Vertex Cover Problem (Dynamic Programming Solution for Tree)

A vertex cover of an undirected graph is a subset of its vertices such that for every edge (u, v) of the graph, either ‘u’ or ‘v’ is in vertex cover. Although the name is Vertex Cover, the set covers all edges of the given graph. 
The problem to find minimum size vertex cover of a graph is NP complete. But it can be solved in polynomial time for trees. In this post a solution for Binary Tree is discussed. The same solution can be extended for n-ary trees.

For example, consider the following binary tree. The smallest vertex cover is {20, 50, 30} and size of the vertex cover is 3. 

The idea is to consider following two possibilities for root and recursively for all nodes down the root. 
1) Root is part of vertex cover: In this case root covers all children edges. We recursively calculate size of vertex covers for left and right subtrees and add 1 to the result (for root).

2) Root is not part of vertex cover: In this case, both children of root must be included in vertex cover to cover all root to children edges. We recursively calculate size of vertex covers of all grandchildren and number of children to the result (for two children of root).

Below are implementation of above idea. 

C

// A naive recursive C implementation for vertex cover problem for a tree
#include <stdio.h>
#include <stdlib.h>

// A utility function to find min of two integers
int min(int x, int y) { return (x < y)? x: y; }

/* A binary tree node has data, pointer to left child and a pointer to
   right child */
struct node
{
    int data;
    struct node *left, *right;
};

// The function returns size of the minimum vertex cover
int vCover(struct node *root)
{
    // The size of minimum vertex cover is zero if tree is empty or there
    // is only one node
    if (root == NULL)
        return 0;
    if (root->left == NULL && root->right == NULL)
        return 0;

    // Calculate size of vertex cover when root is part of it
    int size_incl = 1 + vCover(root->left) + vCover(root->right);

    // Calculate size of vertex cover when root is not part of it
    int size_excl = 0;
    if (root->left)
      size_excl += 1 + vCover(root->left->left) + vCover(root->left->right);
    if (root->right)
      size_excl += 1 + vCover(root->right->left) + vCover(root->right->right);

    // Return the minimum of two sizes
    return min(size_incl, size_excl);
}

// A utility function to create a node
struct node* newNode( int data )
{
    struct node* temp = (struct node *) malloc( sizeof(struct node) );
    temp->data = data;
    temp->left = temp->right = NULL;
    return temp;
}

// Driver program to test above functions
int main()
{
    // Let us construct the tree given in the above diagram
    struct node *root         = newNode(20);
    root->left                = newNode(8);
    root->left->left          = newNode(4);
    root->left->right         = newNode(12);
    root->left->right->left   = newNode(10);
    root->left->right->right  = newNode(14);
    root->right               = newNode(22);
    root->right->right        = newNode(25);

    printf ("Size of the smallest vertex cover is %d ", vCover(root));

    return 0;
}

C++

#include <iostream>
#include <cstdlib>

// A utility function to find min of two integers
int min(int x, int y) { return (x < y)? x: y; }

/* A binary tree node has data, pointer to left child and a pointer to
   right child */
struct node
{
    int data;
    node *left, *right;
};

// The function returns size of the minimum vertex cover
int vCover(node *root)
{
    // The size of minimum vertex cover is zero if tree is empty or there
    // is only one node
    if (root == nullptr)
        return 0;
    if (root->left == nullptr && root->right == nullptr)
        return 0;

    // Calculate size of vertex cover when root is part of it
    int size_incl = 1 + vCover(root->left) + vCover(root->right);

    // Calculate size of vertex cover when root is not part of it
    int size_excl = 0;
    if (root->left)
      size_excl += 1 + vCover(root->left->left) + vCover(root->left->right);
    if (root->right)
      size_excl += 1 + vCover(root->right->left) + vCover(root->right->right);

    // Return the minimum of two sizes
    return min(size_incl, size_excl);
}

// A utility function to create a node
node* newNode( int data )
{
    node* temp = new node;
    temp->data = data;
    temp->left = temp->right = nullptr;
    return temp;
}

// Driver program to test above functions
int main()
{
    // Let us construct the tree given in the above diagram
    node *root         = newNode(20);
    root->left                = newNode(8);
    root->left->left          = newNode(4);
    root->left->right         = newNode(12);
    root->left->right->left   = newNode(10);
    root->left->right->right  = newNode(14);
    root->right               = newNode(22);
    root->right->right        = newNode(25);

    std::cout << "Size of the smallest vertex cover is " << vCover(root) << std::endl;

    return 0;
}

Java

// A naive recursive Java implementation
// for vertex cover problem for a tree

class GFG
{
    // A utility function to find min of two integers
    static int min(int x, int y)
    {
        return (x < y) ? x : y;
    }

    /*
    * A binary tree node has data, pointer
    to left child and a pointer to right
    * child
    */
    static class node
    {
        int data;
        node left, right;
    };

    // The function returns size
    // of the minimum vertex cover
    static int vCover(node root)
    {
        // The size of minimum vertex cover
        // is zero if tree is empty or there
        // is only one node
        if (root == null)
            return 0;
        if (root.left == null && root.right == null)
            return 0;

        // Calculate size of vertex cover
        // when root is part of it
        int size_incl = 1 + vCover(root.left) +
                               vCover(root.right);

        // Calculate size of vertex cover
        // when root is not part of it
        int size_excl = 0;
        if (root.left != null)
            size_excl += 1 + vCover(root.left.left) +
                              vCover(root.left.right);
        if (root.right != null)
            size_excl += 1 + vCover(root.right.left) +
                                vCover(root.right.right);

        // Return the minimum of two sizes
        return Math.min(size_incl, size_excl);
    }

    // A utility function to create a node
    static node newNode(int data)
    {
        node temp = new node();
        temp.data = data;
        temp.left = temp.right = null;
        return temp;
    }

    // Driver code
    public static void main(String[] args)
    {
        // Let us construct tree given in the above diagram
        node root = newNode(20);
        root.left = newNode(8);
        root.left.left = newNode(4);
        root.left.right = newNode(12);
        root.left.right.left = newNode(10);
        root.left.right.right = newNode(14);
        root.right = newNode(22);
        root.right.right = newNode(25);

        System.out.printf("Size of the smallest vertex" +
                            "cover is %d ", vCover(root));

    }
}

// This code is contributed by 29AjayKumar

Python3

# A naive recursive Python3 implementation
# for vertex cover problem for a tree

# A utility function to find min of two integers

# A binary tree node has data, pointer to
# left child and a pointer to right child
class Node:

    def __init__(self, x):

        self.data = x
        self.left = None
        self.right = None

# The function returns size of
# the minimum vertex cover
def vCover(root):

    # The size of minimum vertex cover
    # is zero if tree is empty or there
    # is only one node
    if (root == None):
        return 0

    if (root.left == None and
       root.right == None):
        return 0

    # Calculate size of vertex cover when
    # root is part of it
    size_incl = (1 + vCover(root.left) +
                     vCover(root.right))

    # Calculate size of vertex cover
    # when root is not part of it
    size_excl = 0
    if (root.left):
      size_excl += (1 + vCover(root.left.left) +
                        vCover(root.left.right))
    if (root.right):
      size_excl += (1 + vCover(root.right.left) +
                        vCover(root.right.right))

    # Return the minimum of two sizes
    return min(size_incl, size_excl)

# Driver Code
if __name__ == '__main__':

    # Let us construct the tree
    # given in the above diagram
    root  = Node(20)
    root.left = Node(8)
    root.left.left = Node(4)
    root.left.right = Node(12)
    root.left.right.left = Node(10)
    root.left.right.right = Node(14)
    root.right  = Node(22)
    root.right.right = Node(25)

    print("Size of the smallest vertex cover is", vCover(root))

# This code is contributed by mohit kumar 29

C#

// A naive recursive C# implementation
// for vertex cover problem for a tree
using System;

class GFG
{
    // A utility function to find
    // min of two integers
    static int min(int x, int y)
    {
        return (x < y) ? x : y;
    }

    /*
    * A binary tree node has data, pointer
    to left child and a pointer to right
    * child
    */
    public class node
    {
        public int data;
        public node left, right;
    };

    // The function returns size
    // of the minimum vertex cover
    static int vCover(node root)
    {
        // The size of minimum vertex cover
        // is zero if tree is empty or there
        // is only one node
        if (root == null)
            return 0;
        if (root.left == null &&
            root.right == null)
            return 0;

        // Calculate size of vertex cover
        // when root is part of it
        int size_incl = 1 + vCover(root.left) +
                            vCover(root.right);

        // Calculate size of vertex cover
        // when root is not part of it
        int size_excl = 0;
        if (root.left != null)
            size_excl += 1 + vCover(root.left.left) +
                             vCover(root.left.right);
        if (root.right != null)
            size_excl += 1 + vCover(root.right.left) +
                             vCover(root.right.right);

        // Return the minimum of two sizes
        return Math.Min(size_incl, size_excl);
    }

    // A utility function to create a node
    static node newNode(int data)
    {
        node temp = new node();
        temp.data = data;
        temp.left = temp.right = null;
        return temp;
    }

    // Driver code
    public static void Main(String[] args)
    {
        // Let us construct tree given
        // in the above diagram
        node root = newNode(20);
        root.left = newNode(8);
        root.left.left = newNode(4);
        root.left.right = newNode(12);
        root.left.right.left = newNode(10);
        root.left.right.right = newNode(14);
        root.right = newNode(22);
        root.right.right = newNode(25);

        Console.Write("Size of the smallest vertex" +
                      "cover is {0} ", vCover(root));
    }
}

// This code is contributed by 29AjayKumar

Javascript

<script>
// A naive recursive Javascript implementation
// for vertex cover problem for a tree

    // A utility function to find min of two integers
    function min(x,y)
    {
         return (x < y) ? x : y;
    }

    /*
    * A binary tree node has data, pointer
    to left child and a pointer to right
    * child
    */
    class Node
    {
        constructor(d)
        {
            this.data=d;
            this.left=null;
            this.right=null;
        }
    }

    // The function returns size
    // of the minimum vertex cover
    function vCover(root)
    {
        // The size of minimum vertex cover
        // is zero if tree is empty or there
        // is only one node
        if (root == null)
            return 0;
        if (root.left == null && root.right == null)
            return 0;

        // Calculate size of vertex cover
        // when root is part of it
        let size_incl = 1 + vCover(root.left) +
                               vCover(root.right);

        // Calculate size of vertex cover
        // when root is not part of it
        let size_excl = 0;
        if (root.left != null)
            size_excl += 1 + vCover(root.left.left) +
                              vCover(root.left.right);
        if (root.right != null)
            size_excl += 1 + vCover(root.right.left) +
                                vCover(root.right.right);

        // Return the minimum of two sizes
        return Math.min(size_incl, size_excl);
    }

    // Driver code
    // Let us construct tree given in the above diagram
    root = new Node(20);
    root.left = new Node(8);
    root.left.left = new Node(4);
    root.left.right = new Node(12);
    root.left.right.left = new Node(10);
    root.left.right.right = new Node(14);
    root.right = new Node(22);
    root.right.right = new Node(25);

    document.write("Size of the smallest vertex" +
                            "cover is ", vCover(root));

    // This code is contributed by unknown2108
</script>

Output

Size of the smallest vertex cover is 3 

Time complexity of the above naive recursive approach is exponential. It should be noted that the above function computes the same subproblems again and again. For example, vCover of node with value 50 is evaluated twice as 50 is grandchild of 10 and child of 20.

Since same subproblems are called again, this problem has Overlapping Subproblems property. So Vertex Cover problem has both properties (see this and this) of a dynamic programming problem. Like other typical Dynamic Programming(DP) problems, re-computations of same subproblems can be avoided by storing the solutions to subproblems and solving problems in bottom up manner.

Following is the implementation of Dynamic Programming based solution. In the following solution, an additional field ‘vc’ is added to tree nodes. The initial value of ‘vc’ is set as 0 for all nodes. The recursive function vCover() calculates ‘vc’ for a node only if it is not already set.

C

/* Dynamic programming based program for Vertex Cover problem for
   a Binary Tree */
#include <stdio.h>
#include <stdlib.h>

// A utility function to find min of two integers
int min(int x, int y) { return (x < y)? x: y; }

/* A binary tree node has data, pointer to left child and a pointer to
   right child */
struct node
{
    int data;
    int vc;
    struct node *left, *right;
};

// A memoization based function that returns size of the minimum vertex cover.
int vCover(struct node *root)
{
    // The size of minimum vertex cover is zero if tree is empty or there
    // is only one node
    if (root == NULL)
        return 0;
    if (root->left == NULL && root->right == NULL)
        return 0;

    // If vertex cover for this node is already evaluated, then return it
    // to save recomputation of same subproblem again.
    if (root->vc != 0)
        return root->vc;

    // Calculate size of vertex cover when root is part of it
    int size_incl = 1 + vCover(root->left) + vCover(root->right);

    // Calculate size of vertex cover when root is not part of it
    int size_excl = 0;
    if (root->left)
      size_excl += 1 + vCover(root->left->left) + vCover(root->left->right);
    if (root->right)
      size_excl += 1 + vCover(root->right->left) + vCover(root->right->right);

    // Minimum of two values is vertex cover, store it before returning
    root->vc =  min(size_incl, size_excl);

    return root->vc;
}

// A utility function to create a node
struct node* newNode( int data )
{
    struct node* temp = (struct node *) malloc( sizeof(struct node) );
    temp->data = data;
    temp->left = temp->right = NULL;
    temp->vc = 0; // Set the vertex cover as 0
    return temp;
}

// Driver program to test above functions
int main()
{
    // Let us construct the tree given in the above diagram
    struct node *root         = newNode(20);
    root->left                = newNode(8);
    root->left->left          = newNode(4);
    root->left->right         = newNode(12);
    root->left->right->left   = newNode(10);
    root->left->right->right  = newNode(14);
    root->right               = newNode(22);
    root->right->right        = newNode(25);

    printf ("Size of the smallest vertex cover is %d ", vCover(root));

    return 0;
}

C++

#include <iostream>
#include <stdlib.h>

// A utility function to find min of two integers
int min(int x, int y) { return (x < y)? x: y; }

/* A binary tree node has data, pointer to left child and a pointer to
   right child */
struct node
{
    int data;
    int vc;
    struct node *left, *right;
};

// A memoization based function that returns size of the minimum vertex cover.
int vCover(struct node *root)
{
    // The size of minimum vertex cover is zero if tree is empty or there
    // is only one node
    if (root == NULL)
        return 0;
    if (root->left == NULL && root->right == NULL)
        return 0;

    // If vertex cover for this node is already evaluated, then return it
    // to save recomputation of same subproblem again.
    if (root->vc != 0)
        return root->vc;

    // Calculate size of vertex cover when root is part of it
    int size_incl = 1 + vCover(root->left) + vCover(root->right);

    // Calculate size of vertex cover when root is not part of it
    int size_excl = 0;
    if (root->left)
      size_excl += 1 + vCover(root->left->left) + vCover(root->left->right);
    if (root->right)
      size_excl += 1 + vCover(root->right->left) + vCover(root->right->right);

    // Minimum of two values is vertex cover, store it before returning
    root->vc =  min(size_incl, size_excl);

    return root->vc;
}

// A utility function to create a node
struct node* newNode( int data )
{
    struct node* temp = (struct node *) malloc( sizeof(struct node) );
    temp->data = data;
    temp->left = temp->right = NULL;
    temp->vc = 0; // Set the vertex cover as 0
    return temp;
}

// Driver program to test above functions
int main()
{
    // Let us construct the tree given in the above diagram
    struct node *root         = newNode(20);
    root->left                = newNode(8);
    root->left->left          = newNode(4);
    root->left->right         = newNode(12);
    root->left->right->left   = newNode(10);
    root->left->right->right  = newNode(14);
    root->right               = newNode(22);
    root->right->right        = newNode(25);

    std::cout << "Size of the smallest vertex cover is " << vCover(root) << std::endl;

    return 0;
}

Java

/* Dynamic programming based program for
Vertex Cover problem for a Binary Tree */

class GFG
{
    // A utility function to find min of two integers
    static int min(int x, int y)
    {
        return (x < y) ? x : y;
    }

    /*
    * A binary tree node has data, pointer
    to left child and a pointer to right
    * child
    */
    static class node
    {
        int data;
        int vc;
        node left, right;
    };

    // A memoization based function that returns
    // size of the minimum vertex cover.
    static int vCover(node root)
    {
        // The size of minimum vertex cover is zero
        //  if tree is empty or there is only one node
        if (root == null)
            return 0;
        if (root.left == null && root.right == null)
            return 0;

        // If vertex cover for this node is
        // already evaluated, then return it
        // to save recomputation of same subproblem again.
        if (root.vc != 0)
            return root.vc;

        // Calculate size of vertex cover
        // when root is part of it
        int size_incl = 1 + vCover(root.left) +
                            vCover(root.right);

        // Calculate size of vertex cover
        // when root is not part of it
        int size_excl = 0;
        if (root.left != null)
            size_excl += 1 + vCover(root.left.left) +
                                vCover(root.left.right);
        if (root.right != null)
            size_excl += 1 + vCover(root.right.left) +
                                vCover(root.right.right);

        // Minimum of two values is vertex cover,
        // store it before returning
        root.vc = Math.min(size_incl, size_excl);

        return root.vc;
    }

    // A utility function to create a node
    static node newNode(int data)
    {
        node temp = new node();
        temp.data = data;
        temp.left = temp.right = null;
        temp.vc = 0; // Set the vertex cover as 0
        return temp;
    }

    // Driver code
    public static void main(String[] args)
    {
        // Let us construct tree given in the above diagram
        node root = newNode(20);
        root.left = newNode(8);
        root.left.left = newNode(4);
        root.left.right = newNode(12);
        root.left.right.left = newNode(10);
        root.left.right.right = newNode(14);
        root.right = newNode(22);
        root.right.right = newNode(25);

        System.out.printf("Size of the smallest vertex" +
                            "cover is %d ", vCover(root));
    }
}

// This code is contributed by PrinciRaj1992

C#

/* Dynamic programming based program for
Vertex Cover problem for a Binary Tree */
using System;

class GFG
{
    // A utility function to find
    // min of two integers
    static int min(int x, int y)
    {
        return (x < y) ? x : y;
    }

    /*
    * A binary tree node has data, pointer
    to left child and a pointer to right
    * child
    */
    class node
    {
        public int data;
        public int vc;
        public node left, right;
    };

    // A memoization based function that returns
    // size of the minimum vertex cover.
    static int vCover(node root)
    {
        // The size of minimum vertex cover is zero
        // if tree is empty or there is only one node
        if (root == null)
            return 0;
        if (root.left == null &&
            root.right == null)
            return 0;

        // If vertex cover for this node is
        // already evaluated, then return it
        // to save recomputation of same subproblem again.
        if (root.vc != 0)
            return root.vc;

        // Calculate size of vertex cover
        // when root is part of it
        int size_incl = 1 + vCover(root.left) +
                            vCover(root.right);

        // Calculate size of vertex cover
        // when root is not part of it
        int size_excl = 0;
        if (root.left != null)
            size_excl += 1 + vCover(root.left.left) +
                             vCover(root.left.right);
        if (root.right != null)
            size_excl += 1 + vCover(root.right.left) +
                             vCover(root.right.right);

        // Minimum of two values is vertex cover,
        // store it before returning
        root.vc = Math.Min(size_incl, size_excl);

        return root.vc;
    }

    // A utility function to create a node
    static node newNode(int data)
    {
        node temp = new node();
        temp.data = data;
        temp.left = temp.right = null;
        temp.vc = 0; // Set the vertex cover as 0
        return temp;
    }

    // Driver code
    public static void Main(String[] args)
    {
        // Let us construct tree given in the above diagram
        node root = newNode(20);
        root.left = newNode(8);
        root.left.left = newNode(4);
        root.left.right = newNode(12);
        root.left.right.left = newNode(10);
        root.left.right.right = newNode(14);
        root.right = newNode(22);
        root.right.right = newNode(25);

        Console.Write("Size of the smallest vertex" +
                      "cover is {0} ", vCover(root));
    }
}

// This code is contributed by PrinciRaj1992

Javascript

<script>

/* Dynamic programming based program for
Vertex Cover problem for a Binary Tree */

// A utility function to find min of two integers
function min(x,y)
{
    return (x < y) ? x : y;
}

/*
    * A binary tree node has data, pointer
    to left child and a pointer to right
    * child
    */
class Node
{
    constructor(data)
    {
        this.vc=0;
        this.data=data;
        this.left=this.right=null;
    }
}

// A memoization based function that returns
// size of the minimum vertex cover.
function vCover(root)
{
    // The size of minimum vertex cover is zero
        //  if tree is empty or there is only one node
        if (root == null)
            return 0;
        if (root.left == null && root.right == null)
            return 0;

        // If vertex cover for this node is
        // already evaluated, then return it
        // to save recomputation of same subproblem again.
        if (root.vc != 0)
            return root.vc;

        // Calculate size of vertex cover
        // when root is part of it
        let size_incl = 1 + vCover(root.left) +
                            vCover(root.right);

        // Calculate size of vertex cover
        // when root is not part of it
        let size_excl = 0;
        if (root.left != null)
            size_excl += 1 + vCover(root.left.left) +
                                vCover(root.left.right);
        if (root.right != null)
            size_excl += 1 + vCover(root.right.left) +
                                vCover(root.right.right);

        // Minimum of two values is vertex cover,
        // store it before returning
        root.vc = Math.min(size_incl, size_excl);

        return root.vc;
}

// Driver code
// Let us construct tree given in the above diagram
        let root = new Node(20);
        root.left = new Node(8);
        root.left.left = new Node(4);
        root.left.right = new Node(12);
        root.left.right.left = new Node(10);
        root.left.right.right = new Node(14);
        root.right = new Node(22);
        root.right.right = new Node(25);

        document.write("Size of the smallest vertex " +
                            "cover is ", vCover(root));



// This code is contributed by rag2127

</script>

Python3

# A binary tree node has data, pointer to left child and a pointer to
#   right child


class node:

    def __init__(self):
        # instance fields found by C++ to Python Converter:
        self.data = 0
        self.vc = 0
        self.left = None
        self.right = None


# Driver program to test above functions
def main():
    # Let us construct the tree given in the above diagram
    root = Globals.newNode(20)
    root.left = Globals.newNode(8)
    root.left.left = Globals.newNode(4)
    root.left.right = Globals.newNode(12)
    root.left.right.left = Globals.newNode(10)
    root.left.right.right = Globals.newNode(14)
    root.right = Globals.newNode(22)
    root.right.right = Globals.newNode(25)

    print("Size of the smallest vertex cover is ", end='')
    print(Globals.vCover(root), end='')
    print("\n", end='')


class Globals:
    # A utility function to find min of two integers
    @staticmethod
    def min(x, y):
        if(x < y):
            return x
        else:
            return y

    # A memoization based function that returns size of the minimum vertex cover.

    @staticmethod
    def vCover(root):
        # The size of minimum vertex cover is zero if tree is empty or there
        # is only one node
        if root is None:
            return 0
        if root.left is None and root.right is None:
            return 0

        # If vertex cover for this node is already evaluated, then return it
        # to save recomputation of same subproblem again.
        if root.vc != 0:
            return root.vc

        # Calculate size of vertex cover when root is part of it
        size_incl = 1 + Globals.vCover(root.left) + Globals.vCover(root.right)

        # Calculate size of vertex cover when root is not part of it
        size_excl = 0
        if root.left:
            size_excl += 1 + \
                Globals.vCover(root.left.left) + \
                Globals.vCover(root.left.right)
        if root.right:
            size_excl += 1 + \
                Globals.vCover(root.right.left) + \
                Globals.vCover(root.right.right)

        # Minimum of two values is vertex cover, store it before returning
        root.vc = Globals.min(size_incl, size_excl)

        return root.vc

    # A utility function to create a node
    @staticmethod
    def newNode(data):
        temp = node()
        temp.data = data
        temp.left = temp.right = None
        temp.vc = 0  # Set the vertex cover as 0
        return temp


if __name__ == "__main__":
    main()

Output

Size of the smallest vertex cover is 3 

Time Complexity:
The time complexity of the vCover function is O(n), where n is the number of nodes in the binary tree. This is because each node is visited only once and its vertex cover size is calculated in constant time.

Space Complexity:
The space complexity of the program is O(n), where n is the number of nodes in the binary tree. This is because the space required for the binary tree is proportional to the number of nodes in the tree. Additionally, the memoization technique used in the vCover function requires additional space to store the vertex cover sizes of already evaluated nodes. However, since the depth of the recursion tree is limited by the height of the binary tree, the space complexity of the program is also O(h), where h is the height of the binary tree. In the worst case scenario where the binary tree is skewed, the height of the tree can be as large as n, which would result in a space complexity of O(n).

References:
http://courses.csail.mit.edu/6.006/spring11/lectures/lec21.pdf
Exercise:
Extend the above solution for n-ary trees. 
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

Approach for any general tree :

1. Approach will be same dynamic programming approach as discussed.

2. For every node, if we exclude this node from vertex cover than we have to include its neighbouring nodes,

   and if we include this node in the vertex cover than we will take the minimum of the two possibilities of taking its neighbouring

   nodes in the vertex cover to get minimum vertex cover. 

3. We will store the above information in the dp array.

C++

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

// An utility function to add an edge in the tree
void addEdge(vector<int> adj[], int x, int y)
{
    adj[x].push_back(y);
    adj[y].push_back(x);
}

void dfs(vector<int> adj[], vector<int> dp[], int src,
         int par)
{
    for (auto child : adj[src]) {
        if (child != par)
            dfs(adj, dp, child, src);
    }

    for (auto child : adj[src]) {
        if (child != par) {
            // not including source in the vertex cover
            dp[src][0] += dp[child][1];

            // including source in the vertex cover
            dp[src][1] += min(dp[child][1], dp[child][0]);
        }
    }
}

// function to find minimum size of vertex cover
void minSizeVertexCover(vector<int> adj[], int N)
{
    vector<int> dp[N + 1];

    for (int i = 1; i <= N; i++) {
        // 0 denotes not included in vertex cover
        dp[i].push_back(0);

        // 1 denotes included in vertex cover
        dp[i].push_back(1);
    }

    dfs(adj, dp, 1, -1);

    // printing minimum size vertex cover
    cout << min(dp[1][0], dp[1][1]) << endl;
}

// Driver Code
int main()
{
    /*                          1

                        /            \

                     2                7

               /             \

             3                6

    /        |        \

  4          8          5

 */

    // number of nodes in the tree
    int N = 8;

    // adjacency list representation of the tree
    vector<int> adj[N + 1];

    addEdge(adj, 1, 2);
    addEdge(adj, 1, 7);
    addEdge(adj, 2, 3);
    addEdge(adj, 2, 6);
    addEdge(adj, 3, 4);
    addEdge(adj, 3, 8);
    addEdge(adj, 3, 5);

    minSizeVertexCover(adj, N);

    return 0;
}

Java

// Java implementation for the above approach
import java.util.*;

class GFG {
    // An utility function to add an edge in the tree
    static void addEdge(List<List<Integer> > adj, int x,
                        int y)
    {
        adj.get(x).add(y);
        adj.get(y).add(x);
    }

    static void dfs(List<List<Integer> > adj,
                    List<List<Integer> > dp, int src,
                    int par)
    {
        for (Integer child : adj.get(src)) {
            if (child != par)
                dfs(adj, dp, child, src);
        }

        for (Integer child : adj.get(src)) {
            if (child != par) {
                // not including source in the vertex cover
                dp.get(src).set(0,
                                dp.get(child).get(1)
                                    + dp.get(src).get(0));

                // including source in the vertex cover
                dp.get(src).set(
                    1,
                    dp.get(src).get(1)
                        + Math.min(dp.get(child).get(1),
                                   dp.get(child).get(0)));
            }
        }
    }

    // function to find minimum size of vertex cover
    static void minSizeVertexCover(List<List<Integer> > adj,
                                   int N)
    {
        List<List<Integer> > dp = new ArrayList<>();
        for (int i = 0; i <= N; i++) {
            dp.add(new ArrayList<>());
        }
        for (int i = 1; i <= N; i++) {
            // 0 denotes not included in vertex cover
            dp.get(i).add(0);

            // 1 denotes included in vertex cover
            dp.get(i).add(1);
        }

        dfs(adj, dp, 1, -1);

        // printing minimum size vertex cover
        System.out.println(
            Math.min(dp.get(1).get(0), dp.get(1).get(1)));
    }

    public static void main(String[] args)
    {
        /*                          1

                            /            \

                        2                7

                /             \

                3                6

        /        |        \

    4          8          5

    */

        // number of nodes in the tree
        int N = 8;

        // adjacency list representation of the tree
        List<List<Integer> > adj = new ArrayList<>();
        for (int i = 0; i <= N; i++) {
            adj.add(new ArrayList<>());
        }
        addEdge(adj, 1, 2);
        addEdge(adj, 1, 7);
        addEdge(adj, 2, 3);
        addEdge(adj, 2, 6);
        addEdge(adj, 3, 4);
        addEdge(adj, 3, 8);
        addEdge(adj, 3, 5);

        minSizeVertexCover(adj, N);
    }
}

Python3

# Python3 implementation for the above approach


def addEdge(adj, x, y):
    adj[x].append(y)
    adj[y].append(x)


def dfs(adj, dp, src, par):
    for child in adj[src]:
        if child != par:
            dfs(adj, dp, child, src)

    for child in adj[src]:
        if child != par:
            # not including source in the vertex cover
            dp[src][0] = dp[child][1] + dp[src][0]

            # including source in the vertex cover
            dp[src][1] = dp[src][1] + min(dp[child][1], dp[child][0])


def minSizeVertexCover(adj, N):
    dp = [[0 for j in range(2)] for i in range(N+1)]
    for i in range(1, N+1):
        # 0 denotes not included in vertex cover
        dp[i][0] = 0

        # 1 denotes included in vertex cover
        dp[i][1] = 1

    dfs(adj, dp, 1, -1)

    # printing minimum size vertex cover
    print(min(dp[1][0], dp[1][1]))


# Driver Code
"""
          1
        /   \
      2       7
     / \
    3   6
   /|\
  4 8 5
"""
# number of nodes in the tree
N = 8

# adjacency list representation of the tree
adj = [[] for i in range(N+1)]
addEdge(adj, 1, 2)
addEdge(adj, 1, 7)
addEdge(adj, 2, 3)
addEdge(adj, 2, 6)
addEdge(adj, 3, 4)
addEdge(adj, 3, 8)
addEdge(adj, 3, 5)

minSizeVertexCover(adj, N)

C#

using System;
using System.Collections.Generic;

class GFG
{
    // An utility function to add an edge in the tree
    static void addEdge(List<List<int>> adj, int x, int y)
    {
        adj[x].Add(y);
        adj[y].Add(x);
    }

    static void dfs(List<List<int>> adj, List<List<int>> dp, int src, int par)
    {
        foreach (int child in adj[src])
        {
            if (child != par)
                dfs(adj, dp, child, src);
        }

        foreach (int child in adj[src])
        {
            if (child != par)
            {
                // not including source in the vertex cover
                dp[src][0] = dp[child][1] + dp[src][0];

                // including source in the vertex cover
                dp[src][1] = dp[src][1] + Math.Min(dp[child][1], dp[child][0]);
            }
        }
    }

    // function to find minimum size of vertex cover
    static void minSizeVertexCover(List<List<int>> adj, int N)
    {
        List<List<int>> dp = new List<List<int>>();
        for (int i = 0; i <= N; i++)
        {
            dp.Add(new List<int>());
        }
        for (int i = 1; i <= N; i++)
        {
            // 0 denotes not included in vertex cover
            dp[i].Add(0);

            // 1 denotes included in vertex cover
            dp[i].Add(1);
        }

        dfs(adj, dp, 1, -1);

        // printing minimum size vertex cover
        Console.WriteLine(Math.Min(dp[1][0], dp[1][1]));
    }

    public static void Main(string[] args)
    {
        /*
                            1

                            /            \

                        2                7

                /             \

                3                6

        /        |        \

    4          8          5

    */

        // number of nodes in the tree
        int N = 8;

        // adjacency list representation of the tree
        List<List<int>> adj = new List<List<int>>();
        for (int i = 0; i <= N; i++)
        {
            adj.Add(new List<int>());
        }
        addEdge(adj, 1, 2);
        addEdge(adj, 1, 7);
        addEdge(adj, 2, 3);
        addEdge(adj, 2, 6);
        addEdge(adj, 3, 4);
        addEdge(adj, 3, 8);
        addEdge(adj, 3, 5);

        minSizeVertexCover(adj, N);
    }
}

Javascript

// Javascript implementation for the above approach

      // An utility function to add an edge in the tree
      function addEdge(adj, x, y) {
        adj[x].push(y);
        adj[y].push(x);
      }

      function dfs(adj, dp, src, par) {
        for (let child of adj[src]) {
          if (child != par) dfs(adj, dp, child, src);
        }

        for (let child of adj[src]) {
          if (child != par) {
            // not including source in the vertex cover
            dp[src][0] += dp[child][1];

            // including source in the vertex cover
            dp[src][1] += Math.min(dp[child][1], dp[child][0]);
          }
        }
      }

      // function to find minimum size of vertex cover
      function minSizeVertexCover(adj, N) {
        let dp = Array.from(Array(N + 1), () => new Array());

        for (var i = 1; i <= N; i++) {
          // 0 denotes not included in vertex cover
          dp[i].push(0);

          // 1 denotes included in vertex cover
          dp[i].push(1);
        }

        dfs(adj, dp, 1, -1);

        // printing minimum size vertex cover
        console.log(Math.min(dp[1][0], dp[1][1]));
      }

      // Driver Code

      /*                          1

                                /            \

                             2                7

                       /             \

                     3                6

            /        |        \

          4          8          5

         */

      // number of nodes in the tree
      var N = 8;

      // adjacency list representation of the tree
      let adj = Array.from(Array(N + 1), () => new Array());

      addEdge(adj, 1, 2);
      addEdge(adj, 1, 7);
      addEdge(adj, 2, 3);
      addEdge(adj, 2, 6);
      addEdge(adj, 3, 4);
      addEdge(adj, 3, 8);
      addEdge(adj, 3, 5);

      minSizeVertexCover(adj, N);

Output

3

Time Complexity : O(N)
Auxiliary space : O(N)