feat: add Articulation Points and Bridges algorithm

[Pasan11504]

📝 Add Articulation Points and Bridges Algorithm

Description

This PR implements an algorithm to find Articulation Points (Cut Vertices) and Bridges (Cut Edges) in undirected graphs using a DFS-based approach with discovery times and low-link values.

Articulation Point: A vertex whose removal increases the number of connected components.
Bridge: An edge whose removal increases the number of connected components.

Features

• ✅ Finds articulation points in O(V + E) time
• ✅ Finds bridges in O(V + E) time
• ✅ Single DFS traversal for efficiency
• ✅ Handles disconnected graphs correctly
• ✅ Convenience methods for finding only points or only bridges
• ✅ Comprehensive Javadoc with examples and applications
• ✅ Extensive test coverage with 15 test cases

Algorithm Details

Time Complexity: O(V + E)
Space Complexity: O(V)

Detection Rules:

• Root vertex is an articulation point if it has more than 1 child in DFS tree
• Non-root vertex u is an articulation point if it has a child v where low[v] >= discovery[u]
• Edge (u,v) is a bridge if low[v] > discovery[u]

Test Coverage

The test suite covers:

• Simple graphs with single/multiple articulation points
• Graphs with no articulation points (biconnected components)
• Star graphs (center as articulation point)
• Disconnected graphs
• Edge cases (empty graph, single vertex)
• Edge equality and symmetry validation
• Exception handling for invalid inputs

Real-World Applications

• 🌐 Network reliability analysis (finding critical routers/connections)
• 👥 Social network analysis (identifying influential individuals)
• ⚡ Circuit design (locating critical components)
• 🚗 Transportation networks (finding critical roads/bridges)

Files Changed

• src/main/java/com/thealgorithms/datastructures/graphs/ArticulationPointsAndBridges.java - Main implementation (270 lines)
• src/test/java/com/thealgorithms/datastructures/graphs/ArticulationPointsAndBridgesTest.java - Test suite (371 lines)

Checklist

• I have read the contributing guidelines
• The code follows the repository's style guidelines
• I have performed a self-review of my code
• I have commented my code, particularly in hard-to-understand areas
• I have added comprehensive tests
• The algorithm is not a duplicate of existing implementations

References

• Wikipedia: Biconnected Component
• CP-Algorithms: Bridge Finding
• GeeksforGeeks: Articulation Points

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Hacktoberfest 2025 🎃

Example Usage

// Create graph
int vertices = 5;
List<List<Integer>> graph = new ArrayList<>();
for (int i = 0; i < vertices; i++) {
    graph.add(new ArrayList<>());
}

// Add edges
graph.get(0).add(1); graph.get(1).add(0);
graph.get(1).add(2); graph.get(2).add(1);
graph.get(2).add(3); graph.get(3).add(2);

// Find articulation points and bridges
Result result = ArticulationPointsAndBridges.findArticulationPointsAndBridges(vertices, graph);
System.out.println(result.getArticulationPoints()); // [1, 2]
System.out.println(result.getBridges()); // [(0,1), (1,2), (2,3)]