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. 

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 vertices 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

Javascript

<script>

// Javascript program to find K vertices in
// the graph which are connected to at
// least one of remaining vertices
let N = 200005;

// To store graph
let n, m;
let vis = new Array(N);
for(let i = 0; i < N; i++)
{
    vis[i] = 0;
}

let gr = new Array(N);
let  v = new Array(2);

// Function to add edge
function add_edges(x, y)
{
    gr[x].push(y);
    gr[y].push(x);
}

// Function to find level of each node
function dfs(x, state)
{

    // Push the vertex in respected level
    v[state].push(x);

    // Make vertex visited
    vis[x] = 1;

    // Traverse for all it's child nodes
    for(let i = 0; i < gr[x].length; i++)
    {
        if (vis[gr[x][i]] == 0)
        {
            dfs(gr[x][i], (state ^ 1));
        }
    }
}

// Function to print vertices
function Print_vertices()
{

    // If odd level vertices are less
    if (v[0].length < v[1].length)
    {
        for(let i = 0; i < v[0].length; i++)
        {
            document.write(v[0][i] + " ");
        }
    }

    // If even level vertices are less
    else
    {
        for(let i = 0; i < v[1].length; i++)
        {
            document.write(v[1][i] + " ");
        }
    }
}

// Driver code
n = 4;
m = 3;
for(let i = 0; i < N; i++)
{
    gr[i] = [];
}
for(let i = 0; i < 2; i++)
{
    v[i] = [];
}

// 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 unknown2108

</script>

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.