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Minimize cost to color all the vertices of an Undirected Graph
• Last Updated : 27 Nov, 2020

Given an Undirected Graph consisting of N vertices and M edges, where node values are in the range [1, N], and vertices specified by the array colored[] are colored, the task is to find the minimum color all vertices of the given graph. The cost to color a vertex is given by vCost and the cost to add a new edge between two vertices is given by eCost. If a vertex is colored, then all the vertices that can be reached from that vertex also become colored.

Examples:

Input:N = 3, M = 1, vCost = 3, eCost = 2, colored[] = {1}, source[] = {1} destination[] = {2}
Output:
Explanation:
Vertex 1 is colored and it has an edge with 2.
So, vertex 2 is also colored.
Add an edge between 2 and 3, at a cost of eCost. < vCost.
Hence, the output is 2.

Input: N = 4, M = 2, vCost = 3, eCost = 7, colored[] = {1, 3}, source[] = {1, 2} destination[] = {4, 3}
Output:
Explanation:
Vertex 1 is colored and it has an edge with 4. Hence, vertex 4 is also colored.
Vertex 2 is colored and it has an edge with 3. Hence, vertex 3 is also colored.
Since all the vertices are already colored, therefore, the cost is 0.

Approach:
The idea is to count the number of sub-graphs of uncolored vertices using DFS Traversal
To minimize the cost of coloring an uncolored Subgraph, one of the following needs to be done:

• Color the subgraph
• Add an edge between any colored and uncolored vertex.

Based on the minimum of eCost and vCost, one of the above two steps needs to be chosen.
If the number of uncolored sub-graphs is given by X, then the total cost of coloring all the vertices is given by X×min(eCost, vCost).

Follow the steps below to find the number of uncolored sub-graphs:

1. Perform DFS Traversal on all the colored vertices and mark them visited to identify them as colored.
2. The vertices that are not visited after DFS at step 1 are the uncolored vertices.
3. For each uncolored vertex, mark all the vertices that can be reached from that vertex as visited using DFS.
4. The number of uncolored vertices for which the DFS at step 3 occurs, is the number of sub-graphs X.
5. Calculate the total cost of coloring all the vertices by the formula X×min(eCost, vCost).

Below is the implementation of the above approach:

## C++

 `// C++ Program to implement``// the above approach``#include ``using` `namespace` `std;` `// Function to implement DFS Traversal``// to marks all the vertices visited``// from vertex U``void` `DFS(``int` `U, ``int``* vis, vector<``int``> adj[])``{``    ``// Mark U as visited``    ``vis[U] = 1;` `    ``// Traverse the adjacency list of U``    ``for` `(``int` `V : adj[U]) {``        ``if` `(vis[V] == 0)``            ``DFS(V, vis, adj);``    ``}``}` `// Function to find the minimum cost``// to color all the vertices of graph``void` `minCost(``int` `N, ``int` `M, ``int` `vCost,``             ``int` `eCost, ``int` `sorc[],``             ``vector<``int``> colored,``             ``int` `destination[])``{``    ``// To store adjacency list``    ``vector<``int``> adj[N + 1];` `    ``// Loop through the edges to``    ``// create adjacency list``    ``for` `(``int` `i = 0; i < M; i++) {` `        ``adj[sorc[i]].push_back(destination[i]);``        ``adj[destination[i]].push_back(sorc[i]);``    ``}` `    ``// To check if a vertex of the``    ``// graph is visited``    ``int` `vis[N + 1] = { 0 };` `    ``// Mark visited to all the vertices``    ``// that can be reached by``    ``// colored vertices``    ``for` `(``int` `i = 0; i < colored.size(); i++) {` `        ``// Perform DFS``        ``DFS(colored[i], vis, adj);``    ``}` `    ``// To store count of uncolored``    ``// sub-graphs``    ``int` `X = 0;` `    ``// Loop through vertex to count``    ``// uncolored sub-graphs``    ``for` `(``int` `i = 1; i <= N; i++) {` `        ``// If vertex not visited``        ``if` `(vis[i] == 0) {` `            ``// Increase count of``            ``// uncolored sub-graphs``            ``X++;` `            ``// Perform DFS to mark``            ``// visited to all vertices``            ``// of current sub-graphs``            ``DFS(i, vis, adj);``        ``}``    ``}` `    ``// Calculate minimum cost to color``    ``// all vertices``    ``int` `mincost = X * min(vCost, eCost);` `    ``// Print the result``    ``cout << mincost << endl;``}` `// Driver Code``int` `main()``{` `    ``// Given number of``    ``// vertices and edges``    ``int` `N = 3, M = 1;` `    ``// Given edges``    ``int` `sorc[] = { 1 };``    ``int` `destination[] = { 2 };` `    ``// Given cost of coloring``    ``// and adding an edge``    ``int` `vCost = 3, eCost = 2;` `    ``// Given array of``    ``// colored vertices``    ``vector<``int``> colored = { 1};` `    ``minCost(N, M, vCost, eCost,``            ``sorc, colored, destination);` `    ``return` `0;``}`

## Java

 `// Java program to implement``// the above approach``import` `java.util.*;` `class` `GFG{``    ` `// Function to implement DFS Traversal``// to marks all the vertices visited``// from vertex U``static` `void` `DFS(``int` `U, ``int``[] vis,``                ``ArrayList> adj)``{``    ` `    ``// Mark U as visited``    ``vis[U] = ``1``;`` ` `    ``// Traverse the adjacency list of U``    ``for``(Integer V : adj.get(U))``    ``{``        ``if` `(vis[V] == ``0``)``            ``DFS(V, vis, adj);``    ``}``}`` ` `// Function to find the minimum cost``// to color all the vertices of graph``static` `void` `minCost(``int` `N, ``int` `M, ``int` `vCost,``                    ``int` `eCost, ``int` `sorc[],``                    ``ArrayList colored,``                    ``int` `destination[])``{``    ` `    ``// To store adjacency list``    ``ArrayList> adj = ``new` `ArrayList<>();`` ` `    ``for``(``int` `i = ``0``; i < N + ``1``; i++)``        ``adj.add(``new` `ArrayList());`` ` `    ``// Loop through the edges to``    ``// create adjacency list``    ``for``(``int` `i = ``0``; i < M; i++)``    ``{``        ``adj.get(sorc[i]).add(destination[i]);``        ``adj.get(destination[i]).add(sorc[i]);``    ``}`` ` `    ``// To check if a vertex of the``    ``// graph is visited``    ``int``[] vis = ``new` `int``[N + ``1``];`` ` `    ``// Mark visited to all the vertices``    ``// that can be reached by``    ``// colored vertices``    ``for``(``int` `i = ``0``; i < colored.size(); i++)``    ``{``        ` `        ``// Perform DFS``        ``DFS(colored.get(i), vis, adj);``    ``}`` ` `    ``// To store count of uncolored``    ``// sub-graphs``    ``int` `X = ``0``;`` ` `    ``// Loop through vertex to count``    ``// uncolored sub-graphs``    ``for``(``int` `i = ``1``; i <= N; i++)``    ``{``        ` `        ``// If vertex not visited``        ``if` `(vis[i] == ``0``)``        ``{``            ` `            ``// Increase count of``            ``// uncolored sub-graphs``            ``X++;`` ` `            ``// Perform DFS to mark``            ``// visited to all vertices``            ``// of current sub-graphs``            ``DFS(i, vis, adj);``        ``}``    ``}`` ` `    ``// Calculate minimum cost to color``    ``// all vertices``    ``int` `mincost = X * Math.min(vCost, eCost);`` ` `    ``// Print the result``    ``System.out.println(mincost);``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ` `    ``// Given number of``    ``// vertices and edges``    ``int` `N = ``3``, M = ``1``;``    ` `    ``// Given edges``    ``int` `sorc[] = {``1``};``    ``int` `destination[] = {``2``};``    ` `    ``// Given cost of coloring``    ``// and adding an edge``    ``int` `vCost = ``3``, eCost = ``2``;``    ` `    ``// Given array of``    ``// colored vertices``    ``ArrayList colored = ``new` `ArrayList<>();``    ``colored.add(``1``);``    ` `    ``minCost(N, M, vCost, eCost, sorc,``            ``colored, destination);``}``}` `// This code is contributed by offbeat`

## Python3

 `# Python3 program to implement``# the above approach` `# Function to implement DFS Traversal``# to marks all the vertices visited``# from vertex U``def` `DFS(U, vis, adj):``    ` `    ``# Mark U as visited``    ``vis[U] ``=` `1` `    ``# Traverse the adjacency list of U``    ``for` `V ``in` `adj[U]:``        ``if` `(vis[V] ``=``=` `0``):``            ``DFS(V, vis, adj)` `# Function to find the minimum cost``# to color all the vertices of graph``def` `minCost(N, M, vCost, eCost, sorc,``            ``colored, destination):``                ` `    ``# To store adjacency list``    ``adj ``=` `[[] ``for` `i ``in` `range``(N ``+` `1``)]` `    ``# Loop through the edges to``    ``# create adjacency list``    ``for` `i ``in` `range``(M):``        ``adj[sorc[i]].append(destination[i])``        ``adj[destination[i]].append(sorc[i])` `    ``# To check if a vertex of the``    ``# graph is visited``    ``vis ``=` `[``0``] ``*` `(N ``+` `1``)` `    ``# Mark visited to all the vertices``    ``# that can be reached by``    ``# colored vertices``    ``for` `i ``in` `range``(``len``(colored)):` `        ``# Perform DFS``        ``DFS(colored[i], vis, adj)` `    ``# To store count of uncolored``    ``# sub-graphs``    ``X ``=` `0` `    ``# Loop through vertex to count``    ``# uncolored sub-graphs``    ``for` `i ``in` `range``(``1``, N ``+` `1``):` `        ``# If vertex not visited``        ``if` `(vis[i] ``=``=` `0``):` `            ``# Increase count of``            ``# uncolored sub-graphs``            ``X ``+``=` `1` `            ``# Perform DFS to mark``            ``# visited to all vertices``            ``# of current sub-graphs``            ``DFS(i, vis, adj)` `    ``# Calculate minimum cost to color``    ``# all vertices``    ``mincost ``=` `X ``*` `min``(vCost, eCost)` `    ``# Print the result``    ``print``(mincost)` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:` `    ``# Given number of``    ``# vertices and edges``    ``N ``=` `3``    ``M ``=` `1` `    ``# Given edges``    ``sorc ``=` `[``1``]``    ``destination ``=` `[``2``]` `    ``# Given cost of coloring``    ``# and adding an edge``    ``vCost ``=` `3``    ``eCost ``=` `2` `    ``# Given array of``    ``# colored vertices``    ``colored ``=` `[``1``]` `    ``minCost(N, M, vCost, eCost,``            ``sorc, colored, destination)` `# This code is contributed by mohit kumar 29`

## C#

 `// C# program to implement``// the above approach``using` `System;``using` `System.Collections;``using` `System.Collections.Generic;` `class` `GFG{``    ` `// Function to implement DFS Traversal``// to marks all the vertices visited``// from vertex U``static` `void` `DFS(``int` `U, ``int``[] vis, ArrayList adj)``{``  ` `    ``// Mark U as visited``    ``vis[U] = 1;` `    ``// Traverse the adjacency list of U``    ``foreach``(``int` `V ``in` `(ArrayList)adj[U])``    ``{``        ``if` `(vis[V] == 0)``            ``DFS(V, vis, adj);``    ``}``}` `// Function to find the minimum cost``// to color all the vertices of graph``static` `void` `minCost(``int` `N, ``int` `M, ``int` `vCost,``                    ``int` `eCost, ``int` `[]sorc,``                    ``ArrayList colored,``                    ``int` `[]destination)``{``    ` `    ``// To store adjacency list``    ``ArrayList adj = ``new` `ArrayList();` `    ``for``(``int` `i = 0; i < N + 1; i++)``        ``adj.Add(``new` `ArrayList());` `    ``// Loop through the edges to``    ``// create adjacency list``    ``for``(``int` `i = 0; i < M; i++)``    ``{``        ``((ArrayList)adj[sorc[i]]).Add(destination[i]);``        ``((ArrayList)adj[destination[i]]).Add(sorc[i]);``    ``}` `    ``// To check if a vertex of the``    ``// graph is visited``    ``int``[] vis = ``new` `int``[N + 1];` `    ``// Mark visited to all the vertices``    ``// that can be reached by``    ``// colored vertices``    ``for``(``int` `i = 0; i < colored.Count; i++)``    ``{``        ` `        ``// Perform DFS``        ``DFS((``int``)colored[i], vis, adj);``    ``}``  ` `    ``// To store count of uncolored``    ``// sub-graphs``    ``int` `X = 0;` `    ``// Loop through vertex to count``    ``// uncolored sub-graphs``    ``for``(``int` `i = 1; i <= N; i++)``    ``{``      ` `        ``// If vertex not visited``        ``if` `(vis[i] == 0)``        ``{``            ` `            ``// Increase count of``            ``// uncolored sub-graphs``            ``X++;` `            ``// Perform DFS to mark``            ``// visited to all vertices``            ``// of current sub-graphs``            ``DFS(i, vis, adj);``        ``}``    ``}` `    ``// Calculate minimum cost to color``    ``// all vertices``    ``int` `mincost = X * Math.Min(vCost, eCost);` `    ``// Print the result``    ``Console.Write(mincost);``}` `// Driver code``public` `static` `void` `Main(``string``[] args)``{``    ` `    ``// Given number of``    ``// vertices and edges``    ``int` `N = 3, M = 1;``    ` `    ``// Given edges``    ``int` `[]sorc = {1};``    ``int` `[]destination = {2};``    ` `    ``// Given cost of coloring``    ``// and adding an edge``    ``int` `vCost = 3, eCost = 2;``    ` `    ``// Given array of``    ``// colored vertices``    ``ArrayList colored = ``new` `ArrayList();``    ``colored.Add(1);``    ` `    ``minCost(N, M, vCost, eCost,``            ``sorc, colored,``            ``destination);``}``}` `// This code is contributed by rutvik_56`
Output:
`2`

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

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