Given a connected graph, the task is to find the minimum number of edges to cut/remove to make the given graph disconnected.

A graph is disconnected if there exists at least two vertices of the graph that are not connected by a path.

**Examples:**

Input:

Output:Minimum Number of Edges to Remove = 2

**Approach:**

The approach to this problem is to find the number of paths in the edge disjoint set of paths from a start vertex to an end vertex in the graph. Edge-disjoint set of paths is a set of paths having no common edge between any two paths.

- Pick one node as start node and another node as end node.
- Start BFS from the start node and check if a path exists from start node to end node.
- If yes, then remove all the edges from that path and run BFS again.
- Repeat step 2 and 3 until there’s no path exists from start node to end node.
- Return the number of times a path is deleted.

**Explanation with example:**

**Code:**

## Java

`// Java Program to Find` `// Minimum Number of Edges` `// to Cut to make the` `// Graph Disconnected` ` ` `import` `java.util.*;` ` ` `public` `class` `GFG {` ` ` ` ` `// Function to find the min number of edges` ` ` `public` `static` `int` `minEdgesRemoval(` `int` `[][] edges, ` `int` `n)` ` ` `{` ` ` `// Intialize adjacency list for Graph` ` ` `Map<Integer, List<Integer> > graph` ` ` `= ` `new` `HashMap<Integer, List<Integer> >();` ` ` ` ` `// Initializing starting and ending vertex` ` ` `int` `start = edges[` `0` `][` `0` `];` ` ` `int` `end = edges[` `0` `][` `1` `];` ` ` ` ` `// Create adjacency list of the graph` ` ` `for` `(` `int` `i = ` `0` `; i < n; i++) {` ` ` `int` `n1 = edges[i][` `0` `];` ` ` `int` `n2 = edges[i][` `1` `];` ` ` `List<Integer> li;` ` ` ` ` `// Add edges node 1` ` ` `if` `(graph.containsKey(n1)) {` ` ` `li = graph.get(n1);` ` ` `}` ` ` `else` `{` ` ` `li = ` `new` `ArrayList<Integer>();` ` ` `}` ` ` ` ` `li.add(n2);` ` ` `graph.put(n1, li);` ` ` ` ` `// Add edges node 2` ` ` `if` `(graph.containsKey(n2)) {` ` ` `li = graph.get(n2);` ` ` `}` ` ` `else` `{` ` ` `li = ` `new` `ArrayList<Integer>();` ` ` `}` ` ` ` ` `li.add(n1);` ` ` `graph.put(n2, li);` ` ` `}` ` ` ` ` `// Variable to count the number of paths getting` ` ` `// deleted` ` ` `int` `deleteEdgeCount = ` `0` `;` ` ` ` ` `while` `(` `true` `) {` ` ` ` ` `// bfsTraversalPath is the BFS path from start` ` ` `// to end node It is a map of parent vertex and` ` ` `// child vertex` ` ` `Map<Integer, Integer> bfsTraversalPath = bfs(graph, start);` ` ` ` ` `// If end is present on the path from start node` ` ` `// then delete that path and increment` ` ` `// deleteEdgeCount` ` ` `if` `(bfsTraversalPath.containsKey(end)) {` ` ` ` ` `deleteEdgeCount++;` ` ` `int` `parent = bfsTraversalPath.get(end);` ` ` `int` `currEnd = end;` ` ` ` ` `// Delete all the edges in the current path` ` ` `while` `(parent != -` `1` `)` ` ` `{` ` ` `deleteEdge(graph, parent, currEnd);` ` ` `deleteEdge(graph, currEnd, parent);` ` ` `currEnd = parent;` ` ` `parent = bfsTraversalPath.get(currEnd);` ` ` `}` ` ` `}` ` ` ` ` `// If end is not present in the path` ` ` `// then we have a disconnected graph.` ` ` `else` `{` ` ` `break` `;` ` ` `}` ` ` `}` ` ` ` ` `return` `deleteEdgeCount;` ` ` `}` ` ` ` ` `// Function to delete/remove an edge` ` ` `private` `static` `void` `deleteEdge(Map<Integer, List<Integer> > graph,` ` ` `Integer start, Integer end)` ` ` `{` ` ` ` ` `List<Integer> list = graph.get(start);` ` ` `list.remove(end);` ` ` `}` ` ` ` ` `// Function for BFS Path` ` ` `private` `static` `Map<Integer, Integer>` ` ` `bfs(Map<Integer, List<Integer> > graph, ` `int` `start)` ` ` `{` ` ` ` ` `// Map for BFS Path` ` ` `Map<Integer, Integer> bfsTraversalPath` ` ` `= ` `new` `HashMap<Integer, Integer>();` ` ` ` ` `// Array for marking visited vertex` ` ` `List<Integer> visited = ` `new` `ArrayList<Integer>();` ` ` ` ` `// Array for BFS` ` ` `List<Integer> queue = ` `new` `ArrayList<Integer>();` ` ` ` ` `int` `qStartIndex = ` `0` `;` ` ` ` ` `bfsTraversalPath.put(start, -` `1` `);` ` ` `queue.add(start);` ` ` ` ` `while` `(qStartIndex < queue.size()) ` ` ` `{` ` ` `int` `curr = queue.get(qStartIndex++);` ` ` `visited.add(curr);` ` ` ` ` `for` `(` `int` `k : graph.get(curr))` ` ` `{` ` ` `if` `(!visited.contains(k)) ` ` ` `{` ` ` `queue.add(k);` ` ` `if` `(!bfsTraversalPath.containsKey(k))` ` ` `{` ` ` `bfsTraversalPath.put(k, curr);` ` ` `}` ` ` `}` ` ` `}` ` ` `}` ` ` ` ` `return` `bfsTraversalPath;` ` ` `}` ` ` ` ` `// Driver Code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` ` ` `// Number of edges` ` ` `int` `n = ` `7` `;` ` ` ` ` `// Edge List` ` ` `int` `[][] edges` ` ` `= { { ` `0` `, ` `1` `}, { ` `1` `, ` `2` `}, { ` `2` `, ` `3` `}, { ` `3` `, ` `4` `},` ` ` `{ ` `4` `, ` `0` `}, { ` `4` `, ` `1` `}, { ` `1` `, ` `3` `} };` ` ` ` ` `// Run the function` ` ` `System.out.println(` `"Minimum Number of Edges to Remove = "` ` ` `+ minEdgesRemoval(edges, n));` ` ` `}` `}` |

**Output**

Minimum Number of Edges to Remove = 2

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