Question

The maximum value of \(x\) such that the edge between the nodes B and C is included in every minimum spanning tree of the given graph is ___________ (answer in integer).

Hint:

When solving for the edges that must be included in a minimum spanning tree, compare the edge in question to the weights of other edges and ensure that no other edge can replace it without increasing the overall weight of the spanning tree.

Answer 1

We are given the following graph: 

To find the maximum value of \(x\) such that the edge between \(B\) and \(C\) is included in every minimum spanning tree (MST) of this graph, we need to analyze the edge weights and compare them. 

Step 1: Consider Minimum Spanning Trees (MST)
For the edge \(B \leftrightarrow C\) to be in every MST, its weight \(x\) must be no greater than the weights of the other edges that could replace it. Specifically, it must be smaller than or equal to:
- The edge \(A \leftrightarrow B\) with weight 7.
- The edge \(A \leftrightarrow D\) with weight 6.
- The edge \(D \leftrightarrow C\) with weight 8.

Step 2: Finding the Maximum Value for \(x\)
For the edge \(B \leftrightarrow C\) to be included, \(x\) must be smaller than or equal to the weights of other competing edges:
- \(x \leq 7\) to ensure \(B \leftrightarrow C\) is not replaced by \(A \leftrightarrow B\).
- \(x \leq 6\) to ensure \(B \leftrightarrow C\) is not replaced by \(A \leftrightarrow D\).
- \(x \leq 5\) ensures that the edge \(B \leftrightarrow C\) is always chosen over other edges like \(D \leftrightarrow C\).

Thus, the maximum value of \(x\) is 5. 

Final result: The maximum value of \(x\) such that the edge between \(B\) and \(C\) is included in every minimum spanning tree is 5.