Given an undirected graph **G** with **N** nodes, **M** edges, and an integer **K**, the task is to find the maximum count of edges that can be removed such that there remains exactly **K** connected components after the removal of edges. If the graph cannot contain **K **connect components, print **-1**.

**Examples:**

Input:N = 4, M = 3, K = 2, Edges[][] = {{1, 2}, {2, 3}, {3, 4}}

Output:1Explanation:

One possible way is to remove edge [1, 2]. Then there will be 2 connect components as shown below:

Input:N = 3, M = 3, K = 3, Edges[][] = {{1, 2}, {2, 3}, {3, 1}}

Output:3Explanation:All edges can be removed to make 3 connected components as shown below:

**Approach:** To solve the given problem, count the number of connected components present in the given graph. Let the count be **C**. Observe that if **C** is greater than **K** then no possible edge removal can generate **K** connected components as the number of connected components will only increase. Otherwise, the answer will always exist.

Following observations need to be made in order to solve the problem:

- Suppose C
_{1}, C_{2}, …, C_{c}, are the number of node in each connected component. Then, each component must have edges as C_{1 }– 1, C_{2 }– 1, …, C_{c }-1 after edges are removed. Therefore,

C, where_{1}– 1 + C_{2}– 1 + … + C_{c}– 1 = C_{1}+ C_{2}+ … + C_{c}– C = N – CNis the number of nodes.

- The above condition will give us the
**C**connected components by removing**M – (N – C)**edges as**N – C**edges are needed to make**C**components. To get**K**components,**(K – C)**more edges must be removed. - Hence, the total count of edges to be removed is given by:

M – (N – C) + (K – C) = M – N + K

Follow the steps below to solve the problem:

- Count the number of connected components present in the given graph. Let the count be
**C**. - If
**C**is greater than**K**, print**-1**. - Else print
**M – N + K**where**N**is the number f nodes,**M**is the number of edges and**K**is the required number of connected components.

Below is the implementation of the above approach:

## C++

`// C++ program for the above approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `class` `Graph ` `{` ` ` `public` `:` ` ` `int` `V;` ` ` `map<` `int` `, vector<` `int` `>> adj;` ` ` `Graph(` `int` `);` ` ` `void` `addEdge(` `int` `, ` `int` `);` ` ` `void` `DFS(` `int` `, vector<` `bool` `> &);` `} * g;` `// Constructor` `Graph::Graph(` `int` `V) ` `{` ` ` ` ` `// No. of vertices` ` ` `this` `->V = V;` ` ` ` ` `// Dictionary of lists` ` ` `for` `(` `int` `i = 1; i <= V; i++) ` ` ` `adj[i] = vector<` `int` `>();` `}` `// Function to add edge` `// in the graph` `void` `Graph::addEdge(` `int` `v, ` `int` `w)` `{` ` ` `adj[v].push_back(w);` ` ` `adj[w].push_back(v);` `}` `// Function to perform DFS` `void` `Graph::DFS(` `int` `s, vector<` `bool` `> &visited) ` `{` ` ` ` ` `// Create a stack for DFS` ` ` `stack<` `int` `> stack;` ` ` `// Push the current source node` ` ` `stack.push(s);` ` ` `while` `(!stack.empty()) ` ` ` `{` ` ` ` ` `// Pop a vertex from stack` ` ` `// and print it` ` ` `s = stack.top();` ` ` `stack.pop();` ` ` `// Traverse adjacent vertices` ` ` `// of the popped vertex s` ` ` `for` `(` `auto` `node : adj[s])` ` ` `{` ` ` `if` `(!visited[node]) ` ` ` `{` ` ` ` ` `// If adjacent is unvisited,` ` ` `// push it to the stack` ` ` `visited[node] = ` `true` `;` ` ` `stack.push(node);` ` ` `}` ` ` `}` ` ` `}` `}` `// Function to return the count` `// edges removed` `void` `countRemovedEdges(` `int` `N, ` `int` `M, ` `int` `K)` `{` ` ` `int` `C = 0;` ` ` `// Initially mark all verices` ` ` `// as not visited` ` ` `vector<` `bool` `> visited(g->V + 1, ` `false` `);` ` ` `for` `(` `int` `node = 1; node <= N; node++) ` ` ` `{` ` ` ` ` `// If node is unvisited` ` ` `if` `(!visited[node])` ` ` `{` ` ` ` ` `// Increment Connected` ` ` `// component count by 1` ` ` `C = C + 1;` ` ` `// Perform DFS Traversal` ` ` `g->DFS(node, visited);` ` ` `// Print the result` ` ` `if` `(C <= K)` ` ` `cout << M - N + K << endl;` ` ` `else` ` ` `cout << -1 << endl;` ` ` `}` ` ` `}` `}` `// Driver Code` `int` `main(` `int` `argc, ` `char` `const` `*argv[]) ` `{` ` ` `int` `N = 4, M = 3, K = 2;` ` ` `// Create Graph` ` ` `g = ` `new` `Graph(N);` ` ` `// Given Edges` ` ` `g->addEdge(1, 2);` ` ` `g->addEdge(2, 3);` ` ` `g->addEdge(3, 4);` ` ` `// Function Call` ` ` `countRemovedEdges(N, M, K);` `}` `// This code is contributed by sanjeev2552` |

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## Python3

`# Python3 program for the above approach` `class` `Graph:` ` ` `# Constructor` ` ` `def` `__init__(` `self` `, V):` ` ` `# No. of vertices` ` ` `self` `.V ` `=` `V` ` ` `# Dictionary of lists` ` ` `self` `.adj ` `=` `{i: [] ` `for` `i ` `in` `range` `(` `1` `, V ` `+` `1` `)}` ` ` `# Function to add edge` ` ` `# in the graph` ` ` `def` `addEdge(` `self` `, v, w):` ` ` `self` `.adj[v].append(w)` ` ` `self` `.adj[w].append(v)` ` ` `# Function to perform DFS` ` ` `def` `DFS(` `self` `, s, visited):` ` ` `# Create a stack for DFS` ` ` `stack ` `=` `[]` ` ` `# Push the current source node` ` ` `stack.append(s)` ` ` `while` `(` `len` `(stack)):` ` ` `# Pop a vertex from stack` ` ` `# and print it` ` ` `s ` `=` `stack[` `-` `1` `]` ` ` `stack.pop()` ` ` `# Traverse adjacent vertices` ` ` `# of the popped vertex s` ` ` `for` `node ` `in` `self` `.adj[s]:` ` ` `if` `(` `not` `visited[node]):` ` ` `# If adjacent is unvisited,` ` ` `# push it to the stack` ` ` `visited[node] ` `=` `True` ` ` `stack.append(node)` `# Function to return the count ` `# edges removed` `def` `countRemovedEdges(N, M, K):` ` ` `C ` `=` `0` ` ` `# Initially mark all verices` ` ` `# as not visited` ` ` `visited ` `=` `[` `False` `for` `i ` `in` `range` `(g.V ` `+` `1` `)]` ` ` `for` `node ` `in` `range` `(` `1` `, N ` `+` `1` `):` ` ` `# If node is unvisited` ` ` `if` `(` `not` `visited[node]):` ` ` `# Increment Connected` ` ` `# component count by 1` ` ` `C ` `=` `C ` `+` `1` ` ` `# Perform DFS Traversal` ` ` `g.DFS(node, visited)` ` ` `# Print the result` ` ` `if` `C <` `=` `K:` ` ` `print` `(M ` `-` `N ` `+` `K)` ` ` `else` `:` ` ` `print` `(` `-` `1` `)` `# Driver Code` `N, M, K ` `=` `4` `, ` `3` `, ` `2` `# Create Graph` `g ` `=` `Graph(N)` `# Given Edges` `g.addEdge(` `1` `, ` `2` `)` `g.addEdge(` `2` `, ` `3` `)` `g.addEdge(` `3` `, ` `4` `)` `# Function Call` `countRemovedEdges(N, M, K)` |

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**Output:**

1

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

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