To generate a test case generator for the given problem statement, we need to create a generator that produces valid inputs for the problem. The graph must be connected, with the number of edges \( M \) between \( N-1 \) and \( N \), and there should be no self-loops or multiple edges.

Here's the test case generator in Python:

```python
import random

def generate_connected_graph(N, M):
    from collections import defaultdict, deque

    # Ensure we have a connected graph first using a tree
    edges = []
    for i in range(2, N+1):
        edges.append((random.randint(1, i-1), i))
    
    # If we need more edges, add some randomly
    additional_edges_needed = M - (N - 1)
    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 and (j, i) not in edges]
    random.shuffle(possible_edges)
    
    edges.extend(possible_edges[:additional_edges_needed])
    random.shuffle(edges)
    
    return edges

def tcgen():
    N = random.randint(2, 10**5)
    M = random.randint(N-1, N)
    edges = generate_connected_graph(N, M)
    
    input_str = f"{N} {M}\n"
    for a, b in edges:
        input_str += f"{a} {b}\n"
    
    return input_str.strip()

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

### Explanation:

1. **Generating a Connected Graph:**
   - We first ensure the graph is connected by creating a tree structure (which is inherently connected). This is done by connecting each vertex \( i \) from 2 to \( N \) to a random vertex from 1 to \( i-1 \).
   - If \( M > N-1 \), we add additional edges randomly to satisfy the condition \( N-1 \leq M \leq N \).

2. **Generating the Input String:**
   - The input starts with \( N \) and \( M \).
   - For each edge, we append the vertices \( a \) and \( b \) to the input string.

3. **Randomization:**
   - We shuffle the possible additional edges to ensure randomness.
   - We shuffle the final edge list to avoid any specific order in the output.

This test case generator will produce valid inputs for the given problem, ensuring all constraints are respected.