To create a test case generator for the given problem, we'll generate a tree structure with \(N\) vertices, random edges with lengths, and multiple queries. Here's a detailed step-by-step solution:

1. **Generate Vertices and Edges:**
    - We need to generate \(N-1\) edges to form a tree.
    - Ensure the tree remains connected by using a union-find (disjoint-set) structure or simply connecting a new node to any of the existing nodes.

2. **Generate Edge Lengths:**
    - Each edge length \(c_i\) should be a random integer between 1 and \(10^9\).

3. **Generate Queries:**
    - Each query should consist of two vertices \(x_j\) and \(y_j\) (both different and neither equal to \(K\)).
    - Generate \(Q\) such queries.

4. **Constraints:**
    - \(3 \leq N \leq 10^5\)
    - \(1 \leq c_i \leq 10^9\)
    - \(1 \leq Q \leq 10^5\)
    - Vertices in queries should not be the same and should not be equal to \(K\).

Here's the Python code for the test case generator:

```python
import random

def generate_tree_edges(N):
    edges = []
    for i in range(2, N+1):
        u = i
        v = random.randint(1, i-1)  # Connect to any vertex that is already part of the tree
        c = random.randint(1, 10**9)
        edges.append((u, v, c))
    return edges

def generate_queries(N, Q, K):
    queries = []
    for _ in range(Q):
        while True:
            x = random.randint(1, N)
            y = random.randint(1, N)
            if x != y and x != K and y != K:
                break
        queries.append((x, y))
    return queries

def tcgen():
    N = random.randint(3, 10**5)
    edges = generate_tree_edges(N)
    
    Q = random.randint(1, 10**5)
    K = random.randint(1, N)
    
    queries = generate_queries(N, Q, K)
    
    input_data = f"{N}\n"
    input_data += "\n".join(f"{u} {v} {c}" for u, v, c in edges)
    input_data += f"\n{Q} {K}\n"
    input_data += "\n".join(f"{x} {y}" for x, y in queries)
    
    return input_data

# Example usage
print(tcgen())
```

**Explanation:**

1. **generate_tree_edges(N):** This function generates \(N-1\) edges to form a connected tree. Each edge connects a new vertex \(i\) to any previous vertex \(v\) ensuring connectivity.

2. **generate_queries(N, Q, K):** This function generates \(Q\) queries. It ensures that the vertices \(x\) and \(y\) in each query are different and neither is equal to \(K\).

3. **tcgen():** This function calls the above helper functions to generate the tree and queries, formats them as a single input string, and returns it.

You can place this generator script in your testing environment to generate and verify multiple test cases for the problem.