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 answersInput :
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++ 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 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 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# 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 |
<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> |
<?php // PHP program to find K vertices in // the graph which are connected to at // least one of remaining vertices class GFG {
const N = 200005;
// To store graph
public $n , $m ;
public $vis = array ();
public $gr = array ();
public $v = array ();
public function __construct() {
for ( $i = 0; $i < self::N; $i ++) {
$this ->gr[ $i ] = array ();
}
for ( $i = 0; $i < 2; $i ++) {
$this ->v[ $i ] = array ();
}
}
// Function to add edge
public function add_edges( $x , $y ) {
array_push ( $this ->gr[ $x ], $y );
array_push ( $this ->gr[ $y ], $x );
}
// Function to find level of each node
public function dfs( $x , $state ) {
// Push the vertex in respected level
array_push ( $this ->v[ $state ], $x );
// Make vertex visited
$this ->vis[ $x ] = 1;
// Traverse for all it's child nodes
foreach ( $this ->gr[ $x ] as $i ) {
if ( $this ->vis[ $i ] == 0) {
$this ->dfs( $i , $state ^ 1);
}
}
}
// Function to print vertices
public function Print_vertices() {
// If odd level vertices are less
if ( count ( $this ->v[0]) < count ( $this ->v[1])) {
foreach ( $this ->v[0] as $i ) {
echo $i . " " ;
}
}
// If even level vertices are less
else {
foreach ( $this ->v[1] as $i ) {
echo $i . " " ;
}
}
}
// Driver code
public function main() {
$this ->n = 4;
$this ->m = 3;
// Add edges
$this ->add_edges(1, 2);
$this ->add_edges(2, 3);
$this ->add_edges(3, 4);
// Function call
$this ->dfs(1, 0);
$this ->Print_vertices();
}
} $gfg = new GFG();
$gfg ->main();
// This code is contributed by rajsanghavi9 ?> |
2 4
Time Complexity : O(V+E)
Where V is the number of vertices and E is the number of edges in the graph.