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Cycle Detection

Determines whether a graph contains a cycle.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Graph Traversal
Difficulty: Beginner
Family: Graph Analysis
Tier: Core
Status: Verified
Review: Source Checked
Sources: 2
Formulas: 0
Relationships: 3
Code example: Ready
Visualization: Ready

Overview

Cycle Detection uses traversal state to find edges that prove a loop exists in a directed or undirected graph.

Core Concepts

graph representationcycle statetraversal ordergraph analysisvisited setfrontier expansion

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(V + E)

Average

O(V + E)

Worst

O(V + E)

Space

O(V)

When to Use

Use Cycle Detection when you need to track traversal state.
Use it for dependency validation with inputs like graph.
Use it when relationships are naturally modeled as nodes connected by edges.
Use it when the Graph Analysis tradeoff is acceptable for the problem size.

When Not to Use

Avoid it when the graph is so large that storing every visited node is not practical.
Avoid it when the required output is not cycle found result.

What It Does

Tracks traversal state
Reports when an edge closes a loop

Use Cases

Dependency validation
Deadlock detection
Graph sanity checks

Inputs

Graph

Outputs

Cycle found result
Optional cycle path

Pseudocode

1.Visit nodes with DFS
2.Track nodes currently on the recursion stack
3.If an edge points to an active node, report a cycle
4.Mark nodes complete as DFS returns
5.Continue until all nodes are checked

Canonical Example

Input: graph = {A: [B, C], B: [D], C: [], D: []}, start = A
Output: visited = [A, B, C, D]
Explanation: Cycle Detection: The traversal starts at A and visits reachable nodes by following edges.

Implementation Notes

Keep the input contract explicit: Graph.
Use this step as the implementation spine: Visit nodes with DFS.
Then preserve the next state change: Track nodes currently on the recursion stack.
Keep node ids, edge direction, and visited state explicit in the implementation.
Represent neighbors consistently and keep traversal state separate from the graph data.

Common Pitfalls

Directed and undirected graphs use slightly different cycle checks.
Not defining directed versus undirected edges clearly can change the result.
Forgetting to mark visited nodes before revisiting cycles or shared neighbors.
Returning cycle found result without handling empty, invalid, or boundary inputs.

Interactive Mechanism Walkthrough

Graph Frontier & Edge Relaxation

Show a back edge closing a loop in the graph.

Step 1 of 5: Visit nodes with DFS

cycle-highlight

Notes

Cycle Detection operates in O(V + E) time and O(V) space within the Graph Analysis family. Often compared with Depth-First Search and Topological Sort.