Binary TreesBinary Trees36
  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Inorder Traversal of a Binary Tree using Recursion
  4. 4Inorder Traversal of a Binary Tree using Iteration
  5. 5Postorder Traversal of a Binary Tree Using Recursion
  6. 6Postorder Traversal of a Binary Tree using Iteration
  7. 7Level Order Traversal of a Binary Tree using Recursion
  8. 8Level Order Traversal of a Binary Tree using Iteration
  9. 9Reverse Level Order Traversal of a Binary Tree using Iteration
  10. 10Reverse Level Order Traversal of a Binary Tree using Recursion
  11. 11Find Height of a Binary Tree
  12. 12Find Diameter of a Binary Tree
  13. 13Find Mirror of a Binary Tree
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree
GraphsGraphs46
  1. 1Breadth-First Search in Graphs
  2. 2Depth-First Search in Graphs
  3. 3Number of Provinces in an Undirected Graph
  4. 4Connected Components in a Matrix
  5. 5Rotten Oranges Problem - BFS in Matrix
  6. 6Flood Fill Algorithm - Graph Based
  7. 7Detect Cycle in an Undirected Graph using DFS
  8. 8Detect Cycle in an Undirected Graph using BFS
  9. 9Distance of Nearest Cell Having 1 - Grid BFS
  10. 10Surrounded Regions in Matrix using Graph Traversal
  11. 11Number of Enclaves in Grid
  12. 12Word Ladder - Shortest Transformation using Graph
  13. 13Word Ladder II - All Shortest Transformation Sequences
  14. 14Number of Distinct Islands using DFS
  15. 15Check if a Graph is Bipartite using DFS
  16. 16Topological Sort Using DFS
  17. 17Topological Sort using Kahn's Algorithm
  18. 18Cycle Detection in Directed Graph using BFS
  19. 19Course Schedule - Task Ordering with Prerequisites
  20. 20Course Schedule 2 - Task Ordering Using Topological Sort
  21. 21Find Eventual Safe States in a Directed Graph
  22. 22Alien Dictionary Character Order
  23. 23Shortest Path in Undirected Graph with Unit Distance
  24. 24Shortest Path in DAG using Topological Sort
  25. 25Dijkstra's Algorithm Using Set - Shortest Path in Graph
  26. 26Dijkstra’s Algorithm Using Priority Queue
  27. 27Shortest Distance in a Binary Maze using BFS
  28. 28Path With Minimum Effort in Grid using Graphs
  29. 29Cheapest Flights Within K Stops - Graph Problem
  30. 30Number of Ways to Reach Destination in Shortest Time - Graph Problem
  31. 31Minimum Multiplications to Reach End - Graph BFS
  32. 32Bellman-Ford Algorithm for Shortest Paths
  33. 33Floyd Warshall Algorithm for All-Pairs Shortest Path
  34. 34Find the City With the Fewest Reachable Neighbours
  35. 35Minimum Spanning Tree in Graphs
  36. 36Prim's Algorithm for Minimum Spanning Tree
  37. 37Disjoint Set (Union-Find) with Union by Rank and Path Compression
  38. 38Kruskal's Algorithm - Minimum Spanning Tree
  39. 39Minimum Operations to Make Network Connected
  40. 40Most Stones Removed with Same Row or Column
  41. 41Accounts Merge Problem using Disjoint Set Union
  42. 42Number of Islands II - Online Queries using DSU
  43. 43Making a Large Island Using DSU
  44. 44Bridges in Graph using Tarjan's Algorithm
  45. 45Articulation Points in Graphs
  46. 46Strongly Connected Components using Kosaraju's Algorithm

Articulation Points in Undirected Graphs

Problem Statement

In an undirected connected graph, an articulation point (or cut vertex) is a node that, when removed (along with its edges), increases the number of connected components in the graph.

Problem: Given an undirected graph with V vertices and an adjacency list adj, find all articulation points.

  • Indexing is 0-based, i.e., nodes are numbered from 0 to V-1.
  • There may be self-loops and multiple edges, but you should still identify articulation points based on connectivity.

Examples

Graph (Adjacency List) Articulation Points Description
{0:[1,2], 1:[0,2], 2:[0,1,3], 3:[2,4,5], 4:[3], 5:[3]} [2,3] Removing node 2 or 3 disconnects parts of the graph
{0:[1], 1:[0,2], 2:[1,3], 3:[2]} [1,2] Removing nodes 1 or 2 disconnects the chain
{0:[1,2], 1:[0], 2:[0]} [0] Node 0 connects two leaf nodes
{0:[1], 1:[0]} [] Two nodes connected by a single edge, neither is an articulation point
{0:[1,2], 1:[0,2], 2:[0,1]} [] Triangle graph, fully connected — no articulation points

Visualization Player

Solution

Understanding the Problem

We are given an undirected graph. Our goal is to find all the articulation points in this graph. An articulation point (or cut vertex) is a node that, if removed, increases the number of connected components in the graph.

In simple terms, these are the critical nodes that hold parts of the graph together. If any of them are removed, some nodes may no longer be reachable from others.

This is especially useful in network design and fault-tolerant systems where you want to identify single points of failure.

Step-by-Step Solution with Example

step 1: Represent the graph

We begin by representing the graph using an adjacency list. For example, let's take this graph:

{
  0: [1, 2],
  1: [0, 2],
  2: [0, 1, 3, 5],
  3: [2, 4],
  4: [3],
  5: [2, 6, 8],
  6: [5, 7],
  7: [6],
  8: [5]
}

This graph has 9 nodes (0 to 8), and is connected in such a way that removing some nodes would disconnect parts of it.

step 2: Perform DFS traversal and track timestamps

We use Depth-First Search (DFS) to explore the graph and keep track of:

  • tin[u]: The discovery time of node u
  • low[u]: The lowest discovery time reachable from u or any of its descendants

We initialize both of these arrays with -1 and use a global timer to assign timestamps as we go.

step 3: Apply articulation point conditions during DFS

While traversing each node u, we apply the following logic:

  • Case 1 (Root): If u is the root of DFS and has more than one child, then u is an articulation point.
  • Case 2 (Bridge): If u is not root and there exists a child v such that low[v] >= tin[u], then u is an articulation point.

step 4: Update low values properly

After visiting a neighbor v of u, we update low[u] based on:

  • If v is a child: low[u] = min(low[u], low[v])
  • If v is already visited and is not parent of u: low[u] = min(low[u], tin[v])

step 5: Collect all articulation points

We use a boolean array isArticulation[u] to mark nodes that satisfy the above conditions. After DFS is complete, we collect all nodes marked as articulation points.

Edge Cases

  • Single Node: A graph with only one node has no articulation points.
  • Disconnected Graph: Apply DFS from every unvisited node to ensure all components are covered.
  • Tree Graph: In a tree, all non-leaf nodes (except the root with one child) can be articulation points.
  • Cyclic Graph: Presence of back edges affects the low values and prevents over-marking articulation points.

Finally

The key to solving this problem is understanding how DFS works in combination with discovery and low times. By tracking when we visit a node and the lowest point reachable from it, we can detect where the graph would split if a node is removed.

This method is efficient (O(V + E)) and elegant, using classic DFS with a few smart checks to find critical points in any connected or disconnected undirected graph.

Algorithm Steps

  1. Initialize tin[], low[], visited[], and timer to track DFS discovery times.
  2. Perform DFS traversal starting from any node.
  3. For each unvisited child v of u:
    1. Call DFS recursively on v.
    2. Update low[u] = min(low[u], low[v]).
    3. Check articulation conditions for u.
  4. For visited child v (not parent), update low[u] = min(low[u], tin[v]).
  5. Collect all articulation points from marked vertices.

Code

JavaScript
function findArticulationPoints(V, adj) {
  let timer = 0;
  const tin = new Array(V).fill(-1);
  const low = new Array(V).fill(-1);
  const visited = new Array(V).fill(false);
  const isArticulation = new Array(V).fill(false);

  function dfs(u, parent) {
    visited[u] = true;
    tin[u] = low[u] = timer++;
    let children = 0;

    for (const v of adj[u]) {
      if (v === parent) continue;
      if (visited[v]) {
        low[u] = Math.min(low[u], tin[v]);
      } else {
        dfs(v, u);
        low[u] = Math.min(low[u], low[v]);
        if (low[v] >= tin[u] && parent !== -1) {
          isArticulation[u] = true;
        }
        children++;
      }
    }

    if (parent === -1 && children > 1) {
      isArticulation[u] = true;
    }
  }

  for (let i = 0; i < V; i++) {
    if (!visited[i]) {
      dfs(i, -1);
    }
  }

  const result = [];
  for (let i = 0; i < V; i++) {
    if (isArticulation[i]) result.push(i);
  }
  return result;
}

const graph = {
  0: [1, 2],
  1: [0, 2],
  2: [0, 1, 3],
  3: [2, 4, 5],
  4: [3],
  5: [3]
};

const V = Object.keys(graph).length;
console.log("Articulation Points:", findArticulationPoints(V, graph));

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