Final answer:
In a directed graph, all vertices visited in a path connecting two vertices in the same strongly connected component can be shown using depth-first search.
Step-by-step explanation:
In a directed graph, a strongly connected component (SCC) is a subgraph in which there is a path from every vertex to every other vertex. Therefore, in a directed path connecting two vertices in the same SCC, it means that there is a path from the starting vertex to the ending vertex, and vice versa.
To show this, we can use a technique called depth-first search (DFS). Starting from the first vertex, we explore its neighbors and mark them as visited. We continue this process for each unvisited neighbor until all vertices in the SCC are visited. If the ending vertex is visited, it means that there is a path connecting the two vertices in the SCC.
For example, consider the directed graph shown below:
Using DFS from vertex A, we can visit vertices A, B, C, E, and D. Since vertex D is visited, it shows that there is a path connecting A and D in the same strongly connected component.