Detect Cycle in Undirected 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 an undirected graph, determine whether the graph contains a cycle.
Example 1
- Input
- n = 5, edges = [[0,1],[1,2],[2,3],[3,0]]
- Output
- true
- Why
- The edges form a cycle 0-1-2-3-0.
Example 2
- Input
- n = 4, edges = [[0,1],[1,2],[2,3]]
- Output
- false
- Why
- This is a simple chain with no cycle.
Constraints
- 1 <= n <= 100
- 0 <= edges.length <= 1000
How to think about it
Updated 2026-09-09In an undirected graph, an edge connects both ways. As you traverse, stepping back to the node you just arrived from is just turning around, not a cycle. But reaching any other previously visited vertex proves an alternate path exists between the same endpoints, confirming a cycle.
Approaches, worst first
DFS with parent tracking
time O(V + E) · space O(V)
Maintain a boolean visited array. For each unvisited node, launch DFS passing `(node, parent)`. If any adjacent neighbor is already visited and is not the immediate parent, a cycle is found.
BFS with parent queue
time O(V + E) · space O(V)
Queue tuples of `(current, parent)`. When dequeuing, scan neighbors: if an adjacent node has been visited and differs from the parent, return true immediately.
Disjoint Set UnionWrite this one
time O(V + E * α(V)) · space O(V)
Initialize DSU for n nodes. Iterate through edges: if `find(u) === find(v)`, both endpoints already belong to the same connected component, meaning this edge creates a cycle.
Where people lose marks · 3
- Neglecting disconnected components; you must iterate through every vertex from 0 to n - 1 as a candidate search origin.
- Failing to track the immediate parent, causing the algorithm to treat the edge `u - v` followed by `v - u` as a false cycle.
- Self-loops or parallel edges between the same two nodes must be checked against problem assumptions.
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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