Graph and its representations

Graph is a data structure that consists of following two components:
1. A finite set of vertices also called as nodes.
2. A finite set of ordered pair of the form (u, v) called as edge. The pair is ordered because (u, v) is not same as (v, u) in case of a directed graph(di-graph). The pair of the form (u, v) indicates that there is an edge from vertex u to vertex v. The edges may contain weight/value/cost.

Graphs are used to represent many real-life applications: Graphs are used to represent networks. The networks may include paths in a city or telephone network or circuit network. Graphs are also used in social networks like linkedIn, Facebook. For example, in Facebook, each person is represented with a vertex(or node). Each node is a structure and contains information like person id, name, gender and locale. See this for more applications of graph.

Following is an example of an undirected graph with 5 vertices.

Following two are the most commonly used representations of a graph.
1. Adjacency Matrix
2. Adjacency List
There are other representations also like, Incidence Matrix and Incidence List. The choice of the graph representation is situation specific. It totally depends on the type of operations to be performed and ease of use.

Adjacency Matrix:
Adjacency Matrix is a 2D array of size V x V where V is the number of vertices in a graph. Let the 2D array be adj[][], a slot adj[i][j] = 1 indicates that there is an edge from vertex i to vertex j. Adjacency matrix for undirected graph is always symmetric. Adjacency Matrix is also used to represent weighted graphs. If adj[i][j] = w, then there is an edge from vertex i to vertex j with weight w.

The adjacency matrix for the above example graph is:
Adjacency Matrix Representation

Pros: Representation is easier to implement and follow. Removing an edge takes O(1) time. Queries like whether there is an edge from vertex ‘u’ to vertex ‘v’ are efficient and can be done O(1).

Cons: Consumes more space O(V^2). Even if the graph is sparse(contains less number of edges), it consumes the same space. Adding a vertex is O(V^2) time.
Please see this for a sample Python implementation of adjacency matrix.

Adjacency List:
An array of lists is used. Size of the array is equal to the number of vertices. Let the array be array[]. An entry array[i] represents the list of vertices adjacent to the ith vertex. This representation can also be used to represent a weighted graph. The weights of edges can be represented as lists of pairs. Following is adjacency list representation of the above graph.

Adjacency List Representation of Graph

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Note that in below implementation, we use dynamic arrays (vector in C++/ArrayList in Java) to represent adjacency lists instead of linked list. The vector implementation has advantages of cache friendliness.

C++

// A simple representation of graph using STL 
#include<bits/stdc++.h> 
using namespace std; 
  
// A utility function to add an edge in an 
// undirected graph. 
void addEdge(vector<int> adj[], int u, int v) 
{ 
    adj[u].push_back(v); 
    adj[v].push_back(u); 
} 
  
// A utility function to print the adjacency list 
// representation of graph 
void printGraph(vector<int> adj[], int V) 
{ 
    for (int v = 0; v < V; ++v) 
    { 
        cout << "\n Adjacency list of vertex "
             << v << "\n head "; 
        for (auto x : adj[v]) 
           cout << "-> " << x; 
        printf("\n"); 
    } 
} 
  
// Driver code 
int main() 
{ 
    int V = 5; 
    vector<int> adj[V]; 
    addEdge(adj, 0, 1); 
    addEdge(adj, 0, 4); 
    addEdge(adj, 1, 2); 
    addEdge(adj, 1, 3); 
    addEdge(adj, 1, 4); 
    addEdge(adj, 2, 3); 
    addEdge(adj, 3, 4); 
    printGraph(adj, V); 
    return 0; 
} 

C

// A C Program to demonstrate adjacency list  
// representation of graphs 
#include <stdio.h> 
#include <stdlib.h> 
  
// A structure to represent an adjacency list node 
struct AdjListNode 
{ 
    int dest; 
    struct AdjListNode* next; 
}; 
  
// A structure to represent an adjacency list 
struct AdjList 
{ 
    struct AdjListNode *head;  
}; 
  
// A structure to represent a graph. A graph 
// is an array of adjacency lists. 
// Size of array will be V (number of vertices  
// in graph) 
struct Graph 
{ 
    int V; 
    struct AdjList* array; 
}; 
  
// A utility function to create a new adjacency list node 
struct AdjListNode* newAdjListNode(int dest) 
{ 
    struct AdjListNode* newNode = 
     (struct AdjListNode*) malloc(sizeof(struct AdjListNode)); 
    newNode->dest = dest; 
    newNode->next = NULL; 
    return newNode; 
} 
  
// A utility function that creates a graph of V vertices 
struct Graph* createGraph(int V) 
{ 
    struct Graph* graph =  
        (struct Graph*) malloc(sizeof(struct Graph)); 
    graph->V = V; 
  
    // Create an array of adjacency lists.  Size of  
    // array will be V 
    graph->array =  
      (struct AdjList*) malloc(V * sizeof(struct AdjList)); 
  
    // Initialize each adjacency list as empty by  
    // making head as NULL 
    int i; 
    for (i = 0; i < V; ++i) 
        graph->array[i].head = NULL; 
  
    return graph; 
} 
  
// Adds an edge to an undirected graph 
void addEdge(struct Graph* graph, int src, int dest) 
{ 
    // Add an edge from src to dest.  A new node is  
    // added to the adjacency list of src.  The node 
    // is added at the begining 
    struct AdjListNode* newNode = newAdjListNode(dest); 
    newNode->next = graph->array[src].head; 
    graph->array[src].head = newNode; 
  
    // Since graph is undirected, add an edge from 
    // dest to src also 
    newNode = newAdjListNode(src); 
    newNode->next = graph->array[dest].head; 
    graph->array[dest].head = newNode; 
} 
  
// A utility function to print the adjacency list  
// representation of graph 
void printGraph(struct Graph* graph) 
{ 
    int v; 
    for (v = 0; v < graph->V; ++v) 
    { 
        struct AdjListNode* pCrawl = graph->array[v].head; 
        printf("\n Adjacency list of vertex %d\n head ", v); 
        while (pCrawl) 
        { 
            printf("-> %d", pCrawl->dest); 
            pCrawl = pCrawl->next; 
        } 
        printf("\n"); 
    } 
} 
  
// Driver program to test above functions 
int main() 
{ 
    // create the graph given in above fugure 
    int V = 5; 
    struct Graph* graph = createGraph(V); 
    addEdge(graph, 0, 1); 
    addEdge(graph, 0, 4); 
    addEdge(graph, 1, 2); 
    addEdge(graph, 1, 3); 
    addEdge(graph, 1, 4); 
    addEdge(graph, 2, 3); 
    addEdge(graph, 3, 4); 
  
    // print the adjacency list representation of the above graph 
    printGraph(graph); 
  
    return 0; 
} 

Java

// Java code to demonstrate Graph representation 
// using ArrayList in Java 
  
import java.util.*; 
  
class Graph { 
      
    // A utility function to add an edge in an 
    // undirected graph 
    static void addEdge(ArrayList<ArrayList<Integer> > adj, 
                        int u, int v) 
    { 
        adj.get(u).add(v); 
        adj.get(v).add(u); 
    } 
  
    // A utility function to print the adjacency list 
    // representation of graph 
    static void printGraph(ArrayList<ArrayList<Integer> > adj) 
    { 
        for (int i = 0; i < adj.size(); i++) { 
            System.out.println("\nAdjacency list of vertex" + i); 
            for (int j = 0; j < adj.get(i).size(); j++) { 
                System.out.print(" -> "+adj.get(i).get(j)); 
            } 
            System.out.println(); 
        } 
    } 
  
    // Driver Code 
    public static void main(String[] args) 
    { 
        // Creating a graph with 5 vertices 
        int V = 5; 
        ArrayList<ArrayList<Integer> > adj  
                    = new ArrayList<ArrayList<Integer> >(V); 
          
        for (int i = 0; i < V; i++) 
            adj.add(new ArrayList<Integer>()); 
  
        // Adding edges one by one 
        addEdge(adj, 0, 1); 
        addEdge(adj, 0, 4); 
        addEdge(adj, 1, 2); 
        addEdge(adj, 1, 3); 
        addEdge(adj, 1, 4); 
        addEdge(adj, 2, 3); 
        addEdge(adj, 3, 4); 
          
        printGraph(adj); 
    } 
} 

Python3

""" 
A Python program to demonstrate the adjacency 
list representation of the graph 
"""
  
# A class to represent the adjacency list of the node 
class AdjNode: 
    def __init__(self, data): 
        self.vertex = data 
        self.next = None
  
  
# A class to represent a graph. A graph 
# is the list of the adjacency lists. 
# Size of the array will be the no. of the 
# vertices "V" 
class Graph: 
    def __init__(self, vertices): 
        self.V = vertices 
        self.graph = [None] * self.V 
  
    # Function to add an edge in an undirected graph 
    def add_edge(self, src, dest): 
        # Adding the node to the source node 
        node = AdjNode(dest) 
        node.next = self.graph[src] 
        self.graph[src] = node 
  
        # Adding the source node to the destination as 
        # it is the undirected graph 
        node = AdjNode(src) 
        node.next = self.graph[dest] 
        self.graph[dest] = node 
  
    # Function to print the graph 
    def print_graph(self): 
        for i in range(self.V): 
            print("Adjacency list of vertex {}\n head".format(i), end="") 
            temp = self.graph[i] 
            while temp: 
                print(" -> {}".format(temp.vertex), end="") 
                temp = temp.next
            print(" \n") 
  
  
# Driver program to the above graph class 
if __name__ == "__main__": 
    V = 5
    graph = Graph(V) 
    graph.add_edge(0, 1) 
    graph.add_edge(0, 4) 
    graph.add_edge(1, 2) 
    graph.add_edge(1, 3) 
    graph.add_edge(1, 4) 
    graph.add_edge(2, 3) 
    graph.add_edge(3, 4) 
  
    graph.print_graph() 
  
# This code is contributed by Kanav Malhotra 

C#

// C# code to demonstrate Graph representation  
// using LinkedList in C#  
using System; 
using System.Collections.Generic; 
  
class Graph 
{ 
    // A utility function to add an edge in an  
    // undirected graph  
    static void addEdge(LinkedList<int>[] adj, 
                        int u, int v) 
    { 
        adj[u].AddLast(v); 
        adj[v].AddLast(u); 
    } 
  
    // A utility function to print the adjacency list  
    // representation of graph  
    static void printGraph( LinkedList<int>[]  adj) 
    { 
        for (int i = 0; i < adj.Length; i++) 
        { 
            Console.WriteLine("\nAdjacency list of vertex " + i); 
            Console.Write("head"); 
              
            foreach (var item in adj[i]) 
            {  
                Console.Write(" -> " + item); 
            } 
            Console.WriteLine(); 
        } 
    } 
  
    // Driver Code  
    public static void Main(String[] args) 
    { 
        // Creating a graph with 5 vertices  
        int V = 5; 
        LinkedList<int>[] adj = new LinkedList<int>[V]; 
  
        for (int i = 0; i < V; i++) 
            adj[i] = new LinkedList<int>(); 
  
        // Adding edges one by one  
        addEdge(adj, 0, 1); 
        addEdge(adj, 0, 4); 
        addEdge(adj, 1, 2); 
        addEdge(adj, 1, 3); 
        addEdge(adj, 1, 4); 
        addEdge(adj, 2, 3); 
        addEdge(adj, 3, 4); 
  
        printGraph(adj); 
  
        Console.ReadKey(); 
    } 
} 
  
// This code is contributed by techno2mahi 

Output:

 
 Adjacency list of vertex 0
 head -> 1-> 4

 Adjacency list of vertex 1
 head -> 0-> 2-> 3-> 4

 Adjacency list of vertex 2
 head -> 1-> 3

 Adjacency list of vertex 3
 head -> 1-> 2-> 4

 Adjacency list of vertex 4
 head -> 0-> 1-> 3

Pros: Saves space O(|V|+|E|) . In the worst case, there can be C(V, 2) number of edges in a graph thus consuming O(V^2) space. Adding a vertex is easier.

Cons: Queries like whether there is an edge from vertex u to vertex v are not efficient and can be done O(V).

Reference:
http://en.wikipedia.org/wiki/Graph_%28abstract_data_type%29

This article is compiled by Aashish Barnwal and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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My Personal Notes

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

C++

C

Java

Python3

C#

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