The proposed algorithm does not work in general. Here's why:

The algorithm suggests running a Breadth-First Search (BFS) from the source node $u$ to the intermediate node $x$, and then a separate BFS from node $x$ to the destination node $v$. While BFS finds the shortest path in an unweighted graph, the problem arises from running the two BFS operations independently.

Explanation:

1. First BFS (from $u$ to $x$): This BFS will find the shortest path from $u$ to $x$.
2. Second BFS (from $x$ to $v$): This BFS will find the shortest path from $x$ to $v$.

However, the algorithm does not consider that the shortest path from $u$ to $v$ via $x$ may require you to take a different route from $u$ to $x$ than the one found in the first BFS. In other words, the shortest path from $u$ to $v$ via $x$ is not necessarily the concatenation of the shortest path from $u$ to $x$ and the shortest path from $x$ to $v$.

Example Counterexample:

Consider the following graph:

• $u$ connects directly to $v$ with a distance of 1.
• $u$ connects to $x$ with a distance of 2.
• $x$ connects to $v$ with a distance of 2.

If you use the proposed algorithm:

1. The first BFS finds the shortest path from $u$ to $x$ as having a length of 2.
2. The second BFS finds the shortest path from $x$ to $v$ as having a length of 2.

So, according to this algorithm, the path $u$ → $x$ → $v$ would have a total length of 4. However, the direct path $u$ → $v$ has a length of 1, which is shorter than 4. Therefore, the algorithm fails to find the true shortest path that must pass through $x$.

Conclusion:

The correct answer is no—the algorithm does not necessarily find the shortest path from $u$ to $v$ via $x$.

Would you like to explore more on this topic or have any further questions? Here are some related questions:

1. What is the correct algorithm to find the shortest path through an intermediate node?
2. How does BFS work in an unweighted graph?
3. Can Dijkstra's algorithm be used for the same problem in weighted graphs?
4. What are the complexities of various shortest path algorithms?
5. How does adding weights change the problem and solutions?

Tip: In graph problems, always consider whether intermediate steps in an algorithm might overlook some possibilities that could lead to better solutions.