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
Recommended: Please try your approach on {IDE} first, before moving on to the solution.
Approach:
Initailize 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++
- Java
// 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;
}
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:_ Insert and Deletion of an edge requires O(1) complexity while it takes O(N2) to display the adjacency matrix._
#data structures #graph #matrix #placements #recursion #graph-basics #interview-preparation #picked
