Sure! Here is a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Constraints
    N = random.randint(2, 20)
    M = random.randint(N-1, N*(N-1)//2)
    
    # Generate a spanning tree for vertices 1 to N
    edges = []
    for i in range(2, N+1):
        edges.append((random.randint(1, i-1), i))
    
    # Generate the rest of the edges to ensure we have a connected graph
    all_possible_edges = [(i, j) for i in range(1, N+1) for j in range(i+1, N+1) if (i, j) not in edges]
    while len(edges) < M:
        edge = random.choice(all_possible_edges)
        if edge not in edges:
            edges.append(edge)
            all_possible_edges.remove(edge)
    
    random.shuffle(edges)
    
    return N, M, edges

def format_tc(N, M, edges):
    test_case_str = f"{N} {M}\n"
    for a, b in edges:
        test_case_str += f"{a} {b}\n"
    return test_case_str.strip()

# Example usage:
N, M, edges = tcgen()
print(format_tc(N, M, edges))
```

This Python function `tcgen` generates a random test case according to the specified problem constraints:

1. `N` is chosen randomly between 2 and 20.
2. `M` is chosen randomly between \(N-1\) and \(\frac{N(N-1)}{2}\).
3. The function first generates a spanning tree by sequentially connecting nodes, ensuring that the first \(N-1\) edges form a tree.
4. It then adds random edges to ensure the total number of edges is `M` without creating self-loops or duplicate edges.
5. The `format_tc` function converts the generated data into the required input format.

You can print the test case to see the output and use it for testing your solution.