# Graph and its representations

A graph is a data structure that consists of the 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 the 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. The following two are the most commonly used representations of a graph.
There are other representations also like, Incidence Matrix and Incidence List. The choice of graph representation is situation-specific. It totally depends on the type of operations to be performed and ease of use.

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: 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.

An array of lists is used. The size of the array is equal to the number of vertices. Let the array be an 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 the adjacency list representation of the above graph. ## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

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

## C++

 `// A simple representation of graph using STL ` `#include ` `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 ` `#include ` ` `  `// 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 beginning ` `    ``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 > 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 > adj) ` `    ``{ ` `        ``for` `(``int` `i = ``0``; i < adj.size(); i++) { ` `            ``System.out.println(``"\nAdjacency list of vertex"` `+ i); ` `            ``System.out.print(``"head"``); ` `            ``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 > adj  ` `                    ``= ``new` `ArrayList >(V); ` `         `  `        ``for` `(``int` `i = ``0``; i < V; i++) ` `            ``adj.add(``new` `ArrayList()); ` ` `  `        ``// 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:

```

head -> 0-> 2-> 3-> 4

```

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).

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