What does Strongly Connected Components mean?Why a single vertex can be a compoent?

In a directed graph, a strongly connected component (SCC) is a subgraph where every vertex in the subgraph is reachable from every other vertex in the subgraph. In other words, if we start at any vertex in the SCC, we can reach any other vertex in the SCC by following a path of directed edges.

A single vertex can be considered an SCC if it has a self-loop, i.e., an edge that starts and ends at the same vertex. In this case, the vertex is strongly connected to itself and forms a single-vertex SCC. A single-vertex SCC can also occur if there are no edges that connect the vertex to any other vertex in the graph.

It is important to note that SCCs can overlap, meaning that a vertex can belong to multiple SCCs. For example, consider a graph with a self-loop on vertex 1 and a directed edge from vertex 1 to vertex 2. In this case, vertex 1 is a single-vertex SCC, and the subgraph containing vertices 1 and 2 is also an SCC. Therefore, vertex 1 belongs to two SCCs.

Finding SCCs is an important problem in graph theory and has many applications in computer science, such as in compilers, software verification, and network analysis.

有向图的强连通分量(Strongly connected component) - 知乎 (zhihu.com)