A vertex with the highest post order number lies in a source SCC

Lemma

Suppose we have a directed graph and be the result of running the DFS to find connected components in an undirected graph algorithm. Then the vertex with the largest post order number lies in a source strongly connected component in the strongly connected component graph.

Proof

Let be the strongly connected component with the vertex of highest value.

Suppose some other strongly connected component were to have an edge connecting to in the strongly connected component graph.

From the Lemma has a vertex with higher post order than all vertices in . This is a contradiction as contains the vertex with highest post order.