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 than carry out normal Kruskal’s algorithm. This gives us optimal spanning tree.

Java

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// 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;
  
        // Numbr 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);
    }
}

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C#

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// 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;
  
        // Numbr 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

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Output:

3 2
3 5
3 4
1 2


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