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)

To determine the maximum integer value of $x$ such that the edge $(B, C)$ is included in every Minimum Spanning Tree (MST), we can analyze the cuts and cycles of the graph.

Using Kruskal's algorithm:

1. The edge with the smallest weight is $(B, D)$ with weight $1$.

2. The next smallest edge is $(A, C)$ with weight $3$.

Selecting these two edges creates two disconnected components:

To form a spanning tree on the 4 vertices, exactly one edge crossing the cut between $\{B, D\}$ and $\{A, C\}$ must be chosen. The edges crossing this cut are:

• $(B, C)$ with weight $x$

• $(A, D)$ with weight $6$

• $(A, B)$ with weight $7$

• $(C, D)$ with weight $8$

For edge $(B, C)$ to be present in every MST, it must be the strictly unique minimum-weight edge crossing this cut:

If $x = 6$, there is a tie with edge $(A, D)$, which means an MST could choose $(A, D)$ instead of $(B, C)$, so $(B, C)$ would not be in *every* MST.

Therefore, the maximum integer value of $x$ is 5.