,1],[0,2],[1,2],[2,3],[3,4],[2,5],[5,6]]
- Output
- [0,1,2,3,4,5,6]
- Why
- All nodes eventually reach terminal nodes; none are in a cycle.
Example 2
- Input
- n = 4, edges = [[0,1],[0,2],[1,3],[2,3]]
- Output
- [0,1,2,3]
- Why
- Every node eventually reaches node 3 which has no outgoing edges.
Constraints
- 1 <= n <= 10^4
- 0 <= edges.length <= 10^4
How to think about it
Updated 2026-09-29A node is unsafe if and only if it can reach a directed cycle. Reversing all edges turns terminal nodes (out-degree 0) into source nodes (in-degree 0). Running Kahn's algorithm on this reversed graph peels away nodes whose outgoing paths can only lead to terminal states, leaving behind only nodes trapped in or leading to cycles.
Approaches, worst first
DFS three-color cycle detection
time O(V + E) · space O(V + E)
Color nodes: 0 for unvisited, 1 for active on recursion stack, 2 for verified safe. If DFS encounters a node colored 1, a cycle is detected. If all outgoing edges lead to nodes eventually colored 2, mark current as 2. Collect all nodes colored 2.
Reverse graph topological peelWrite this one
time O(V + E) · space O(V + E)
Reverse every directed edge and compute out-degrees in the original graph (in-degrees in reversed). Enqueue all nodes with original out-degree 0. Dequeue nodes, decrement original out-degrees of their predecessors, and enqueue when 0. Sort the collected safe nodes.
Where people lose marks · 3
- The problem requires returning safe nodes sorted in ascending order; DFS discovery order will not be sorted naturally without an explicit sort or sequential collection.
- A node with no outgoing edges at all is an immediate terminal node and is always safe.
- Not caching unsafe results; re-exploring components from scratch leads to exponential backtracking.
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.
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