How long does it take to determine if an undirected graph contains a vertex that is connected to no other vertices. i. if you use an adjacency matrix ii. if you use adjacency lists.

To determine if an undirected graph contains a vertex that is connected to no other vertices, also known as an isolated vertex, we can use either an adjacency matrix or adjacency lists to represent the graph. The time complexity of the operation will depend on which representation is used.

i. Using an Adjacency Matrix

An adjacency matrix is a 2D array A of size V x V where V is the number of vertices in the graph. Each cell A[i][j] indicates whether there is an edge between vertex i and vertex j. If A[i][j] is 1 (or any non-zero value), there is an edge; if it is 0, there is no edge.

To find an isolated vertex, we need to check each row of the matrix to see if all its entries are 0. If we find such a row, the corresponding vertex is isolated.

Step by Step Approach:

1. Iterate over each vertex i.
2. For each vertex i, iterate over all other vertices j.
3. Check if A[i][j] is 0 for all j. If it is, vertex i is isolated.
4. If we find an isolated vertex, we can stop and return the result.

Time Complexity:

The time complexity of this approach is O(V^2), where V is the number of vertices. This is because we need to check every cell in the row for each vertex.

Example:

Consider a graph with 4 vertices represented by the following adjacency matrix:

A = | 0 1 0 0 |
    | 1 0 1 1 |
    | 0 1 0 0 |
    | 0 1 0 0 |

In this case, vertex 0 is an isolated vertex because the entire first row consists of 0s (except for the diagonal entry, which is always 0).

ii. Using Adjacency Lists

An adjacency list is an array of lists L, where each list L[i] contains the neighbors of vertex i.

To find an isolated vertex, we simply need to check if any list L[i] is empty. If it is, vertex i is isolated.

Step by Step Approach:

1. Iterate over each vertex i.
2. Check if the list L[i] is empty.
3. If L[i] is empty, vertex i is isolated.
4. If we find an isolated vertex, we can stop and return the result.

Time Complexity:

The time complexity of this approach is O(V), where V is the number of vertices. This is because we only need to check if each list is empty, which is an O(1) operation.

Example:

Consider a graph with 4 vertices represented by the following adjacency lists:

L[0] -> []
L[1] -> [2, 3]
L[2] -> [1]
L[3] -> [1]

In this case, vertex 0 is an isolated vertex because the list L[0] is empty.

Comparison Table

Aspect	Adjacency Matrix	Adjacency Lists
Data Structure	2D array	Array of lists
Time to Find Isolation	O(V^2)	O(V)
Space Complexity	O(V^2)	O(V + E)
Ease of Finding Isolation	Need to check entire row	Just check if list is empty

In summary, using adjacency lists is more efficient for finding isolated vertices in an undirected graph, both in terms of time complexity and space complexity, especially when the graph is sparse (i.e., has relatively few edges compared to the number of vertices).