Spanning Tree With Maximum Degree (Using Kruskal’s Algorithm)

Given an undirected unweighted connected graph consisting of n vertices and m edges. The task is to find any spanning tree of this graph such that the maximum degree over all vertices is maximum possible. The order in which you print the output edges does not matter and an edge can be printed in reverse also i.e. (u, v) can also be printed as (v, u).

Examples:

Input:
        1
       / \
      2   5
       \ /
        3
        |
        4
Output:
3 2
3 5
3 4
1 2
The maximum degree over all vertices 
is of vertex 3 which is 3 and is 
maximum possible.

Input:
        1
       /
      2 
     / \ 
    5   3
        |
        4
Output:
2 1
2 5
2 3
3 4

Prerequisite: Kruskal Algorithm to find Minimum Spanning Tree

Approach: The given problem can be solved using Kruskal’s algorithm to find the Minimum Spanning tree. 
We find the vertex which has maximum degree in the graph. At first we will perform the union of all the edges which are incident to this vertex and then carry out normal Kruskal’s algorithm. This gives us optimal spanning tree. 

C++

#include<bits/stdc++.h>
using namespace std;

// par and sz will store the parent
// and rank of particular node
// in the Union Find Algorithm
vector<int> par,sz;

// Find function of Union Find Algorithm
int find(int x)
{
    if(par[x]!=x)
        par[x]=find(par[x]);
    return par[x];
}

// Union function of Union Find Algorithm
void Union(int u, int v)
{
    int x = find(u);
    int y = find(v);
    if (x == y)
        return;
    if (sz[x] > sz[y])
        par[y] = x;
    else if (sz[x] < sz[y])
        par[x] = y;
    else {
        par[x] = y;
        sz[y]++;
    }
}

// Function to find the required spanning tree
void findSpanningTree(vector<int> deg,int n,int m,vector<vector<int>> g)
{
    par.resize(n+1);
    sz.resize(n+1);
    // Initialising parent of a node
    // by itself
    for (int i = 1; i <= n; i++)
        par[i] = i;

    // Variable to store the node
    // with maximum degree
    int max = 1;

    // Finding the node with maximum degree
    for (int i = 2; i <= n; i++)
        if (deg[i] > deg[max])
            max = i;

    // Union of all edges incident
    // on vertex with maximum degree
    for (int v : g[max]) {
        cout << max << " " << v << '\n';
        Union(max, v);
    }

    // Carrying out normal Kruskal Algorithm
    for (int u = 1; u <= n; u++) {
        for (int v : g[u]) {
            int x = find(u);
            int y = find(v);
            if (x == y)
                continue;
            Union(x, y);
            cout << u << " " << v << '\n';
        }
    }
}

int main()
{
    // Number of nodes
    int n = 5;

    // Number of edges
    int m = 5;

    // store the graph
    vector<vector<int>> g(n+1);

    // store the degree
    // of each node in the graph
    vector<int> deg(n+1);

    // add edges and update degrees
    g[1].push_back(2);
    g[2].push_back(1);
    deg[1]++;
    deg[2]++;
    g[1].push_back(5);
    g[5].push_back(1);
    deg[1]++;
    deg[5]++;
    g[2].push_back(3);
    g[3].push_back(2);
    deg[2]++;
    deg[3]++;
    g[5].push_back(3);
    g[3].push_back(5);
    deg[3]++;
    deg[5]++;
    g[3].push_back(4);
    g[4].push_back(3);
    deg[3]++;
    deg[4]++;

    findSpanningTree(deg, n, m, g);
    return 0;
}

Java

// Java implementation of the approach
import java.util.*;
public class GFG {

    // par and rank will store the parent
    // and rank of particular node
    // in the Union Find Algorithm
    static int par[], rank[];

    // Find function of Union Find Algorithm
    static int find(int x)
    {
        if (par[x] != x)
            par[x] = find(par[x]);
        return par[x];
    }

    // Union function of Union Find Algorithm
    static void union(int u, int v)
    {
        int x = find(u);
        int y = find(v);
        if (x == y)
            return;
        if (rank[x] > rank[y])
            par[y] = x;
        else if (rank[x] < rank[y])
            par[x] = y;
        else {
            par[x] = y;
            rank[y]++;
        }
    }

    // Function to find the required spanning tree
    static void findSpanningTree(int deg[], int n,
                                 int m, ArrayList<Integer> g[])
    {
        par = new int[n + 1];
        rank = new int[n + 1];

        // Initialising parent of a node
        // by itself
        for (int i = 1; i <= n; i++)
            par[i] = i;

        // Variable to store the node
        // with maximum degree
        int max = 1;

        // Finding the node with maximum degree
        for (int i = 2; i <= n; i++)
            if (deg[i] > deg[max])
                max = i;

        // Union of all edges incident
        // on vertex with maximum degree
        for (int v : g[max]) {
            System.out.println(max + " " + v);
            union(max, v);
        }

        // Carrying out normal Kruskal Algorithm
        for (int u = 1; u <= n; u++) {
            for (int v : g[u]) {
                int x = find(u);
                int y = find(v);
                if (x == y)
                    continue;
                union(x, y);
                System.out.println(u + " " + v);
            }
        }
    }

    // Driver code
    public static void main(String args[])
    {
        // Number of nodes
        int n = 5;

        // Number of edges
        int m = 5;

        // ArrayList to store the graph
        ArrayList<Integer> g[] = new ArrayList[n + 1];
        for (int i = 1; i <= n; i++)
            g[i] = new ArrayList<>();

        // Array to store the degree
        // of each node in the graph
        int deg[] = new int[n + 1];

        // Add edges and update degrees
        g[1].add(2);
        g[2].add(1);
        deg[1]++;
        deg[2]++;
        g[1].add(5);
        g[5].add(1);
        deg[1]++;
        deg[5]++;
        g[2].add(3);
        g[3].add(2);
        deg[2]++;
        deg[3]++;
        g[5].add(3);
        g[3].add(5);
        deg[3]++;
        deg[5]++;
        g[3].add(4);
        g[4].add(3);
        deg[3]++;
        deg[4]++;

        findSpanningTree(deg, n, m, g);
    }
}

Python3

# Python3 implementation of the approach
from typing import List

# par and rank will store the parent
# and rank of particular node
# in the Union Find Algorithm
par = []
rnk = []

# Find function of Union Find Algorithm
def find(x: int) -> int:

    global par

    if (par[x] != x):
        par[x] = find(par[x])

    return par[x]

# Union function of Union Find Algorithm
def Union(u: int, v: int) -> None:

    global par, rnk

    x = find(u)
    y = find(v)

    if (x == y):
        return
    if (rnk[x] > rnk[y]):
        par[y] = x
    elif (rnk[x] < rnk[y]):
        par[x] = y
    else:
        par[x] = y
        rnk[y] += 1

# Function to find the required spanning tree
def findSpanningTree(deg: List[int], n: int, m: int,
                     g: List[List[int]]) -> None:

    global rnk, par

    # Initialising parent of a node
    # by itself
    par = [i for i in range(n + 1)]
    rnk = [0] * (n + 1)

    # Variable to store the node
    # with maximum degree
    max = 1

    # Finding the node with maximum degree
    for i in range(2, n + 1):
        if (deg[i] > deg[max]):
            max = i

    # Union of all edges incident
    # on vertex with maximum degree
    for v in g[max]:
        print("{} {}".format(max, v))
        Union(max, v)

    # Carrying out normal Kruskal Algorithm
    for u in range(1, n + 1):
        for v in g[u]:
            x = find(u)
            y = find(v)

            if (x == y):
                continue

            Union(x, y)
            print("{} {}".format(u, v))

# Driver code
if __name__ == "__main__":

    # Number of nodes
    n = 5

    # Number of edges
    m = 5

    # ArrayList to store the graph
    g = [[] for _ in range(n + 1)]

    # Array to store the degree
    # of each node in the graph
    deg = [0] * (n + 1)

    # Add edges and update degrees
    g[1].append(2)
    g[2].append(1)
    deg[1] += 1
    deg[2] += 1
    g[1].append(5)
    g[5].append(1)
    deg[1] += 1
    deg[5] += 1
    g[2].append(3)
    g[3].append(2)
    deg[2] += 1
    deg[3] += 1
    g[5].append(3)
    g[3].append(5)
    deg[3] += 1
    deg[5] += 1
    g[3].append(4)
    g[4].append(3)
    deg[3] += 1
    deg[4] += 1

    findSpanningTree(deg, n, m, g)

# This code is contributed by sanjeev2552

C#

// C# implementation of the approach
using System;
using System.Collections.Generic;

class GFG
{

    // par and rank will store the parent
    // and rank of particular node
    // in the Union Find Algorithm
    static int []par;
    static int []rank;

    // Find function of Union Find Algorithm
    static int find(int x)
    {
        if (par[x] != x)
            par[x] = find(par[x]);
        return par[x];
    }

    // Union function of Union Find Algorithm
    static void union(int u, int v)
    {
        int x = find(u);
        int y = find(v);
        if (x == y)
            return;
        if (rank[x] > rank[y])
            par[y] = x;
        else if (rank[x] < rank[y])
            par[x] = y;
        else {
            par[x] = y;
            rank[y]++;
        }
    }

    // Function to find the required spanning tree
    static void findSpanningTree(int []deg, int n,
                                int m, List<int> []g)
    {
        par = new int[n + 1];
        rank = new int[n + 1];

        // Initialising parent of a node
        // by itself
        for (int i = 1; i <= n; i++)
            par[i] = i;

        // Variable to store the node
        // with maximum degree
        int max = 1;

        // Finding the node with maximum degree
        for (int i = 2; i <= n; i++)
            if (deg[i] > deg[max])
                max = i;

        // Union of all edges incident
        // on vertex with maximum degree
        foreach (int v in g[max])
        {
            Console.WriteLine(max + " " + v);
            union(max, v);
        }

        // Carrying out normal Kruskal Algorithm
        for (int u = 1; u <= n; u++)
        {
            foreach (int v in g[u])
            {
                int x = find(u);
                int y = find(v);
                if (x == y)
                    continue;
                union(x, y);
                Console.WriteLine(u + " " + v);
            }
        }
    }

    // Driver code
    public static void Main(String []args)
    {

        // Number of nodes
        int n = 5;

        // Number of edges
        int m = 5;

        // ArrayList to store the graph
        List<int> []g = new List<int>[n + 1];
        for (int i = 1; i <= n; i++)
            g[i] = new List<int>();

        // Array to store the degree
        // of each node in the graph
        int []deg = new int[n + 1];

        // Add edges and update degrees
        g[1].Add(2);
        g[2].Add(1);
        deg[1]++;
        deg[2]++;
        g[1].Add(5);
        g[5].Add(1);
        deg[1]++;
        deg[5]++;
        g[2].Add(3);
        g[3].Add(2);
        deg[2]++;
        deg[3]++;
        g[5].Add(3);
        g[3].Add(5);
        deg[3]++;
        deg[5]++;
        g[3].Add(4);
        g[4].Add(3);
        deg[3]++;
        deg[4]++;

        findSpanningTree(deg, n, m, g);
    }
}

// This code has been contributed by 29AjayKumar

Javascript

<script>

    // JavaScript implementation of the approach

    // par and rank will store the parent
    // and rank of particular node
    // in the Union Find Algorithm
    let par, rank;

    // Find function of Union Find Algorithm
    function find(x)
    {
        if (par[x] != x)
            par[x] = find(par[x]);
        return par[x];
    }

    // Union function of Union Find Algorithm
    function union(u, v)
    {
        let x = find(u);
        let y = find(v);
        if (x == y)
            return;
        if (rank[x] > rank[y])
            par[y] = x;
        else if (rank[x] < rank[y])
            par[x] = y;
        else {
            par[x] = y;
            rank[y]++;
        }
    }

    // Function to find the required spanning tree
    function findSpanningTree(deg, n, m, g)
    {
        par = new Array(n + 1);
        rank = new Array(n + 1);

        // Initialising parent of a node
        // by itself
        for (let i = 1; i <= n; i++)
            par[i] = i;

        // Variable to store the node
        // with maximum degree
        let max = 1;

        // Finding the node with maximum degree
        for (let i = 2; i <= n; i++)
            if (deg[i] > deg[max])
                max = i;

        // Union of all edges incident
        // on vertex with maximum degree
        for (let v = 0; v < g[max].length; v++) {
            document.write(max + " " + g[max][v] + "</br>");
            union(max, g[max][v]);
        }

        // Carrying out normal Kruskal Algorithm
        for (let u = 1; u <= n; u++) {
            for (let v = 0; v < g[u].length; v++) {
                let x = find(u);
                let y = find(g[u][v]);
                if (x == y)
                    continue;
                union(x, y);
                document.write(u + " " + g[u][v] + "</br>");
            }
        }
    }

    // Number of nodes
    let n = 5;

    // Number of edges
    let m = 5;

    // ArrayList to store the graph
    let g = new Array(n + 1);
    for (let i = 1; i <= n; i++)
      g[i] = [];

    // Array to store the degree
    // of each node in the graph
    let deg = new Array(n + 1);
    deg.fill(0);

    // Add edges and update degrees
    g[1].push(2);
    g[2].push(1);
    deg[1]++;
    deg[2]++;
    g[1].push(5);
    g[5].push(1);
    deg[1]++;
    deg[5]++;
    g[2].push(3);
    g[3].push(2);
    deg[2]++;
    deg[3]++;
    g[5].push(3);
    g[3].push(5);
    deg[3]++;
    deg[5]++;
    g[3].push(4);
    g[4].push(3);
    deg[3]++;
    deg[4]++;

    findSpanningTree(deg, n, m, g);

</script>

Output:

3 2
3 5
3 4
1 2

Time Complexity: O(N * logN), where N is the total number of nodes in the graph.
Auxiliary Space: O(N)