Sure, I can help you write a test case generator for the problem statement "Problem D: 2D is an abbreviation of Decision Diagram".

Here's a Python script that generates random test cases according to the given problem constraints:

```python
import random

def generate_truth_table(N):
    table_size = 2 ** N
    truth_table = ''.join(random.choice('01') for _ in range(table_size))
    return truth_table

def tcgen():
    N = random.randint(1, 10)
    bit_line = generate_truth_table(N)
    return N, bit_line

# Example usage
for _ in range(5):  # Generate 5 test cases
    N, bit_line = tcgen()
    print(f"N: {N}")
    print(f"bit_line: {bit_line}")
    print()
```

### Explanation:
1. **Randomly Select `N`:** We select `N` randomly within the range [1, 10] using `random.randint(1, 10)`.
2. **Generate Truth Table:** We generate a truth table of size \(2^N\) consisting of '0' and '1' characters. The function `generate_truth_table` creates this string of length \(2^N\).
3. **Return Test Case:** The function `tcgen` returns a tuple containing `N` and the generated `bit_line`.
4. **Example Usage:** The script includes a loop to generate and print 5 test cases for demonstration.

This generator ensures that the test cases adhere to the constraints provided in the problem statement. You can adjust the number of test cases generated by modifying the range in the example usage loop.