Here's a Python function to generate test cases for the DFS problem described:

```python
import random

def generate_adjacency_list(n):
    adjacency_list = []
    for u in range(1, n + 1):
        k = random.randint(0, n - 1)  # Degree of vertex u
        if k == 0:
            adjacency_list.append((u, 0, []))
        else:
            neighbors = sorted(random.sample(range(1, n + 1), k))
            adjacency_list.append((u, k, neighbors))
    return adjacency_list

def dfs_tcgen():
    n = random.randint(1, 100)  # Number of vertices
    adjacency_list = generate_adjacency_list(n)
    
    input_str = f"{n}\n"
    for u, k, neighbors in adjacency_list:
        input_str += f"{u} {k} {' '.join(map(str, neighbors))}\n"
    
    return input_str

# Example of generating a test case
print(dfs_tcgen())
```

### Explanation:
1. **`generate_adjacency_list(n)`**:
    - This helper function generates an adjacency list for a graph with `n` vertices.
    - Each vertex `u` has a degree `k` which is a random integer between `0` and `n-1`.
    - If `k == 0`, it means vertex `u` has no neighbors.
    - Otherwise, `k` neighbors are randomly selected from the range `[1, n]` and are sorted to ensure they are in ascending order.

2. **`dfs_tcgen()`**:
    - Generates a random number `n` representing the number of vertices.
    - Calls `generate_adjacency_list(n)` to generate the adjacency list for these vertices.
    - Constructs the input string for the DFS problem based on the adjacency list.
    - The input string is then returned, which can be used for testing.

This function should provide a wide variety of test cases to check the correctness and efficiency of a DFS implementation.