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



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 Program to demonstrate adjacency list 
// representation of graphs
import java.util.LinkedList;
  
public class GFG 
{
    // A user define class to represent a graph.
    // A graph is an array of adjacency lists.
    // Size of array will be V (number of vertices 
    // in graph)
    static class Graph
    {
        int V;
        LinkedList<Integer> adjListArray[];
          
        // constructor 
        Graph(int V)
        {
            this.V = V;
              
            // define the size of array as 
            // number of vertices
            adjListArray = new LinkedList[V];
              
            // Create a new list for each vertex
            // such that adjacent nodes can be stored
            for(int i = 0; i < V ; i++){
                adjListArray[i] = new LinkedList<>();
            }
        }
    }
      
    // Adds an edge to an undirected graph
    static void addEdge(Graph graph, int src, int dest)
    {
        // Add an edge from src to dest. 
        graph.adjListArray[src].add(dest);
          
        // Since graph is undirected, add an edge from dest
        // to src also
        graph.adjListArray[dest].add(src);
    }
       
    // A utility function to print the adjacency list 
    // representation of graph
    static void printGraph(Graph graph)
    {       
        for(int v = 0; v < graph.V; v++)
        {
            System.out.println("Adjacency list of vertex "+ v);
            System.out.print("head");
            for(Integer pCrawl: graph.adjListArray[v]){
                System.out.print(" -> "+pCrawl);
            }
            System.out.println("\n");
        }
    }
       
    // Driver program to test above functions
    public static void main(String args[])
    {
        // create the graph given in above figure
        int V = 5;
        Graph graph = new Graph(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);
    }
}
// This code is contributed by Sumit Ghosh

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


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

Related Post:
Graph representation using STL for competitive programming | Set 1 (DFS of Unweighted and Undirected)
Graph implementation using STL for competitive programming | Set 2 (Weighted graph)

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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