Complexities of Graph Operations

Time Complexities

Below, you can find the time complexities for the 4 basic graph functions.

Note that, in this table, V means the total number of vertices and E means the total number of edges in the Graph.

Operation	Adjacency List	Adjacency Matrix
Add Vertex	O(1)	O(V2)
Remove Vertex	O(V+E)	O(V2)
Add Edge	O(1)	O(1)
Remove Edge	O(E)	O(1)
Search	O(V)	O(1)
Breadth First Search(BFS)	O(V+E)	O($V^2$)
Depth First Search(DFS)	O(V+E)	O(V2)

Adjacency List

• Adding an edge in adjacency lists takes constant time as we only need to insert at the head node of the corresponding vertex.

• Removing an edge takes O(E) time because, in the worst case, all the edges could be at a single vertex and hence, we would have to traverse all E edges to reach the last one.

• Removing a vertex takes O(V + E) time because we have to delete all its edges and then reindex the rest of the list one step back in order to fill the deleted spot.

• Searching an edge between a pair of vertices can take up to O(V) if all V nodes are present at a certain index and we have to traverse them.

Adjacency Matrix

• Edge operations are performed in constant time as we only need to manipulate the value in the particular cell.

• Vertex operations are performed in O(V²) since we need to add rows and columns. We will also need to fill all the new cells.

• Searching an edge is O(1) because we can access each edge by indexing.

Comparison

Both representations are suitable for different situations. If your application frequently manipulates vertices, the adjacency list is a better choice.

If you are dealing primarily with edges, the adjacency matrix is the more efficient approach.

Keep these complexities in mind because they will give you a better idea about the time complexities of the several algorithms we’ll see in this section.