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Find K vertices in the graph which are connected to at least one of remaining vertices
• Last Updated : 07 Jun, 2021

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 :
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 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 ``using` `namespace` `std;``#define N 200005` `// To store graph``int` `n, m, vis[N];``vector<``int``> gr[N];``vector<``int``> v;` `// 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.size() < v.size()) {``        ``for` `(``auto` `i : v)``            ``cout << i << ``" "``;``    ``}``    ``// If even level vertices are less``    ``else` `{``        ``for` `(``auto` `i : v)``            ``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[] gr = ``new` `Vector[N];``    ``static` `Vector[] 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();``        ``}``        ``for` `(``int` `i = ``0``; i < ``2``; i++)``        ``{``            ``v[i] = ``new` `Vector();``        ``}``        ` `        ``// 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``>;` `    ``// 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.Count < v.Count)``        ``{``            ``foreach` `(``int` `i ``in` `v)``            ``{``                ``Console.Write(i + ``" "``);``            ``}``        ``}``        ` `        ``// If even level vertices are less``        ``else``        ``{``            ``foreach` `(``int` `i ``in` `v)``            ``{``                ``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

 ``
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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