Minimum number of edges required to be removed from an Undirected Graph to make it acyclic

Given an undirected graph consisting of N nodes containing values from the range [1, N] and M edges in a matrix Edges[][], the task is to determine the minimum number of edges required to be removed such that the resulting graph does not contain any cycle.

Examples:

Input: N = 3, M = 3, edges[][] = [[1, 2], [2, 3], [3, 1]]
 

Output: 1
Explanation:
Removing any one of the edges will make the graph acyclic. Therefore, at least one edge needs to be removed.



Input: N = 3, M = 2, edges[][] = [[1, 2], [2, 3]]
 

Output: 0
Explanation: Graph is already acyclic. Therefore, no edge removal is required.

Naive Approach: The simplest approach is to try deleting all possible combination of sequence of edges from the given graph one by one and for each combination, count the number of removals required to make the graph acyclic. Finally, among these combinations, choose the one which deletes the minimum number of edges to obtain an acyclic graph. 
Time Complexity: O(M!) 
Auxiliary Space: O(N + M)

Efficient Approach: The above approach can be optimized based on the following observations:

  1. A graph is acyclic when it is a Tree or a forest of trees(disconnected groups of trees).
  2. A tree with C nodes will have (C – 1) edges.
  3. If there are K connected components from C1 to CK, then minimum number of edges to be removed is equal to:

    M – (C1 – 1) – (C2 – 1) … (Ck -1 ) 
    => M – (C1 + … + CK) + K 
    => M – N + K

Follow the steps below to solve the problem:

  1. Find the number of connected components from the given graph using DFS.
  2. Considering the count of connected components to be K, then print M – N + K as the required minimum number of edges to be removed to make the resulting graph acyclic.

Below is the implementation of the above approach:

C++

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// C++ Program to implement
// the above approach
  
#include <bits/stdc++.h>
using namespace std;
  
// Stores the adjacency list
vector<int> vec[100001];
  
// Stores if a vertex is
// visited or not
bool vis[100001];
int cc = 1;
  
// Function to perform DFS Traversal
// to count the number and size of
// all connected components
void dfs(int node)
{
    // Mark the current node as visited
    vis[node] = true;
  
    // Traverse the adjacency list
    // of the current node
    for (auto x : vec[node]) {
  
        // For every unvisited node
        if (!vis[x]) {
            cc++;
  
            // Recursive DFS Call
            dfs(x);
        }
    }
}
  
// Function to add undirected
// edge in the graph
void addEdge(int u, int v)
{
    vec[u].push_back(v);
    vec[v].push_back(u);
}
  
// Function to calculate minimum
// number of edges to be removed
void minEdgeRemoved(int N, int M,
                    int Edges[][2])
{
  
    // Create Adjacency list
    for (int i = 0; i < M; i++) {
        addEdge(Edges[i][0],
                Edges[i][1]);
    }
  
    memset(vis, false, sizeof(vis));
    int k = 0;
  
    // Iterate over all the nodes
    for (int i = 1; i <= N; i++) {
        if (!vis[i]) {
            cc = 1;
            dfs(i);
            k++;
        }
    }
  
    // Print the final count
    cout << M - N + k << endl;
}
  
// Driver Code
int main()
{
    int N = 3, M = 2;
  
    int Edges[][2] = { { 1, 2 }, { 2, 3 } };
  
    minEdgeRemoved(N, M, Edges);
}

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Java

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// Java program to implement
// the above approach
import java.util.*;
  
class GFG{
  
// Stores the adjacency list
@SuppressWarnings("unchecked")
static Vector<Integer> []vec = new Vector[100001];
  
// Stores if a vertex is
// visited or not
static boolean []vis = new boolean[100001];
static int cc = 1;
  
// Function to perform DFS Traversal
// to count the number and size of
// all connected components
static void dfs(int node)
{
      
    // Mark the current node as visited
    vis[node] = true;
  
    // Traverse the adjacency list
    // of the current node
    for(int x : vec[node])
    {
          
        // For every unvisited node
        if (!vis[x])
        {
            cc++;
  
            // Recursive DFS call
            dfs(x);
        }
    }
}
  
// Function to add undirected
// edge in the graph
static void addEdge(int u, int v)
{
    vec[u].add(v);
    vec[v].add(u);
}
  
// Function to calculate minimum
// number of edges to be removed
static void minEdgeRemoved(int N, int M,
                           int Edges[][])
{
      
    // Create Adjacency list
    for(int i = 0; i < M; i++)
    {
        addEdge(Edges[i][0],
                Edges[i][1]);
    }
  
    int k = 0;
  
    // Iterate over all the nodes
    for(int i = 1; i <= N; i++)
    {
        if (!vis[i])
        {
            cc = 1;
            dfs(i);
            k++;
        }
    }
  
    // Print the final count
    System.out.print(M - N + k + "\n");
}
  
// Driver Code
public static void main(String[] args)
{
    int N = 3, M = 2;
  
    int Edges[][] = { { 1, 2 }, { 2, 3 } };
      
    for(int i = 0; i < vec.length; i++)
        vec[i] = new Vector<Integer>();
          
    minEdgeRemoved(N, M, Edges);
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 program to implement
# the above approach
  
# Stores the adjacency list
vec = [[] for i in range(100001)]
  
# Stores if a vertex is
# visited or not
vis = [False] * 100001
cc = 1
  
# Function to perform DFS Traversal
# to count the number and size of
# all connected components
def dfs(node):
      
    global cc
      
    # Mark the current node as visited
    vis[node] = True
  
    # Traverse the adjacency list
    # of the current node
    for x in vec[node]:
  
        # For every unvisited node
        if (vis[x] == 0):
            cc += 1
  
            # Recursive DFS Call
            dfs(x)
  
# Function to add undirected
# edge in the graph
def addEdge(u, v):
      
    vec[u].append(v)
    vec[v].append(u)
  
# Function to calculate minimum
# number of edges to be removed
def minEdgeRemoved(N, M, Edges):
      
    global cc
  
    # Create Adjacency list
    for i in range(M):
        addEdge(Edges[i][0], Edges[i][1])
  
    # memset(vis, false, sizeof(vis))
    k = 0
  
    # Iterate over all the nodes
    for i in range(1, N + 1):
        if (not vis[i]):
            cc = 1
            dfs(i)
            k += 1
  
    # Print the final count
    print(M - N + k)
  
# Driver Code
if __name__ == '__main__':
      
    N = 3
    M = 2
  
    Edges = [ [ 1, 2 ], [ 2, 3 ] ]
  
    minEdgeRemoved(N, M, Edges)
  
# This code is contributed by mohit kumar 29

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C#

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// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
  
class GFG{
  
// Stores the adjacency list
static List<int> []vec = new List<int>[100001];
  
// Stores if a vertex is
// visited or not
static bool []vis = new bool[100001];
static int cc = 1;
  
// Function to perform DFS Traversal
// to count the number and size of
// all connected components
static void dfs(int node)
{
      
    // Mark the current node as visited
    vis[node] = true;
  
    // Traverse the adjacency list
    // of the current node
    foreach(int x in vec[node])
    {
          
        // For every unvisited node
        if (!vis[x])
        {
            cc++;
  
            // Recursive DFS call
            dfs(x);
        }
    }
}
  
// Function to add undirected
// edge in the graph
static void addEdge(int u, int v)
{
    vec[u].Add(v);
    vec[v].Add(u);
}
  
// Function to calculate minimum
// number of edges to be removed
static void minEdgeRemoved(int N, int M,
                           int [,]Edges)
{
      
    // Create Adjacency list
    for(int i = 0; i < M; i++)
    {
        addEdge(Edges[i, 0],
                Edges[i, 1]);
    }
  
    int k = 0;
  
    // Iterate over all the nodes
    for(int i = 1; i <= N; i++)
    {
        if (!vis[i])
        {
            cc = 1;
            dfs(i);
            k++;
        }
    }
  
    // Print the readonly count
    Console.Write(M - N + k + "\n");
}
  
// Driver Code
public static void Main(String[] args)
{
    int N = 3, M = 2;
  
    int [,]Edges = { { 1, 2 }, { 2, 3 } };
      
    for(int i = 0; i < vec.Length; i++)
        vec[i] = new List<int>();
          
    minEdgeRemoved(N, M, Edges);
}
}
  
// This code is contributed by 29AjayKumar

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Output: 

0

Time Complexity: O(N + M)
Auxiliary Space: O(N + M)

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