Prerequisites: Linked List, Graph Data Structure

In this article, adding and removing a vertex is discussed in a given adjacency list representation.

Let the Directed Graph be:

The graph can be represented in the Adjacency List representation as:

It is a Linked List representation where the head of the linked list is a vertex in the graph and all the connected nodes are the vertices to which the first vertex is connected. For example, from the graph, it is clear that vertex 0 is connected to vertex 4, 3 and 1. The same is representated in the adjacency list(or Linked List) representation.

Approach:

The idea is to represent the graph as a list of linked lists where the head of the linked list is the vertex and all the connected linked lists are the vertices to which it is connected.
• Adding a Vertex in the Graph: To add a vertex in the graph, the adjacency list can be iterated to the place where the insertion is required and the new node can be created using linked list implementation. For example, if 5 needs to be added between vertex 2 and vertex 3 such that vertex 3 points to vertex 5 and vertex 5 points to vertex 2, then a new edge is created between vertex 5 and vertex 3 and a new edge is created from vertex 5 and vertex 2. After adding the vertex, the adjacency list changes to:
  
• Removing a Vertex in the Graph: To delete a vertex in the graph, iterate through the list of each vertex if an edge is present or not. If the edge is present, then delete the vertex in the same way as delete is performed in a linked list. For example, the adjacency list translates to the below list if vertex 4 is deleted from the list:
  

  Below is the implementation of the above approach:

  
  

  
  
  

  # Python implementation of the above approach # Implementing Linked List representation class AdjNode(object):     def __init__(self, data):         self.vertex = data         self.next = None    # Adjacency List representation class AdjList(object):        def __init__(self, vertices):         self.v = vertices         self.graph =[None]*self.v    # Function to add an edge from a source vertex  # to a destination vertex     def addedge(self, source, destination):         node = AdjNode(destination)         node.next = self.graph         self.graph= node    # Function to call the above function.     def addvertex(self, vk, source, destination):         graph.addedge(source, vk)          graph.addedge(vk, destination)    # Function to print the graph     def print_graph(self):         for i in range(self.v):             print(i, end ="")             temp = self.graph[i]             while temp:                print("->", temp.vertex, end ="")                temp = temp.next             print("\n")    # Function to delete a vertex     def delvertex(self, k):    # Iterating through all the vertices of the graph         for i in range(self.v):             temp = self.graph[i]             if i == k:                 while temp:                     self.graph[i]= temp.next                     temp = self.graph[i]                                    # Delete the vertex              # using linked list concept                     if temp:                 if temp.vertex == k:                     self.graph[i]= temp.next                     temp = None             while temp:                 if temp.vertex == k:                     break                 prev = temp                 temp = temp.next                if temp == None:                 continue                prev.next = temp.next             temp = None           # Driver code if __name__ == "__main__":        V = 6     graph = AdjList(V)      graph.addedge(0, 1)     graph.addedge(0, 3)     graph.addedge(0, 4)     graph.addedge(1, 2)     graph.addedge(3, 2)     graph.addedge(4, 3)    print("Initial adjacency list") graph.print_graph()     # Add vertex graph.addvertex(5, 3, 2) print("Adjacency list after adding vertex") graph.print_graph()     # Delete vertex graph.delvertex(4) print("Adjacency list after deleting vertex") graph.print_graph()
  Output:

  Initial adjacency list
  0-> 4-> 3-> 1
  
  1-> 2
  
  2
  
  3-> 2
  
  4-> 3
  
  5
  
  Adjacency list after adding vertex
  0-> 4-> 3-> 1
  
  1-> 2
  
  2
  
  3-> 5-> 2
  
  4-> 3
  
  5-> 2
  
  Adjacency list after deleting vertex
  0-> 3-> 1
  
  1-> 2
  
  2
  
  3-> 5-> 2
  
  4
  
  5-> 2
  

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