Find a set of at most N/2 nodes from a Graph such that all remaining nodes are directly connected to one of the chosen nodes

Given an integer N, representing the number of nodes present in an undirected graph, with each node valued from 1 to N, and a 2D array Edges[][], representing the pair of vertices connected by an edge, the task is to find a set of at most N/2 nodes such that nodes that are not present in the set, are connected adjacent to any one of the nodes present in the set.

Examples :

Input: N = 4, Edges[][2] = {{2, 3}, {1, 3}, {4, 2}, {1, 2}}
Output: 3 2
Explanation: Connections specified in the above graph are as follows:
      1
 /       \
2   –   3
|
4
Selecting the set {2, 3} satisfies the required conditions.

Input: N = 5, Edges[][2] = {{2, 1}, {3, 1}, {3, 2}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}, {5, 3}, {5, 4}}
Output: 1

Approach: The given problem can be solved based on the following observations:

• Assume a node to be the source node, then the distance of each vertex from the source node will be either odd or even.
• Split the nodes into two different sets based on parity, the size of at least one of the sets will not exceed N/2. Since each node of some parity is connected to at least one node of opposite parity, the criteria of choosing at most N/2 nodes is satisfied.

Follow the steps below to solve the problem:

• Assume any vertex to be the source node.
• Initialize two sets, say evenParity and oddParity, to store the nodes having even and odd distances from the source node respectively.
• Perform BFS Traversal on the given graph and split the vertices into two different sets depending on the parity of their distances from the source:
  • If the distance of each connected node from the source node is odd, then insert the current node in the set oddParity.
  • If the distance of each connected node from the source node is even,  then insert the current node in the set evenParity.
• After completing the above steps, print the elements of the set with the minimum size.

C++

// C++ program for the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to add an edge
// to the adjacency list
void addEdge(vector<vector<int> >& adj,
             int u, int v)
{
    adj[u].push_back(v);
    adj[v].push_back(u);
}

// Function to perform BFS
// traversal on a given graph
vector<vector<int> > BFS(
    int N, int source,
    vector<vector<int> > adjlist)
{
    // Stores the distance of each
    // node from the source node
    int dist[N + 1];

    vector<vector<int> > vertex_set;

    // Update the distance of all
    // vertices from source as -1
    memset(dist, -1, sizeof dist);

    // Assign two separate vectors
    // for parity odd and even parities
    vertex_set.assign(2, vector<int>(0));

    // Perform BFS Traversal
    queue<int> Q;

    // Push the source node
    Q.push(source);
    dist = 0;

    // Iterate until queue becomes empty
    while (!Q.empty()) {

        // Get the front node
        // present in the queue
        int u = Q.front();
        Q.pop();

        // Push the node into vertex_set
        vertex_set[dist[u] % 2].push_back(u);

        // Check if the adjacent
        // vertices are visited
        for (int i = 0;
             i < (int)adjlist[u].size(); i++) {

            // Adjacent node
            int v = adjlist[u][i];

            // If the node v is unvisited
            if (dist[v] == -1) {

                // Update the distance
                dist[v] = dist[u] + 1;

                // Enqueue the node v
                Q.push(v);
            }
        }
    }

    // Return the possible set of nodes
    return vertex_set;
}

// Function to find a set of vertices
// of at most N/2 nodes such that each
// unchosen node is connected adjacently
// to one of the nodes in the set
void findSet(int N,
             vector<vector<int> > adjlist)
{
    // Source vertex
    int source = 1;

    // Store the vertex set
    vector<vector<int> > vertex_set
        = BFS(N, source, adjlist);

    // Stores the index
    // with minimum size
    int in = 0;

    if (vertex_set[1].size()
        < vertex_set[0].size())
        in = 1;

    // Print the nodes present in the set
    for (int node : vertex_set[in]) {
        cout << node << " ";
    }
}

// Driver Code
int main()
{
    int N = 5;
    int M = 8;
    vector<vector<int> > adjlist;
    adjlist.assign(N + 1, vector<int>(0));

    // Graph Formation
    addEdge(adjlist, 2, 5);
    addEdge(adjlist, 2, 1);
    addEdge(adjlist, 5, 1);
    addEdge(adjlist, 4, 5);
    addEdge(adjlist, 1, 4);
    addEdge(adjlist, 2, 4);
    addEdge(adjlist, 3, 4);
    addEdge(adjlist, 3, 5);

    // Function Call to print the
    // set of at most N / 2 nodes
    findSet(N, adjlist);

    return 0;
}

Python3

# Python3 program for the above approach
from collections import deque

# Function to add an edge
# to the adjacency list
def addEdge(adj, u, v):
    adj[u].append(v)
    adj[v].append(u)
    return adj

# Function to perform BFS
# traversal on a given graph
def BFS(N, source, adjlist):

    # Stores the distance of each
    # node from the source node
    dist = [-1]*(N + 1)
    vertex_set = [[] for i in range(2)]

    # Perform BFS Traversal
    Q = deque()

    # Push the source node
    Q.append(source)
    dist = 0

    # Iterate until queue becomes empty
    while len(Q) > 0:

        # Get the front node
        # present in the queue
        u = Q.popleft()

        # Push the node into vertex_set
        vertex_set[dist[u] % 2].append(u)

        # Check if the adjacent
        # vertices are visited
        for i in range(len(adjlist[u])):

            # Adjacent node
            v = adjlist[u][i]

            # If the node v is unvisited
            if (dist[v] == -1):

                # Update the distance
                dist[v] = dist[u] + 1

                # Enqueue the node v
                Q.append(v)

    # Return the possible set of nodes
    return vertex_set

# Function to find a set of vertices
# of at most N/2 nodes such that each
# unchosen node is connected adjacently
# to one of the nodes in the set
def findSet(N, adjlist):

    # Source vertex
    source = 1

    # Store the vertex set
    vertex_set = BFS(N, source, adjlist)

    # Stores the index
    # with minimum size
    inn = 0

    if (len(vertex_set[1]) < len(vertex_set[0])):
        inn = 1

    # Print the nodes present in the set
    for node in vertex_set[inn]:
        print(node, end=" ")

# Driver Code
if __name__ == '__main__':
    N = 5
    M = 8
    adjlist = [[] for i in range(N+1)]

    # Graph Formation
    adjlist = addEdge(adjlist, 2, 5)
    adjlist = addEdge(adjlist, 2, 1)
    adjlist = addEdge(adjlist, 5, 1)
    adjlist = addEdge(adjlist, 4, 5)
    adjlist = addEdge(adjlist, 1, 4)
    adjlist = addEdge(adjlist, 2, 4)
    adjlist = addEdge(adjlist, 3, 4)
    adjlist = addEdge(adjlist, 3, 5)

    # Function Call to print the
    # set of at most N / 2 nodes
    findSet(N, adjlist)

# This code is contributed by mohit kumar 29.

Java

import java.util.LinkedList;
import java.util.Queue;

class Main {
static void addEdge(LinkedList<Integer>[] adj, int u, int v) {
adj[u].add(v);
adj[v].add(u);
}

static LinkedList<Integer>[] BFS(int N, int source, LinkedList<Integer>[] adjlist) {
    int[] dist = new int[N+1];
    for (int i = 0; i <= N; i++) {
        dist[i] = -1;
    }
    LinkedList<Integer>[] vertex_set = new LinkedList[2];
    vertex_set[0] = new LinkedList<Integer>();
    vertex_set[1] = new LinkedList<Integer>();

    // Perform BFS Traversal
    Queue<Integer> Q = new LinkedList<Integer>();

    // Push the source node
    Q.add(source);
    dist = 0;

    // Iterate until queue becomes empty
    while (Q.size() > 0) {
        int u = Q.remove();
        vertex_set[dist[u] % 2].add(u);

        for (Integer v : adjlist[u]) {
            // If the node v is unvisited
            if (dist[v] == -1) {
                // Update the distance
                dist[v] = dist[u] + 1;

                // Enqueue the node v
                Q.add(v);
            }
        }
    }
    // Return the possible set of nodes
    return vertex_set;
}

// Function to find a set of vertices
// of at most N/2 nodes such that each
// unchosen node is connected adjacently
// to one of the nodes in the set
static void findSet(int N, LinkedList<Integer>[] adjlist) {
    // Source vertex
    int source = 1;

    // Store the vertex set
    LinkedList<Integer>[] vertex_set = BFS(N, source, adjlist);

    // Stores the index
    // with minimum size
    int inn = 0;

    if (vertex_set[1].size() < vertex_set[0].size()) {
        inn = 1;
    }

    // Print the nodes present in the set
    for (int node : vertex_set[inn]) {
        System.out.print(node + " ");
    }
}

public static void main(String[] args) {
    int N = 5;
    int M = 8;
    LinkedList<Integer>[] adjlist = new LinkedList[N+1];
    for (int i = 0; i <= N; i++) {
        adjlist[i] = new LinkedList<Integer>();
    }

    // Graph Formation
    addEdge(adjlist, 2, 5);
    addEdge(adjlist, 2, 1);
    addEdge(adjlist, 5, 1);
    addEdge(adjlist, 4, 5);
    addEdge(adjlist, 1, 4);
    addEdge(adjlist, 2, 4);
    addEdge(adjlist, 3, 4);
    addEdge(adjlist, 3, 5);

    findSet(N, adjlist);
 }
}

Javascript

// This function adds an edge to an adjacency list
function addEdge(adj, u, v) {
    adj[u].push(v); // Add v to the adjacency list of u
    adj[v].push(u); // Add u to the adjacency list of v
    return adj; // Return the updated adjacency list
}

// This function performs a BFS on a graph represented as an adjacency list
function BFS(N, source, adjlist) {
    let dist = new Array(N + 1).fill(-1); // Array to store the distance of each vertex from the source
    let vertex_set = [[], []]; // Two sets to store the vertices with even and odd distances from the source

    let Q = []; // Queue to store the vertices to be visited

    Q.push(source); // Enqueue the source vertex
    dist = 0; // The distance of the source vertex from itself is 0

    while (Q.length > 0) { // Continue BFS until the queue is empty
        let u = Q.shift(); // Dequeue a vertex from the queue

        vertex_set[dist[u] % 2].push(u); // Add the vertex to the set corresponding to its distance from the source

        for (let i = 0; i < adjlist[u].length; i++) { // Visit all neighbors of the current vertex
            let v = adjlist[u][i]; // Get the i-th neighbor of u

            if (dist[v] === -1) { // If v has not been visited yet
                dist[v] = dist[u] + 1; // Update the distance of v from the source
                Q.push(v); // Enqueue v for later processing
            }
        }
    }

    return vertex_set; // Return the two sets of vertices
}

// This function finds the vertex set with fewer vertices with odd distance from the source
function findSet(N, adjlist) {
    let source = 1; // The source vertex for BFS

    let vertex_set = BFS(N, source, adjlist); // Find the sets of vertices with even and odd distances from the source

    let inn = 0; // Index of the set with fewer vertices with odd distance from the source

    if (vertex_set[1].length < vertex_set[0].length) { // If the set of vertices with odd distance is smaller
        inn = 1; // Set the index to 1
    }

    for (let node of vertex_set[inn]) { // For each node in the smaller set
        console.log(node); // Output the node
    }
}

let N = 5; // Number of vertices
let M = 8; // Number of edges
let adjlist = new Array(N + 1).fill([]); // Initialize an empty adjacency list for each vertex

// Add the edges to the graph
adjlist = addEdge(adjlist, 2, 5);
adjlist = addEdge(adjlist, 2, 1);
adjlist = addEdge(adjlist, 5, 1);
adjlist = addEdge(adjlist, 4, 5);
adjlist = addEdge(adjlist, 1, 4);
adjlist = addEdge(adjlist, 2, 4);
adjlist = addEdge(adjlist, 3, 4);
adjlist = addEdge(adjlist, 3, 5);

findSet(N, adjlist); // Find the set of vertices with fewer vertices with odd distance from the source vertex

C#

using System;
using System.Collections.Generic;

class Program
{
    // Function to add an edge
    // to the adjacency list
    static void AddEdge(List<List<int>> adj, int u, int v)
    {
        adj[u].Add(v);
        adj[v].Add(u);
    }

    // Function to perform BFS
    // traversal on a given graph
    static List<List<int>> BFS(int N, int source, List<List<int>> adjlist)
    {
        // Stores the distance of each
        // node from the source node
        int[] dist = new int[N + 1];
        for (int i = 0; i < dist.Length; i++)
        {
            dist[i] = -1;
        }

        List<List<int>> vertex_set = new List<List<int>>();
        // Assign two separate lists
        // for parity odd and even parities
        vertex_set.Add(new List<int>());
        vertex_set.Add(new List<int>());

        // Perform BFS Traversal
        Queue<int> Q = new Queue<int>();
        // Push the source node
        Q.Enqueue(source);
        dist = 0;

        // Iterate until queue becomes empty
        while (Q.Count > 0)
        {
            // Get the front node
            // present in the queue
            int u = Q.Dequeue();

            // Push the node into vertex_set
            vertex_set[dist[u] % 2].Add(u);

            // Check if the adjacent
            // vertices are visited
            for (int i = 0; i < adjlist[u].Count; i++)
            {
                // Adjacent node
                int v = adjlist[u][i];

                // If the node v is unvisited
                if (dist[v] == -1)
                {
                    // Update the distance
                    dist[v] = dist[u] + 1;

                    // Enqueue the node v
                    Q.Enqueue(v);
                }
            }
        }

        // Return the possible set of nodes
        return vertex_set;
    }

    // Function to find a set of vertices
    // of at most N/2 nodes such that each
    // unchosen node is connected adjacently
    // to one of the nodes in the set
    static void FindSet(int N, List<List<int>> adjlist)
    {
        // Source vertex
        int source = 1;

        // Store the vertex set
        List<List<int>> vertex_set = BFS(N, source, adjlist);

        // Stores the index
        // with minimum size
        int inIndex = 0;
        if (vertex_set[1].Count < vertex_set[0].Count)
        {
            inIndex = 1;
        }

        // Print the nodes present in the set
        foreach (int node in vertex_set[inIndex])
        {
            Console.Write(node + " ");
        }
    }

    // Driver Code
    static void Main(string[] args)
    {
        int N = 5;
        int M = 8;
        List<List<int>> adjlist = new List<List<int>>();
        for (int i = 0; i < N + 1; i++)
        {
            adjlist.Add(new List<int>());
        }

        // Graph Formation
        AddEdge(adjlist, 2, 5);
        AddEdge(adjlist, 2, 1);
        AddEdge(adjlist, 5, 1);
        AddEdge(adjlist, 4, 5);
        AddEdge(adjlist, 1, 4);
        AddEdge(adjlist, 2, 4);
        AddEdge(adjlist, 3, 4);
        AddEdge(adjlist, 3, 5);

    // Function Call to print the
    // set of at most N / 2 nodes
    FindSet(N, adjlist);
    }
}

Output:

1 3

Time Complexity: O(N + M)
Auxiliary Space: O(N + M)