Bridges in Graph
A hard Graph problem included in Striver A2Z. Below: the roles whose interviews prioritise this topic, and how to practise it.
- Topic
- Graph
- Sheets
- 1
- Core for
- 11 roles
- Platform
- LeetCode
The problem
Given an undirected connected graph with n nodes and a list of edges, find all critical connections (bridges) — edges whose removal increases the number of connected components.
Example 1
- Input
- n = 4, edges = [[0,1],[1,2],[2,0],[1,3]]
- Output
- [[1,3]]
- Why
- Edge [1,3] is the only bridge. Removing it disconnects node 3 from the rest.
Example 2
- Input
- n = 2, edges = [[0,1]]
- Output
- [[0,1]]
- Why
- The single edge is a bridge.
Constraints
- 1 <= n <= 10^5
- 0 <= edges.length <= 2 * 10^5
How to think about it
Updated 2026-09-09An edge (u, v) is a bridge if and only if no back-edge exists anywhere in v's subtree that can reach u or an ancestor of u. By tracking discovery timestamp `tin[u]` and the earliest reachable timestamp `low[v]` during a depth-first search tree traversal, the condition `low[v] > tin[u]` identifies a bridge in a single pass without needing to simulate edge removals.
Approaches, worst first
Individual edge removal simulation
time O(E * (V + E)) · space O(V + E)
Iterate through every edge, temporarily delete it from the graph, and execute a BFS or DFS traversal from node 0 to check whether all n vertices remain reachable. If the visited count drops below n, the edge is critical. Testing each edge with a separate traversal costs quadratic time.
Tarjan bridge-finding DFSWrite this one
time O(V + E) · space O(V + E)
Assign each node a discovery time `tin[u]`. Let `low[u]` be the minimum discovery time reachable via at most one back-edge. In DFS, for every neighbor v != parent: if unvisited, recurse and update `low[u] = min(low[u], low[v])`. If `low[v] > tin[u]`, record `[u, v]` as a critical connection.
Where people lose marks · 3
- Passing the immediate parent vertex during DFS is essential to avoid treating the bidirectional tree edge `u - parent` as a back-edge.
- The strict inequality `low[v] > tin[u]` is required for bridges; using `>=` falsely flags edges in simple cycles.
- Parallel edges between the same two nodes require tracking edge IDs or count of connections, because traversing a duplicate edge is a valid alternative path.
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.
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