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	$O(1)$	$O(V^2)$
Remove Vertex	$O(V+E)$	$O(V^2)$
Add Edge	$O(1)$	$O(1)$
Remove Edge	$O(E)$	$O(1)$

Adjacency List

• Addition operations in adjacency lists take 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 scenario, all the edges could be at a single vertex, meaning 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 re-index the rest of the array one step back in order to fill the deleted spot.

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 $O(V^2)$ since we need to add ​rows and columns. We will also need to fill all the new cells.

Comparison

Both representations are suitable for different situations. If your model 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.