Hi I am struggling to prove the following question regarding strongly connected components in a directed graph. Any help would be appreciated, thanks in advance.
Prove that a directed graph with two vertices can only reach each other if their strongly connected components can as well
Define the relation $x\to y$ to be true when there is a path that starts in $x$ and ends in $y$ (via other vertices if necessary). Define also $x\leftrightarrow y$ to mean $x\to y$ and $y\to x$ (not necessarily the reverse path).
Then $x\leftrightarrow y$ is an equivalence relation (check!), and the equivalence classes $SC(x)$ are called strong components.
If $u\to v$, and $x\in SC(u)$, $y\in SC(v)$, then there are paths $x\to u$, $u\to x$, $y\to v$, and $v\to y$. So $x\to u\to v\to y$ gives a path from $x$ to $y$, hence $x\to y$.
For the converse, suppose $SC(u)$ reaches $SC(v)$. Then there are $x\in SC(u)$, $y\in SC(v)$ such that $x\to y$. But then $u\to x\to y\to v$ connects $u$ to $v$.