Find K vertices in the graph which are connected to at least one of remaining vertices

Given a connected graph with N vertices. The task is to select k(k must be less than or equals to n/2, not necessarily minimum) vertices from the graph such that all these selected vertices are connected to at least one of the non selected vertex. In case of multiple answers print any one of them.

Examples:

Input :

Output : 1
Vertex 1 is connected to all other non selected vertices. Here
{1, 2}, {2, 3}, {3, 4}, {1, 3}, {1, 4}, {2, 4} are also the valid answers

Input :

Output : 1 3
Vertex 1, 3 are connected to all other non selected vertices. {2, 4} is also a valid answer.

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Efficient Approach: An efficient way is to find vertices which are even level and odd level using simple dfs or bfs function. Then if the verices at odd level are less than the vertices at even level then print odd level vertices. Otherwise, print even level vertices.

Below is the implementation of the above approach:

C++

// C++ program to find K vertices in  
// the graph which are connected to at  
// least one of remaining vertices 
#include <bits/stdc++.h> 
using namespace std; 
#define N 200005 
  
// To store graph 
int n, m, vis[N]; 
vector<int> gr[N]; 
vector<int> v[2]; 
  
// Function to add edge 
void add_edges(int x, int y) 
{ 
    gr[x].push_back(y); 
    gr[y].push_back(x); 
} 
  
// Function to find level of each node 
void dfs(int x, int state) 
{ 
    // Push the vertex in respected level 
    v[state].push_back(x); 
  
    // Make vertex visited 
    vis[x] = 1; 
  
    // Traverse for all it's child nodes 
    for (auto i : gr[x]) 
        if (vis[i] == 0) 
            dfs(i, state ^ 1); 
} 
  
// Function to print vertices 
void Print_vertices() 
{ 
    // If odd level vertices are less 
    if (v[0].size() < v[1].size()) { 
        for (auto i : v[0]) 
            cout << i << " "; 
    } 
    // If even level vertices are less 
    else { 
        for (auto i : v[1]) 
            cout << i << " "; 
    } 
} 
  
// Driver code 
int main() 
{ 
    int n = 4, m = 3; 
  
    // Add edges 
    add_edges(1, 2); 
    add_edges(2, 3); 
    add_edges(3, 4); 
  
    // Function call 
    dfs(1, 0); 
  
    Print_vertices(); 
  
    return 0; 
} 

Java

// Java program to find K vertices in  
// the graph which are connected to at  
// least one of remaining vertices 
import java.util.*; 
  
class GFG 
{ 
  
    static final int N = 200005; 
  
    // To store graph 
    static int n, m; 
    static int[] vis = new int[N]; 
    static Vector<Integer>[] gr = new Vector[N]; 
    static Vector<Integer>[] v = new Vector[2]; 
  
    // Function to add edge 
    static void add_edges(int x, int y)  
    { 
        gr[x].add(y); 
        gr[y].add(x); 
    } 
  
    // Function to find level of each node 
    static void dfs(int x, int state)  
    { 
        // Push the vertex in respected level 
        v[state].add(x); 
  
        // Make vertex visited 
        vis[x] = 1; 
  
        // Traverse for all it's child nodes 
        for (int i : gr[x]) 
        { 
            if (vis[i] == 0) 
            { 
                dfs(i, state ^ 1); 
            } 
        } 
    } 
  
    // Function to print vertices 
    static void Print_vertices()  
    { 
        // If odd level vertices are less 
        if (v[0].size() < v[1].size())  
        { 
            for (int i : v[0])  
            { 
                System.out.print(i + " "); 
            } 
        }  
          
        // If even level vertices are less 
        else 
        { 
            for (int i : v[1])  
            { 
                System.out.print(i + " "); 
            } 
        } 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        n = 4; 
        m = 3; 
        for (int i = 0; i < N; i++) 
        { 
            gr[i] = new Vector<Integer>(); 
        } 
        for (int i = 0; i < 2; i++)  
        { 
            v[i] = new Vector<Integer>(); 
        } 
          
        // Add edges 
        add_edges(1, 2); 
        add_edges(2, 3); 
        add_edges(3, 4); 
  
        // Function call 
        dfs(1, 0); 
  
        Print_vertices(); 
    } 
} 
  
// This code is contributed by 29AjayKumar 

Python3

# Python3 program to find K vertices in 
# the graph which are connected to at 
# least one of remaining vertices 
  
N = 200005
  
# To store graph 
n, m, =0,0
vis=[0 for i in range(N)] 
gr=[[] for i in range(N)] 
v=[[] for i in range(2)] 
  
# Function to add edge 
def add_edges(x, y): 
    gr[x].append(y) 
    gr[y].append(x) 
  
# Function to find level of each node 
def dfs(x, state): 
  
    # Push the vertex in respected level 
    v[state].append(x) 
  
    # Make vertex visited 
    vis[x] = 1
  
    # Traverse for all it's child nodes 
    for i in gr[x]: 
        if (vis[i] == 0): 
            dfs(i, state ^ 1) 
  
  
# Function to prvertices 
def Print_vertices(): 
  
    # If odd level vertices are less 
    if (len(v[0]) < len(v[1])): 
        for i in v[0]: 
            print(i,end=" ") 
    # If even level vertices are less 
    else: 
        for i in v[1]: 
            print(i,end=" ") 
  
# Driver code 
  
n = 4
m = 3
  
# Add edges 
add_edges(1, 2) 
add_edges(2, 3) 
add_edges(3, 4) 
  
# Function call 
dfs(1, 0) 
  
Print_vertices() 
  
# This code is contributed by mohit kumar 29 

C#

    // C# program to find K vertices in  
// the graph which are connected to at  
// least one of remaining vertices 
using System; 
using System.Collections.Generic; 
  
class GFG 
{ 
    static readonly int N = 200005; 
  
    // To store graph 
    static int n, m; 
    static int[] vis = new int[N]; 
    static List<int>[] gr = new List<int>[N]; 
    static List<int>[] v = new List<int>[2]; 
  
    // Function to add edge 
    static void add_edges(int x, int y)  
    { 
        gr[x].Add(y); 
        gr[y].Add(x); 
    } 
  
    // Function to find level of each node 
    static void dfs(int x, int state)  
    { 
        // Push the vertex in respected level 
        v[state].Add(x); 
  
        // Make vertex visited 
        vis[x] = 1; 
  
        // Traverse for all it's child nodes 
        foreach (int i in gr[x]) 
        { 
            if (vis[i] == 0) 
            { 
                dfs(i, state ^ 1); 
            } 
        } 
    } 
  
    // Function to print vertices 
    static void Print_vertices()  
    { 
        // If odd level vertices are less 
        if (v[0].Count < v[1].Count)  
        { 
            foreach (int i in v[0])  
            { 
                Console.Write(i + " "); 
            } 
        }  
          
        // If even level vertices are less 
        else
        { 
            foreach (int i in v[1])  
            { 
                Console.Write(i + " "); 
            } 
        } 
    } 
  
    // Driver code 
    public static void Main(String[] args) 
    { 
        n = 4; 
        m = 3; 
        for (int i = 0; i < N; i++) 
        { 
            gr[i] = new List<int>(); 
        } 
        for (int i = 0; i < 2; i++)  
        { 
            v[i] = new List<int>(); 
        } 
          
        // Add edges 
        add_edges(1, 2); 
        add_edges(2, 3); 
        add_edges(3, 4); 
  
        // Function call 
        dfs(1, 0); 
  
        Print_vertices(); 
    } 
} 
  
// This code is contributed by Rajput-Ji 

Output:

2 4

Time Complexity : O(V+E)
Where V is the number of vertices and E is the number of edges in the graph.

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

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