To generate test cases for this problem, we need to generate the following:

1. Two integers \( A \) and \( B \) representing the dimensions of the distance matrix.
2. A matrix \( d \) of dimensions \( A \times B \) where each element \( d_{x,y} \) is an integer between 1 and 100.

Additionally, we need to ensure the generated graph satisfies the conditions mentioned in the problem statement, or we should report "Impossible" if it's not possible to construct such a graph.

Here's a Python function that generates these test cases:

```python
import random

def generate_test_case():
    # Generate A and B
    A = random.randint(1, 10)
    B = random.randint(1, 10)

    # Generate the distance matrix d
    d = [[random.randint(1, 100) for _ in range(B)] for _ in range(A)]

    return A, B, d

# Function to print the test case
def print_test_case():
    A, B, d = generate_test_case()
    
    print(A, B)
    for row in d:
        print(' '.join(map(str, row)))

print_test_case()
```

This function generates the test case by:
1. Randomly selecting values for \( A \) and \( B \).
2. Creating a distance matrix \( d \) with dimensions \( A \times B \) containing integers between 1 and 100.
3. Printing the test case in the required format.

Here's how the output might look when you run the function:

```
3 4
23 45 67 23
12 34 56 78
90 12 34 56
```

In a real competitive programming scenario, you would use this function to generate various test cases and validate them against your solution to ensure it handles all edge cases and constraints effectively.