Call the Graph the set which composed of several vertices $P_1,\ \cdots P_2$ and several edges $($segments$)$ connecting two points among these vertices. Now let $G$ be a graph with 9 vertices and satisfies the following condition.

Condition: Even if we select any five points from the vertices in $G,$ there exist at least two edges whose endpoints are included in the set of 5 points.

What is the minimum possible numbers of edges satisfying the condition?
To solve this problem, we need to determine the minimum number of edges in a graph \( G \) with 9 vertices such that any selection of 5 vertices includes at least two edges among them.

1. **Define the problem in terms of graph theory:**
   - Let \( G \) be a graph with 9 vertices.
   - We need to ensure that any subset of 5 vertices contains at least two edges.

2. **Use the concept of Turán's theorem:**
   - Turán's theorem provides a way to determine the maximum number of edges in a graph that does not contain a complete subgraph \( K_r \). However, we need to ensure that any subset of 5 vertices contains at least two edges, which is a different problem.

3. **Consider the minimum number of edges required:**
   - We need to find the minimum number of edges such that any 5 vertices have at least two edges among them.

4. **Calculate the minimum number of edges step-by-step:**
   - Let \( x_n \) be the minimum number of edges for a graph with \( n \) vertices.
   - We start with \( x_5 = 2 \) because any 5 vertices must have at least two edges.
   - For \( n = 6 \), we need to ensure that any 5 vertices out of 6 have at least two edges. This implies \( x_6 \ge 3 \).
   - For \( n = 7 \), we need to ensure that any 5 vertices out of 7 have at least two edges. This implies \( x_7 \ge 5 \).
   - For \( n = 8 \), we need to ensure that any 5 vertices out of 8 have at least two edges. This implies \( x_8 \ge 7 \).
   - For \( n = 9 \), we need to ensure that any 5 vertices out of 9 have at least two edges. This implies \( x_9 \ge 9 \).

5. **Verify the minimum number of edges:**
   - We need to check if 9 edges are sufficient to satisfy the condition.
   - Consider a graph with 9 vertices and 9 edges. If we can construct such a graph where any 5 vertices have at least two edges, then 9 is the minimum number of edges.

6. **Construct a possible graph:**
   - One possible construction is to create a cycle graph \( C_9 \) with 9 vertices and 9 edges. In a cycle graph, each vertex is connected to two other vertices, forming a single cycle.
   - In a cycle graph \( C_9 \), any 5 vertices will always include at least two edges because removing any 4 vertices leaves a subgraph with at least two edges.

Therefore, the minimum number of edges required to satisfy the condition is 9.

The final answer is \(\boxed{9}\).