C++ Program to Represent Graph Using Incidence Matrix

What is incidence matrix?

An incidence matrix is a mathematical representation of a graph that shows the relationship between vertices and edges. In other words, it is a matrix M of size v x e, where v = number of vertices and e = number of edges. For example, if the graph has an edge number n from vertex i to vertex j, then in the incidence matrix at i^th row and n^th column it will be 1.

Graph

The image below represent a simple undirected graph with 6 vertices and 8 edges.

Incidence Matrix

The incidence matrix of the above graph is shown below.

Algorithm

The following are the steps to represent a graph using an incidence matrix:

• Step 1: Start
• Step 2: Input the number of vertices V and number of edges E in the graph
• Step 3: Initialize a V x E matrix (incMatrix) and set all values to 0.
• Step 4: For each edge, repeat the steps 5,6 and 7.
• Step 5: Input the pair of vertices (u, v) that the edge connects
• Step 6: If the graph is undirected,set incMatrix[u][edge] = 1 and set incMatrix[v][edge] = 1
• Step 7: If the graph is directed,set incMatrix[u][edge] = -1 (edge leaves u) and set incMatrix[v][edge] = 1 (edge enters v)
• Step 8: Display the incidence matrix.
• Step 9: End

C++ Program for Incidence Matrix Representation of a Graph

In the code below we will be implementing the Incidence Matrix Representation of a Graph in C++:

#include <iostream>
using namespace std;

int main() {
    int V, E;
    bool isDirected;

    // Step 1 & 2: Input vertices and edges
    cout << "Enter number of vertices: ";
    cin >> V;
    cout << "Enter number of edges: ";
    cin >> E;

    cout << "Is the graph directed? (1 for Yes, 0 for No): ";
    cin >> isDirected;

    // Step 3: Initialize incidence matrix
    int incMatrix[V][E];
    for (int i = 0; i < V; i++)
        for (int j = 0; j < E; j++)
            incMatrix[i][j] = 0;

    // Step 4: Input edges
    for (int e = 0; e < E; e++) {
        int u, v;
        cout << "Enter edge " << e + 1 << " (format: u v): ";
        cin >> u >> v;

        if (isDirected) {
            incMatrix[u][e] = -1; // from u
            incMatrix[v][e] = 1;  // to v
        } else {
            incMatrix[u][e] = 1;
            incMatrix[v][e] = 1;
        }
    }

    // Step 5: Display the incidence matrix
    cout << "\nIncidence Matrix:\n";
    for (int i = 0; i < V; i++) {
        for (int j = 0; j < E; j++) {
            cout << incMatrix[i][j] << " ";
        }
        cout << endl;
    }

    // Step 6: End
    return 0;
}

Sample Output

Here is a sample output of the above program.