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To solve this problem, we need to determine the value of \( x \) such that the minimum spanning tree (MST) of the given graph has a total length of 65, with the condition that the edge labeled \( x \) is included in the MST.

### Approach
1. **Understanding the MST Requirement:** A minimum spanning tree connects all vertices with the smallest possible sum of edge weights. We need to select edges in a way that avoids cycles while minimizing the total weight.
  
2. **Initial Setup:** We know the total length of the MST is 65 and that \( x \) is included. By examining the weights of other edges, we can determine which edges might be part of the MST.

3. **Edge Selection:** We start with the lowest-weight edges, ensuring we meet the condition of an MST by avoiding cycles. 

4. **Finding \( x \):** Using the MST total of 65 as a constraint, we include \( x \) in the MST and select other edges that minimize the total.

I'll analyze the graph and perform the calculations to identify \( x \). Let me process this and I'll be right back with the detailed solution.### Solution Steps

#### Step 1: List All Edge Weights
From the graph:
- The edges have weights: 8, 9, 9, 7, 12, 13, 13, 14, 15, 16, and 20, along with \( x \).

#### Step 2: Minimum Spanning Tree Condition
The goal is to include the edge labeled \( x \) in the MST and have a total MST length of 65.

#### Step 3: Select the Smallest Edges to Build the MST
To form an MST, we should start by selecting edges in increasing order of weight, avoiding cycles, and ensuring that we use \( x \) as one of the edges.

#### Step 4: Trial Values for \( x \)
We are given options for \( x \): 14, 13, 12, 11, and 10. We’ll test each option to see which one allows the MST to total exactly 65.

#### Step 5: Calculate the MST for \( x = 12 \)
1. Select the smallest edges first, including \( x = 12 \).
2. MST edges (chosen to avoid cycles) could be: 8, 7, 9, 9, 12 (for \( x \)), 13, and 7 more.

After running the

Given that the edge labelled x is included in the minimum spanning tree, what is the value of x if the minimum spanning tree for the graph below has a length of 65?

Explore how to determine the value of an edge (x) in a minimum spanning tree (MST) with a total weight constraint of 65. This problem involves applying concepts from graph theory, including Kruskal's and Prim's algorithms for MST construction.

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