Sure, I can help you write a test case generator for this problem. The generator will need to create a connected graph with \(N\) vertices and \(M\) edges where \(N-1 \leq M \leq N\). Here's a Python function that generates such test cases:

```python
import random

def tcgen():
    N = random.randint(2, 10**5)
    M = random.randint(N-1, N)
    
    edges = []
    # To ensure the graph is connected, we start by building a spanning tree
    for i in range(2, N+1):
        edges.append((random.randint(1, i-1), i))
    
    # If more edges are needed, we add them randomly
    additional_edges_needed = M - (N - 1)
    while additional_edges_needed > 0:
        u = random.randint(1, N)
        v = random.randint(1, N)
        if u != v and (u, v) not in edges and (v, u) not in edges:
            edges.append((u, v))
            additional_edges_needed -= 1
    
    random.shuffle(edges)
    
    return N, M, edges

# Generating a test case
N, M, edges = tcgen()
print(N, M)
for edge in edges:
    print(edge[0], edge[1])
```

This function ensures that:

1. \(N\) is a random number between 2 and \(10^5\).
2. \(M\) is a random number between \(N-1\) and \(N\).
3. The edges are generated such that the graph is always connected. Initially, a spanning tree is created to ensure connectivity. If more edges are required, they are added randomly while avoiding self-loops and multiple edges.

Here is a step-by-step explanation of the code:

1. The function `tcgen()` generates random values for \(N\) and \(M\).
2. It initially creates a spanning tree to ensure the graph is connected by connecting each vertex \(i\) (where \(i\) ranges from 2 to \(N\)) to a random previous vertex.
3. If \(M > N-1\), additional edges are added randomly between any two different vertices until the total number of edges is \(M\).
4. The edges are then shuffled to avoid any pattern that might bias the test cases.
5. The function returns \(N\), \(M\), and the list of edges.

With this generator, you can easily create multiple test cases to validate your solution for the given problem.