Add and Remove Edge in Adjacency Matrix representation of a Graph

Prerequisites: Graph and its representations
Given an adjacency matrix g[][] of a graph consisting of N vertices, the task is to modify the matrix after insertion of all edges[] and removal of edge between vertices (X, Y). In an adjacency matrix, if an edge exists between vertices i and j of the graph, then g[i][j] = 1 and g[j][i] = 1. If no edge exists between these two vertices, then g[i][j] = 0 and g[j][i] = 0.

Examples:

Input: N = 6, Edges[] = {{0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 3}, {2, 3}, {2, 4}, {2, 5}, {3, 5}}, X = 2, Y = 3 
Output:
Adjacency matrix after edge insertion:
0 1 1 1 1 0 
1 0 0 1 0 0 
1 0 0 1 1 1 
1 1 1 0 0 1 
1 0 1 0 0 0 
0 0 1 1 0 0
Adjacency matrix after edge removal:
0 1 1 1 1 0 
1 0 0 1 0 0 
1 0 0 0 1 1 
1 1 0 0 0 1 
1 0 1 0 0 0 
0 0 1 1 0 0 
Explanation:
The graph and the corresponding adjacency matrix after insertion of edges:

The graph after removal and adjacency matrix after removal of edge between vertex X and Y:

Input: N = 6, Edges[] = {{0, 1}, {0, 2}, {0, 3}, {0, 4}, {1, 3}, {2, 3}, {2, 4}, {2, 5}, {3, 5}}, X = 3, Y = 5 
Output:
Adjacency matrix after edge insertion:
0 1 1 1 1 0 
1 0 0 1 0 0 
1 0 0 1 1 1 
1 1 1 0 0 1 
1 0 1 0 0 0 
0 0 1 1 0 0
Adjacency matrix after edge removal:
0 1 1 1 1 0 
1 0 0 1 0 0 
1 0 0 1 1 1 
1 1 1 0 0 0 
1 0 1 0 0 0 
0 0 1 0 0 0

Approach:
Initialize a matrix of dimensions N x N and follow the steps below:

• Inserting an edge: To insert an edge between two vertices suppose i and j, set the corresponding values in the adjacency matrix equal to 1, i.e. g[i][j]=1 and g[j][i]=1 if both the vertices i and j exists.
• Removing an edge: To remove an edge between two vertices suppose i and j, set the corresponding values in the adjacency matrix equal to 0. That is, set g[i][j]=0 and g[j][i]=0 if both the vertices i and j exists.

Below is the implementation of the above approach:

C++

// C++ program to add and remove edge
// in the adjacency matrix of a graph

#include <iostream>
using namespace std;

class Graph {
private:
    // Number of vertices
    int n;

    // Adjacency matrix
    int g[10][10];

public:
    // Constructor
    Graph(int x)
    {
        n = x;

        // Initializing each element of the
        // adjacency matrix to zero
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                g[i][j] = 0;
            }
        }
    }

    // Function to display adjacency matrix
    void displayAdjacencyMatrix()
    {
        // Displaying the 2D matrix
        for (int i = 0; i < n; i++) {
            cout << "\n";
            for (int j = 0; j < n; j++) {
                cout << " " << g[i][j];
            }
        }
    }

    // Function to update adjacency
    // matrix for edge insertion
    void addEdge(int x, int y)
    {
        // Checks if the vertices
        // exist in the graph
        if ((x < 0) || (x >= n)) {
            cout << "Vertex" << x
                 << " does not exist!";
        }
        if ((y < 0) || (y >= n)) {
            cout << "Vertex" << y
                 << " does not exist!";
        }

        // Checks if it is a self edge
        if (x == y) {
            cout << "Same Vertex!";
        }

        else {
            // Insert edge
            g[y][x] = 1;
            g[x][y] = 1;
        }
    }

    // Function to update adjacency
    // matrix for edge removal
    void removeEdge(int x, int y)
    {
        // Checks if the vertices
        // exist in the graph
        if ((x < 0) || (x >= n)) {
            cout << "Vertex" << x
                 << " does not exist!";
        }
        if ((y < 0) || (y >= n)) {
            cout << "Vertex" << y
                 << " does not exist!";
        }

        // Checks if it is a self edge
        if (x == y) {
            cout << "Same Vertex!";
        }

        else {
            // Remove edge
            g[y][x] = 0;
            g[x][y] = 0;
        }
    }
};

// Driver Code
int main()
{
    int N = 6, X = 2, Y = 3;

    Graph obj(N);

    // Adding edges to the graph
    obj.addEdge(0, 1);
    obj.addEdge(0, 2);
    obj.addEdge(0, 3);
    obj.addEdge(0, 4);
    obj.addEdge(1, 3);
    obj.addEdge(2, 3);
    obj.addEdge(2, 4);
    obj.addEdge(2, 5);
    obj.addEdge(3, 5);

    cout << "Adjacency matrix after"
         << " edge insertions:\n";
    obj.displayAdjacencyMatrix();

    obj.removeEdge(X, Y);

    cout << "\nAdjacency matrix after"
         << " edge removal:\n";
    obj.displayAdjacencyMatrix();

    return 0;
}

Java

// Java program to add and remove edge
// in the adjacency matrix of a graph

class Graph {

    // Number of vertices
    private int n;

    // Adjacency matrix
    private int[][] g = new int[10][10];

    // Constructor
    Graph(int x)
    {
        this.n = x;

        // Initializing each element of the
        // adjacency matrix to zero
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < n; ++j) {
                g[i][j] = 0;
            }
        }
    }

    // Function to display adjacency matrix
    public void displayAdjacencyMatrix()
    {
        // Displaying the 2D matrix
        for (int i = 0; i < n; ++i) {
            System.out.println();
            for (int j = 0; j < n; ++j) {
                System.out.print(" " + g[i][j]);
            }
        }

        System.out.println();
    }

    // Function to update adjacency
    // matrix for edge insertion
    public void addEdge(int x, int y)
    {
        // Checks if the vertices exists
        if ((x < 0) || (x >= n)) {
            System.out.printf("Vertex " + x
                              + " does not exist!");
        }
        if ((y < 0) || (y >= n)) {
            System.out.printf("Vertex " + y
                              + " does not exist!");
        }

        // Checks if it is a self edge
        if (x == y) {
            System.out.println("Same Vertex!");
        }

        else {
            // Insert edge
            g[y][x] = 1;
            g[x][y] = 1;
        }
    }

    // Function to update adjacency
    // matrix for edge removal
    public void removeEdge(int x, int y)
    {
        // Checks if the vertices exists
        if ((x < 0) || (x >= n)) {
            System.out.printf("Vertex " + x
                              + " does not exist!");
        }
        if ((y < 0) || (y >= n)) {
            System.out.printf("Vertex " + y
                              + " does not exist!");
        }

        // Checks if it is a self edge
        if (x == y) {
            System.out.println("Same Vertex!");
        }

        else {
            // Remove edge
            g[y][x] = 0;
            g[x][y] = 0;
        }
    }
}

// Driver Code
class Main {
    public static void main(String[] args)
    {

        int N = 6, X = 2, Y = 3;
        Graph obj = new Graph(N);

        // Inserting edges
        obj.addEdge(0, 1);
        obj.addEdge(0, 2);
        obj.addEdge(0, 3);
        obj.addEdge(0, 4);
        obj.addEdge(1, 3);
        obj.addEdge(2, 3);
        obj.addEdge(2, 4);
        obj.addEdge(2, 5);
        obj.addEdge(3, 5);

        System.out.println("Adjacency matrix after"
                           + " edge insertions:");
        obj.displayAdjacencyMatrix();

        obj.removeEdge(2, 3);

        System.out.println("\nAdjacency matrix after"
                           + " edge removal:");
        obj.displayAdjacencyMatrix();
    }
}

Python3

# Python3 program to add and remove edge
# in adjacency matrix representation of a graph

class Graph:

    # Number of vertices
    __n = 0

    # Adjacency matrix
    __g = [[0 for x in range(10)]
              for y in range(10)]

    # Constructor
    def __init__(self, x):
        self.__n = x

        # Initializing each element of
        # the adjacency matrix to zero
        for i in range(0, self.__n):
            for j in range(0, self.__n):
                self.__g[i][j] = 0

    # Function to display adjacency matrix
    def displayAdjacencyMatrix(self):

        # Displaying the 2D matrix
        for i in range(0, self.__n):
            print()
            for j in range(0, self.__n):
                print("", self.__g[i][j], end = "")

    # Function to update adjacency
    # matrix for edge insertion
    def addEdge(self, x, y):

        # Checks if the vertices
        # exist in the graph
        if (x < 0) or (x >= self.__n):
            print("Vertex {} does not exist!".format(x))
        if (y < 0) or (y >= self.__n):
            print("Vertex {} does not exist!".format(y))

        # Checks if it is a self edge
        if(x == y):
            print("Same Vertex!")

        else:

            # Adding edge between the vertices
            self.__g[y][x] = 1
            self.__g[x][y] = 1

    # Function to update adjacency
    # matrix for edge removal
    def removeEdge(self, x, y):

        # Checks if the vertices
        # exist in the graph
        if (x < 0) or (x >= self.__n):
            print("Vertex {} does not exist!".format(x))
        if (y < 0) or (y >= self.__n):
            print("Vertex {} does not exist!".format(y))

        # Checks if it is a self edge
        if(x == y):
            print("Same Vertex!")

        else:

            # Remove edge from between
            # the vertices
            self.__g[y][x] = 0
            self.__g[x][y] = 0

# Driver code

# Creating an object of class Graph
obj = Graph(6);

# Adding edges to the graph
obj.addEdge(0, 1)
obj.addEdge(0, 2)
obj.addEdge(0, 3)
obj.addEdge(0, 4)
obj.addEdge(1, 3)
obj.addEdge(2, 3)
obj.addEdge(2, 4)
obj.addEdge(2, 5)
obj.addEdge(3, 5)

# Edges added to the adjacency matrix
print("Adjacency matrix after "
      "edge insertions:\n")
obj.displayAdjacencyMatrix();

# Removing the edge between vertices
# "2" and "3" from the graph
obj.removeEdge(2, 3);

# The adjacency matrix after
# removing the edge
print("\nAdjacency matrix after "
      "edge removal:\n")
obj.displayAdjacencyMatrix();

# This code is contributed by amarjeet_singh

C#

// C# program to add and remove edge
// in adjacency matrix representation
// of a graph
using System;

class Graph{

// Number of vertices
private int n;

// Adjacency matrix
private int[,] g = new int[10, 10];

// Constructor
public Graph(int x)
{
    this.n = x;

    // Initializing each element of
    // the adjacency matrix to zero
    for(int i = 0; i < n; ++i)
    {
        for(int j = 0; j < n; ++j)
        {
            g[i, j] = 0;
        }
    }
}

// Function to display adjacency matrix
public void displayAdjacencyMatrix()
{

    // Displaying the 2D matrix
    for(int i = 0; i < n; ++i)
    {
        Console.WriteLine();
        for(int j = 0; j < n; ++j)
        {
            Console.Write(" " + g[i, j]);
        }
    }
}

// Function to update adjacency
// matrix for edge insertion
public void addEdge(int x, int y)
{

    // Checks if the vertices exist
    // in the graph
    if ((x < 0) || (x >= n))
    {
        Console.WriteLine("Vertex {0} does " +
                          "not exist!", x);
    }
    if ((y < 0) || (y >= n))
    {
        Console.WriteLine("Vertex {0} does " +
                          "not exist!", y);
    }

    // Checks if it is a self edge
    if (x == y)
    {
        Console.WriteLine("Same Vertex!");
    }

    else
    {

        // Adding edge between the vertices
        g[y, x] = 1;
        g[x, y] = 1;
    }
}

// Function to update adjacency
// matrix for edge removal
public void removeEdge(int x, int y)
{

    // Checks if the vertices exist
    // in the graph
    if ((x < 0) || (x >= n))
    {
        Console.WriteLine("Vertex {0} does" +
                          "not exist!", x);
    }
    if ((y < 0) || (y >= n))
    {
        Console.WriteLine("Vertex {0} does" +
                          "not exist!", y);
    }

    // Checks if it is a self edge
    if (x == y)
    {
        Console.WriteLine("Same Vertex!");
    }

    else
    {

        // Remove edge from between
        // the vertices
        g[y, x] = 0;
        g[x, y] = 0;
    }
}
}

class GFG{

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

    // Creating an object of class Graph
    Graph obj = new Graph(6);

    // Adding edges to the graph
    obj.addEdge(0, 1);
    obj.addEdge(0, 2);
    obj.addEdge(0, 3);
    obj.addEdge(0, 4);
    obj.addEdge(1, 3);
    obj.addEdge(2, 3);
    obj.addEdge(2, 4);
    obj.addEdge(2, 5);
    obj.addEdge(3, 5);

    // Edges added to the adjacency matrix
    Console.WriteLine("Adjacency matrix after " +
                      "edge insertions:\n");
    obj.displayAdjacencyMatrix();

    // Removing the edge between vertices
    // "2" and "3" from the graph
    obj.removeEdge(2, 3);

    // The adjacency matrix after
    // removing the edge
    Console.WriteLine("\nAdjacency matrix after " +
                      "edge removal:");
    obj.displayAdjacencyMatrix();
}
}

// This code is contributed by amarjeet_singh

Javascript

<script>

// Javascript program to add and remove edge
// in adjacency matrix representation
// of a graph

// Number of vertices
var n = 0;

// Adjacency matrix
var g = Array.from(Array(10), ()=>Array(10).fill(0));

// Constructor
function initialize(x)
{
    n = x;

    // Initializing each element of
    // the adjacency matrix to zero
    for(var i = 0; i < n; ++i)
    {
        for(var j = 0; j < n; ++j)
        {
            g[i][j] = 0;
        }
    }
}

// Function to display adjacency matrix
function displayAdjacencyMatrix()
{

    // Displaying the 2D matrix
    for(var i = 0; i < n; ++i)
    {
        document.write("<br>");
        for(var j = 0; j < n; ++j)
        {
            document.write(" " + g[i][j]);
        }
    }
}

// Function to update adjacency
// matrix for edge insertion
function addEdge(x, y)
{

    // Checks if the vertices exist
    // in the graph
    if ((x < 0) || (x >= n))
    {
        document.write(`Vertex ${x} does not exist!`);
    }
    if ((y < 0) || (y >= n))
    {
        document.write(`Vertex ${y} does not exist!`);
    }

    // Checks if it is a self edge
    if (x == y)
    {
        document.write("Same Vertex!<br>");
    }

    else
    {

        // Adding edge between the vertices
        g[y][x] = 1;
        g[x][y] = 1;
    }
}

// Function to update adjacency
// matrix for edge removal
function removeEdge(x, y)
{

    // Checks if the vertices exist
    // in the graph
    if ((x < 0) || (x >= n))
    {
        document.write(`Vertex ${x} does not exist!`);
    }
    if ((y < 0) || (y >= n))
    {
        document.write(`Vertex ${y} does not exist!`);
    }

    // Checks if it is a self edge
    if (x == y)
    {
        document.write("Same Vertex!<br>");
    }

    else
    {

        // Remove edge from between
        // the vertices
        g[y][x] = 0;
        g[x][y] = 0;
    }
}


// Driver code
// Creating an object of class Graph
initialize(6);
// Adding edges to the graph
addEdge(0, 1);
addEdge(0, 2);
addEdge(0, 3);
addEdge(0, 4);
addEdge(1, 3);
addEdge(2, 3);
addEdge(2, 4);
addEdge(2, 5);
addEdge(3, 5);
// Edges added to the adjacency matrix
document.write("Adjacency matrix after " +
                  "edge insertions:<br>");
displayAdjacencyMatrix();
// Removing the edge between vertices
// "2" and "3" from the graph
removeEdge(2, 3);

// The adjacency matrix after
// removing the edge
document.write("<br>Adjacency matrix after " +
                  "edge removal:<br>");
displayAdjacencyMatrix();


</script>

Output:

Adjacency matrix after edge insertions:

 0 1 1 1 1 0
 1 0 0 1 0 0
 1 0 0 1 1 1
 1 1 1 0 0 1
 1 0 1 0 0 0
 0 0 1 1 0 0
Adjacency matrix after edge removal:

 0 1 1 1 1 0
 1 0 0 1 0 0
 1 0 0 0 1 1
 1 1 0 0 0 1
 1 0 1 0 0 0
 0 0 1 1 0 0

Time Complexity: Insertion and Deletion of an edge requires O(1) complexity while it takes O(N²) to display the adjacency matrix. 
Auxiliary Space: O(N²)