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Minimize the number of weakly connected nodes
• Difficulty Level : Hard
• Last Updated : 31 Dec, 2020

Given an undirected graph, task is to find the minimum number of weakly connected nodes after converting this graph into directed one.

Weakly Connected Nodes : Nodes which are having 0 indegree(number of incoming edges).

Prerequisite : BFS traversal

Examples :

```Input : 4 4
0 1
1 2
2 3
3 0
Output : 0 disconnected components

Input : 6 5
1 2
2 3
4 5
4 6
5 6
Output : 1 disconnected components```

Explanation : Approach : We find a node which helps in traversing maximum nodes in a single walk. To cover all possible paths, DFS graph traversal technique is used for this.

Do the above steps to traverse the graph. Now, iterate through graph again and check which nodes are having 0 indegree.

## C++

 `// C++ code to minimize the number``// of weakly connected nodes``#include ``using` `namespace` `std;` `// Set of nodes which are traversed``// in each launch of the DFS``set<``int``> node;``vector<``int``> Graph;` `// Function traversing the graph using DFS``// approach and updating the set of nodes``void` `dfs(``bool` `visit[], ``int` `src)``{``    ``visit[src] = ``true``;``    ``node.insert(src);``    ``int` `len = Graph[src].size();``    ``for` `(``int` `i = 0; i < len; i++)   ``        ``if` `(!visit[Graph[src][i]])       ``            ``dfs(visit, Graph[src][i]);``}` `// building a undirected graph``void` `buildGraph(``int` `x[], ``int` `y[], ``int` `len){` `    ``for` `(``int` `i = 0; i < len; i++)``    ``{``        ``int` `p = x[i];``        ``int` `q = y[i];``        ``Graph[p].push_back(q);``        ``Graph[q].push_back(p);``    ``}``}` `// computes the minimum number of disconnected``// components when a bi-directed graph is``// converted to a undirected graph``int` `compute(``int` `n)``{``    ``// Declaring and initializing``    ``// a visited array``    ``bool` `visit[n + 5];``    ``memset``(visit, ``false``, ``sizeof``(visit));``    ``int` `number_of_nodes = 0;` `    ``// We check if each node is``    ``// visited once or not``    ``for` `(``int` `i = 0; i < n; i++)``    ``{``        ``// We only launch DFS from a``        ``// node iff it is unvisited.``        ``if` `(!visit[i]) {` `            ``// Clearing the set of nodes``            ``// on every relaunch of DFS``            ``node.clear();``            ` `            ``// relaunching DFS from an``            ``// unvisited node.``            ``dfs(visit, i);``           ` `            ``// iterating over the node set to count the``            ``// number of nodes visited after making the``            ``// graph directed and storing it in the``            ``// variable count. If count / 2 == number``            ``// of nodes - 1, then increment count by 1.``            ``int` `count = 0;        ``            ``for` `(``auto` `it = node.begin(); it != node.end(); ++it)``                ``count += Graph[(*it)].size();``        ` `            ``count /= 2;       ``            ``if` `(count == node.size() - 1)``               ``number_of_nodes++;``        ``}``    ``}``    ``return` `number_of_nodes;``}` `//Driver function``int` `main()``{``    ``int` `n = 6,m = 4;``    ``int` `x[m + 5] = {1, 1, 4, 4};``    ``int` `y[m+5] = {2, 3, 5, 6};``    ` `    ``/*For given x and y above, graph is as below :``        ``1-----2         4------5``        ``|               |``        ``|               |``        ``|               |``        ``3               6``        ` `        ``// Note : This code will work for``        ``// connected graph also as :``        ``1-----2``        ``|     | \``        ``|     |  \``        ``|     |   \``        ``3-----4----5``    ``*/``    ` `    ``// Building graph in the form of a adjacency list``    ``buildGraph(x, y, n);``    ``cout << compute(n) << ``" weakly connected nodes"``;``    ` `    ``return` `0;``}`

## Python3

 `# Python3 code to minimize the number``# of weakly connected nodes`` ` `# Set of nodes which are traversed``# in each launch of the DFS``node ``=` `set``()``Graph ``=` `[[] ``for` `i ``in` `range``(``10001``)]`` ` `# Function traversing the graph using DFS``# approach and updating the set of nodes``def` `dfs(visit, src):` `    ``visit[src] ``=` `True``    ``node.add(src)``    ``llen ``=` `len``(Graph[src])``    ` `    ``for` `i ``in` `range``(llen):``        ``if` `(``not` `visit[Graph[src][i]]):``            ``dfs(visit, Graph[src][i])` `# Building a undirected graph``def` `buildGraph(x, y, llen):``    ` `    ``for` `i ``in` `range``(llen):``        ``p ``=` `x[i]``        ``q ``=` `y[i]``        ``Graph[p].append(q)``        ``Graph[q].append(p)``    ` `# Computes the minimum number of disconnected``# components when a bi-directed graph is``# converted to a undirected graph``def` `compute(n):` `    ``# Declaring and initializing``    ``# a visited array``    ``visit ``=` `[``False` `for` `i ``in` `range``(n ``+` `5``)]``    ` `    ``number_of_nodes ``=` `0`` ` `    ``# We check if each node is``    ``# visited once or not``    ``for` `i ``in` `range``(n):``        ` `        ``# We only launch DFS from a``        ``# node iff it is unvisited.``        ``if` `(``not` `visit[i]):`` ` `            ``# Clearing the set of nodes``            ``# on every relaunch of DFS``            ``node.clear()``             ` `            ``# Relaunching DFS from an``            ``# unvisited node.``            ``dfs(visit, i)``            ` `            ``# Iterating over the node set to count the``            ``# number of nodes visited after making the``            ``# graph directed and storing it in the``            ``# variable count. If count / 2 == number``            ``# of nodes - 1, then increment count by 1.``            ``count ``=` `0`     `            ` `            ``for` `it ``in` `node:``                ``count ``+``=` `len``(Graph[(it)])``         ` `            ``count ``/``/``=` `2``            ` `            ``if` `(count ``=``=` `len``(node) ``-` `1``):``               ``number_of_nodes ``+``=` `1``        ` `    ``return` `number_of_nodes` `# Driver code``if` `__name__``=``=``'__main__'``:``    ` `    ``n ``=` `6``    ``m ``=` `4``    ``x ``=` `[ ``1``, ``1``, ``4``, ``4``, ``0``, ``0``, ``0``, ``0``, ``0` `]``    ``y ``=` `[ ``2``, ``3``, ``5``, ``6``, ``0``, ``0``, ``0``, ``0``, ``0` `]``     ` `    ``'''For given x and y above, graph is as below :``        ``1-----2         4------5``        ``|               |``        ``|               |``        ``|               |``        ``3               6``         ` `        ``# Note : This code will work for``        ``# connected graph also as :``        ``1-----2``        ``|     | \``        ``|     |  \``        ``|     |   \``        ``3-----4----5``    ``'''``     ` `    ``# Building graph in the form of a adjacency list``    ``buildGraph(x, y, n)``    ` `    ``print``(``str``(compute(n)) ``+``          ``" weakly connected nodes"``)``     ` `# This code is contributed by rutvik_56`

## C#

 `// C# code to minimize the number``// of weakly connected nodes``using` `System;``using` `System.Collections;``using` `System.Collections.Generic;` `class` `GFG{``    ` `// Set of nodes which are traversed``// in each launch of the DFS``static` `HashSet<``int``> node = ``new` `HashSet<``int``>();``static` `List<``int``> []Graph = ``new` `List<``int``>;`` ` `// Function traversing the graph using DFS``// approach and updating the set of nodes``static` `void` `dfs(``bool` `[]visit, ``int` `src)``{``    ``visit[src] = ``true``;``    ``node.Add(src);``    ``int` `len = Graph[src].Count;``    ` `    ``for``(``int` `i = 0; i < len; i++)   ``        ``if` `(!visit[Graph[src][i]])       ``            ``dfs(visit, Graph[src][i]);``}`` ` `// Building a undirected graph``static` `void` `buildGraph(``int` `[]x, ``int` `[]y, ``int` `len)``{``    ``for``(``int` `i = 0; i < len; i++)``    ``{``        ``int` `p = x[i];``        ``int` `q = y[i];``        ``Graph[p].Add(q);``        ``Graph[q].Add(p);``    ``}``}`` ` `// Computes the minimum number of disconnected``// components when a bi-directed graph is``// converted to a undirected graph``static` `int` `compute(``int` `n)``{``    ` `    ``// Declaring and initializing``    ``// a visited array``    ``bool` `[]visit = ``new` `bool``[n + 5];``    ``Array.Fill(visit, ``false``);` `    ``int` `number_of_nodes = 0;`` ` `    ``// We check if each node is``    ``// visited once or not``    ``for``(``int` `i = 0; i < n; i++)``    ``{``        ` `        ``// We only launch DFS from a``        ``// node iff it is unvisited.``        ``if` `(!visit[i])``        ``{``            ` `            ``// Clearing the set of nodes``            ``// on every relaunch of DFS``            ``node.Clear();``             ` `            ``// Relaunching DFS from an``            ``// unvisited node.``            ``dfs(visit, i);``            ` `            ``// Iterating over the node set to count the``            ``// number of nodes visited after making the``            ``// graph directed and storing it in the``            ``// variable count. If count / 2 == number``            ``// of nodes - 1, then increment count by 1.``            ``int` `count = 0;        ``            ``foreach``(``int` `it ``in` `node)``                ``count += Graph[(it)].Count;``         ` `            ``count /= 2;``            ` `            ``if` `(count == node.Count - 1)``               ``number_of_nodes++;``        ``}``    ``}``    ``return` `number_of_nodes;``}`` ` `// Driver Code``static` `void` `Main(``string` `[]args)``{``    ``int` `n = 6;``    ``for``(``int` `i = 0; i < 10001; i++)``    ``{``        ``Graph[i] = ``new` `List<``int``>();``    ``}``    ``int` `[]x = { 1, 1, 4, 4, 0, 0, 0, 0 };``    ``int` `[]y = { 2, 3, 5, 6, 0, 0, 0, 0 };``     ` `    ``/*For given x and y above, graph is as below :``        ``1-----2         4------5``        ``|               |``        ``|               |``        ``|               |``        ``3               6``         ` `        ``// Note : This code will work for``        ``// connected graph also as :``        ``1-----2``        ``|     | \``        ``|     |  \``        ``|     |   \``        ``3-----4----5``    ``*/``     ` `    ``// Building graph in the form of a adjacency list``    ``buildGraph(x, y, n);``    ``Console.Write(compute(n) +``    ``" weakly connected nodes"``);``}``}` `// This code is contributed by pratham76`
Output:
`2 weakly connected nodes`

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