在图中找到与剩余顶点中的至少一个相连的 K 个顶点

给定一个有N个顶点的连通图。任务是从图中选择 k（k 必须小于或等于 n/2，不一定是最小值）个顶点，以便所有这些选定的顶点都连接到至少一个未选定的顶点。如果有多个答案，请打印其中任何一个。

例子：

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. 
 

有效的方法：一种有效的方法是使用简单的 dfs 或 bfs函数找到偶数级和奇数级的顶点。然后，如果奇数级别的顶点小于偶数级别的顶点，则打印奇数级别的顶点。否则，打印偶数级顶点。

下面是上述方法的实现：

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 gr[N];
vector 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[] 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[] gr = new List[N];
    static List[] v = new List[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();
        }
        for (int i = 0; i < 2; i++)
        {
            v[i] = new List();
        }
         
        // 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

 
// 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
 

输出：

2 4

时间复杂度： O(V+E)
其中 V 是顶点数，E 是图中的边数。

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