Sure, let's write a test case generator for the provided competitive programming problem statement about the graph and minimum spanning tree.

To generate test cases, we need to consider the constraints and the requirements mentioned:

- \(2 \leq N \leq 20\)
- \(N-1 \leq M \leq N(N-1)/2\)
- The subgraph consisting of Vertex 1,2,\ldots,N and Edge 1,2,\ldots,N-1 must be a spanning tree.
- The graph is simple, connected, and undirected without self-loops or multiple edges.

Here's the Python code for generating test cases based on the above constraints:

```python
import random

def generate_spanning_tree(N):
    """ Generate a spanning tree with N vertices. """
    edges = []
    for i in range(2, N + 1):
        u = i
        v = random.randint(1, i - 1)
        edges.append((u, v))
    return edges

def generate_additional_edges(N, M, current_edges):
    """ Generate additional edges to ensure the graph has M edges in total. """
    all_possible_edges = [(i, j) for i in range(1, N + 1) for j in range(i + 1, N + 1)]
    existing_edges_set = set(current_edges)
    remaining_edges = [edge for edge in all_possible_edges if edge not in existing_edges_set]
    
    additional_edges_needed = M - (N - 1)
    additional_edges = random.sample(remaining_edges, additional_edges_needed)
    
    return additional_edges

def tcgen():
    # Step 1: Randomly choose N
    N = random.randint(2, 20)
    
    # Step 2: Randomly choose M such that N-1 <= M <= N(N-1)/2
    min_edges = N - 1
    max_edges = (N * (N - 1)) // 2
    M = random.randint(min_edges, max_edges)
    
    # Step 3: Generate a spanning tree
    spanning_tree_edges = generate_spanning_tree(N)
    
    # Step 4: Generate additional edges to reach a total of M edges
    additional_edges = generate_additional_edges(N, M, spanning_tree_edges)
    
    # Combine all edges
    all_edges = spanning_tree_edges + additional_edges
    
    # Formatting output
    result = [f"{N} {M}"]
    for edge in all_edges:
        result.append(f"{edge[0]} {edge[1]}")
        
    return "\n".join(result)

# Generate a test case and print it
print(tcgen())
```

Explanation:
1. **generate_spanning_tree(N)**: Generates a spanning tree with \(N\) vertices ensuring that there are \(N-1\) edges.
2. **generate_additional_edges(N, M, current_edges)**: Generates additional edges to ensure the total number of edges is \(M\). It avoids duplicate and self-loops.
3. **tcgen()**: The main function that combines the above steps to generate a complete test case.

This generator ensures that the graph is always connected and adheres to the constraints mentioned in the problem statement.