Certainly! Here's a Python function that generates test cases for the given problem statement about Takahashi and Aoki's game on a grid:

```python
import random

def tcgen():
    # Constraints
    H = random.randint(1, 2 * 10**5)
    W = random.randint(1, 2 * 10**5)
    N = random.randint(0, 2 * 10**5)
    
    # Generate N unique obstacles
    obstacles = set()
    while len(obstacles) < N:
        x = random.randint(1, H)
        y = random.randint(1, W)
        if (x, y) != (1, 1):  # (1,1) should not have an obstacle
            obstacles.add((x, y))
    
    # Convert set to sorted list for consistent output
    obstacles = sorted(list(obstacles))
    
    # Format the output as described
    output = f"{H} {W} {N}\n"
    for (x, y) in obstacles:
        output += f"{x} {y}\n"
    
    return output.strip()

# Example usage
if __name__ == "__main__":
    print(tcgen())
```

This function `tcgen` will generate random test cases based on the constraints provided:

1. `H` and `W` are chosen randomly between 1 and \(2 \times 10^5\).
2. `N` is chosen randomly between 0 and \(2 \times 10^5\).
3. Obstacles are generated as unique pairs that are not equal to (1,1) to ensure the start position is always free.

The set ensures that no duplicate obstacles are generated. Finally, the obstacles are formatted into the required input format.

You can run the `tcgen` function multiple times to generate different test cases for your competitive programming problem.